# LaTeX2HTML 2002-2-1 (1.71)
# Associate images original text with physical files.


$key = q/x=x_0+v_g,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img389.png"
 ALT="$x=x_0+v_g t$">|; 

$key = q/xrightarrow-infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img964.png"
 ALT="$x\rightarrow-\infty$">|; 

$key = q/psi_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2826.png"
 ALT="$\psi_-$">|; 

$key = q/{displaymath}f(theta,phi)=f(theta).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2904.png"
 ALT="\begin{displaymath}
f(\theta, \phi) = f(\theta).
\end{displaymath}">|; 

$key = q/{displaymath}left|frac{m}{2pi,hbar^2}intfrac{exp(,{{rm{i},k,r')}{r'},V({{bf{r}'),d^3{{bf{r'}right|ll1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2897.png"
 ALT="\begin{displaymath}
\left\vert \frac{m}{2\pi  \hbar^2} \int \frac{ \exp( {\rm i}  k  r')}{r'}
 V({\bf r}') d^3{\bf r'} \right\vert \ll 1.
\end{displaymath}">|; 

$key = q/{displaymath}left(frac{d^2}{dz^2}-2,frac{d}{dz}-frac{l,(l+1)}{z^2}+frac{zeta}{z}right)Z=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1593.png"
 ALT="\begin{displaymath}
\left(\frac{d^2}{dz^2} -2 \frac{d}{dz} - \frac{l (l+1)}{z^2} + \frac{\zeta}{z}\right) Z = 0.
\end{displaymath}">|; 

$key = q/{{bf{r}equiv(x_1,x_2,x_3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1498.png"
 ALT="${\bf r}\equiv (x_1, x_2, x_3)$">|; 

$key = q/{displaymath}{{bf{S}={{bf{S}_1+{{bf{S}_2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img2029.png"
 ALT="\begin{displaymath}
{\bf S} ={\bf S}_1 + {\bf S}_2
\end{displaymath}">|; 

$key = q/m=n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2166.png"
 ALT="$m=n$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter03slashfig02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="281" BORDER="0"
 SRC="|."$dir".q|img189.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter03/fig02.eps}}
\end{figure}">|; 

$key = q/displaystyle-frac{{{rm{e}^{-beta,x}}{beta^2},(1+beta,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2747.png"
 ALT="$\displaystyle -\frac{{\rm e}^{-\beta x}}{\beta^2} (1+\beta x),$">|; 

$key = q/{displaymath}psi^{(2)}_{l,1slash2;j,m+1slash2}=alpha,Y_{l,m},chi_++beta,Y_{l,m+1},chi_-.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="37" BORDER="0"
 SRC="|."$dir".q|img1931.png"
 ALT="\begin{displaymath}
\psi^{(2)}_{l,1/2;j,m+1/2} = \alpha Y_{l,m} \chi_+ + \beta 
Y_{l,m+1} \chi_-.
\end{displaymath}">|; 

$key = q/{displaymath}V({{bf{r})=-frac{e^2}{4pi,epsilon_0,|{{bf{r}|},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1581.png"
 ALT="\begin{displaymath}
V({\bf r}) = - \frac{e^2}{4\pi \epsilon_0 \vert{\bf r}\vert},
\end{displaymath}">|; 

$key = q/psi_a';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img673.png"
 ALT="$\psi_a'$">|; 

$key = q/{displaymath}sum_n|c_n|^{,2}=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2694.png"
 ALT="\begin{displaymath}
\sum_n \vert c_n\vert^{ 2} = 1.
\end{displaymath}">|; 

$key = q/{displaymath}H_0,psi_n=E_n,psi_n,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2148.png"
 ALT="\begin{displaymath}
H_0 \psi_n = E_n \psi_n,
\end{displaymath}">|; 

$key = q/{displaymath}psi^{(2)}_{l-1slash2,m+1slash2}=left(frac{l-m}{2,l+1}right)^{1slash{2,l+1}right)^{1slash2}psi^{(1)}_{m+1,-1slash2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="508" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1956.png"
 ALT="\begin{displaymath}
\psi^{(2)}_{l-1/2,m+1/2} = \left(\frac{l-m}{2 l+1}\right)^{...
...-\left(\frac{l+m+1}{2 l+1}\right)^{1/2}\psi^{(1)}_{m+1,-1/2}.
\end{displaymath}">|; 

$key = q/-2,-1,0,+1,+2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1435.png"
 ALT="$-2,-1,0,+1,+2$">|; 

$key = q/w,dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img139.png"
 ALT="$w dt$">|; 

$key = q/delta_{ij},a_i,b_jequiva_i,b_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1515.png"
 ALT="$\delta_{ij} a_i b_j
\equiv a_i b_i$">|; 

$key = q/{displaymath}E_0leqlanglepsi|H|psirangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2689.png"
 ALT="\begin{displaymath}
E_0 \leq \langle\psi\vert H\vert\psi\rangle.
\end{displaymath}">|; 

$key = q/N+dN;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img933.png"
 ALT="$N+dN$">|; 

$key = q/{displaymath}R=left(frac{k-q}{k+q}right)^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1057.png"
 ALT="\begin{displaymath}
R = \left(\frac{k-q}{k+q}\right)^2,
\end{displaymath}">|; 

$key = q/{displaymath}c_n(t)=delta_{ni}-frac{i}{hbar}int_0^tH_{ni}(t'),exp(,{{rm{i},omega_{ni},t'),dt'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="334" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2478.png"
 ALT="\begin{displaymath}
c_n(t) = \delta_{ni} -\frac{i}{\hbar}\int_0^t H_{ni}(t') \exp( {\rm i} \omega_{ni} t') dt'.
\end{displaymath}">|; 

$key = q/displaystylelangleprangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img535.png"
 ALT="$\displaystyle \langle p\rangle$">|; 

$key = q/10^{-16},{{rm{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img396.png"
 ALT="$10^{-16} {\rm s}$">|; 

$key = q/f(u_1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img103.png"
 ALT="$f(u_1)$">|; 

$key = q/mbox{{boldmath{{epsilon{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2541.png"
 ALT="$\mbox{\boldmath$\epsilon$}$">|; 

$key = q/displaystyleP_2(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2431.png"
 ALT="$\displaystyle P_2(t)$">|; 

$key = q/{displaymath}E=frac{l^2,pi^2,hbar^2}{2,m,a^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1243.png"
 ALT="\begin{displaymath}
E = \frac{l^2 \pi^2 \hbar^2}{2 m a^2}.
\end{displaymath}">|; 

$key = q/{displaymath}{{rm{i},hbar,frac{partialpsi}{partialt}=sum_mleft({{rm{i},hbar,fracight)expleft(-{{rm{i},E_m,tslashhbarright)psi_m,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="418" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2404.png"
 ALT="\begin{displaymath}
{\rm i} \hbar \frac{\partial\psi}{\partial t} =
\sum_m\le...
...c_m E_m\right)
\exp\left(-{\rm i} E_m t/\hbar\right)\psi_m,
\end{displaymath}">|; 

$key = q/psi(x,y,z,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img183.png"
 ALT="$\psi(x,y,z,t)$">|; 

$key = q/n'neqn;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2263.png"
 ALT="$n'\neq n$">|; 

$key = q/{displaymath}{{bf{U}=e,|{{bf{E}|left(array{{cc}0&langle2,0,0|z|2,1,0rangle[0.5ex]langle2,1,0|z|2,0,0rangle&0array{right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="395" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img2272.png"
 ALT="\begin{displaymath}
{\bf U} = e \vert{\bf E}\vert\left(\begin{array}{cc}
0&amp; \la...
...]
\langle 2,1,0\vert z\vert 2,0,0\rangle&amp;0
\end{array}\right),
\end{displaymath}">|; 

$key = q/m_s=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2003.png"
 ALT="$m_s=1/2$">|; 

$key = q/E_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img238.png"
 ALT="$E_y$">|; 

$key = q/chi_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1722.png"
 ALT="$\chi_2$">|; 

$key = q/displaystylel,(l+1),hbar^{,2},Y_{l,m}(theta,phi).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1365.png"
 ALT="$\displaystyle l (l+1) \hbar^{ 2} Y_{l,m}(\theta,\phi).$">|; 

$key = q/displaystyle-{{rm{i},hbar,frac{partial}{partialphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1359.png"
 ALT="$\displaystyle -{\rm i} \hbar \frac{\partial}{\partial\phi},$">|; 

$key = q/{displaymath}x=d,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img172.png"
 ALT="\begin{displaymath}
x= d,
\end{displaymath}">|; 

$key = q/{displaymath}alpha_{nlm}=2,e^2sum_{n',l'=lpm1}frac{|langlen,l,m|z|n',l',mrangle|^2}{E_{n'l'm}-E_{nlm}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img2217.png"
 ALT="\begin{displaymath}
\alpha_{nlm} = 2 e^2\sum_{n',l'=l\pm 1}
\frac{\vert\langle n,l,m\vert z\vert n',l',m\rangle\vert^2}{E_{n'l'm}-E_{nlm}}.
\end{displaymath}">|; 

$key = q/A,BneqB,A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img622.png"
 ALT="$A B\neq B A$">|; 

$key = q/L^2slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img758.png"
 ALT="$L^2/3$">|; 

$key = q/{displaymath}psi(x)=psi_0,{{rm{e}^{-(x-x_0)^{,2}slash(4,sigma^2)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img462.png"
 ALT="\begin{displaymath}
\psi(x) = \psi_0 {\rm e}^{-(x-x_0)^{ 2}/(4 \sigma^2)}.
\end{displaymath}">|; 

$key = q/i,j,k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1503.png"
 ALT="$i,j,k$">|; 

$key = q/I({{bf{r}_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2734.png"
 ALT="$I({\bf r}_2)$">|; 

$key = q/l+l'+2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2212.png"
 ALT="$l+l'+2$">|; 

$key = q/y=(k-k_0)slash(2,{mitDelta}k);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img373.png"
 ALT="$y=(k-k_0)/(2 {\mit\Delta}k)$">|; 

$key = q/{displaymath}k',a=sqrt{k^2,a^2+frac{2,m,|V_0|,a^2}{hbar^2}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img3008.png"
 ALT="\begin{displaymath}
k' a = \sqrt{ k^2  a^2 + \frac{2  m  \vert V_0\vert  a^2}{\hbar^2}},
\end{displaymath}">|; 

$key = q/{displaymath}chi=sum_ac_a,chi_{a}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1740.png"
 ALT="\begin{displaymath}
\chi = \sum_a c_a  \chi_{a}.
\end{displaymath}">|; 

$key = q/lambdaslashD;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img412.png"
 ALT="$\lambda/D$">|; 

$key = q/A_1=A-4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img909.png"
 ALT="$A_1=A-4$">|; 

$key = q/B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img549.png"
 ALT="$B$">|; 

$key = q/{displaymath}{{bf{S}_pcdot{{bf{S}_e=frac{1}{2},(S^2-S_e^{,2}-S_p^{,2}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img2377.png"
 ALT="\begin{displaymath}
{\bf S}_p\cdot{\bf S}_e = \frac{1}{2} (S^2-S_e^{ 2}-S_p^{ 2}).
\end{displaymath}">|; 

$key = q/{displaymath}{{rm{e}^{,{{rm{i},phi}equivcosphi+{{rm{i},sinphi,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img211.png"
 ALT="\begin{displaymath}
{\rm e}^{ {\rm i} \phi} \equiv \cos\phi + {\rm i} \sin\phi,
\end{displaymath}">|; 

$key = q/{{bf{r}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2708.png"
 ALT="${\bf r}_2$">|; 

$key = q/displaystylea,psi_a,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img662.png"
 ALT="$\displaystyle a \psi_a,$">|; 

$key = q/t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img141.png"
 ALT="$t$">|; 

$key = q/V_0=0.75,E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$V_0 = 0.75 E$">|; 

$key = q/{displaymath}Hequivfrac{p^2}{2,m}+V(x).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img557.png"
 ALT="\begin{displaymath}
H \equiv \frac{p^2}{2 m} + V(x).
\end{displaymath}">|; 

$key = q/displaystyleS_z,chi_{s_1,s_2;s,m_s}^{(2)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2056.png"
 ALT="$\displaystyle S_z  \chi_{s_1,s_2;s,m_s}^{(2)}$">|; 

$key = q/displaystylefrac{1}{k^2}ointdphiint_{-1}^{1}dmusum_lsum_{l'}(2,l+1),(2,l'+1)exp[,{{rm{i},(delta_l-delta_{l'})];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="437" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2938.png"
 ALT="$\displaystyle \frac{1}{k^2} \oint d\phi \int_{-1}^{1} d\mu
\sum_l \sum_{l'} (2 l+1) (2 l'+1)
\exp[ {\rm i} (\delta_l-\delta_{l'})]$">|; 

$key = q/k'=k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2857.png"
 ALT="$k'=k$">|; 

$key = q/J_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1880.png"
 ALT="$J_z$">|; 

$key = q/{displaymath}epsilon_{ijk}=left{array{{rcl}0&mbox{hspace{1cm}}&mbox{if{i,j,k{notareanti-cyclicpermutationof{1,2,3{}array{right..{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="549" HEIGHT="84" BORDER="0"
 SRC="|."$dir".q|img1504.png"
 ALT="\begin{displaymath}
\epsilon_{ijk} = \left\{
\begin{array}{rcl}
0 &amp;\mbox{\hspace...
... are anti-cyclic permutation of $1,2,3$}
\end{array}\right. .
\end{displaymath}">|; 

$key = q/x^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img757.png"
 ALT="$x^2$">|; 

$key = q/{figure}{epsfysize{=2.8in{centerline{epsffile{Chapter05slashfig03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="304" BORDER="0"
 SRC="|."$dir".q|img866.png"
 ALT="\begin{figure}
\epsfysize =2.8in
\centerline{\epsffile{Chapter05/fig03.eps}}
\end{figure}">|; 

$key = q/{displaymath}H_{1,2}(Z)=-frac{hbar^2}{2,m_e},nabla^{,2}_{1,2}-frac{Z,e^2}{4pi,epsilon_0,r_{1,2}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2757.png"
 ALT="\begin{displaymath}
H_{1,2}(Z) = -\frac{\hbar^2}{2 m_e} \nabla^{ 2}_{1,2} -\frac{Z e^2}{4\pi \epsilon_0 r_{1,2}}
\end{displaymath}">|; 

$key = q/exp(,y^2slash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1018.png"
 ALT="$\exp( y^2/2)$">|; 

$key = q/R_ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1286.png"
 ALT="$R_\ast$">|; 

$key = q/{displaymath}E=2int_0^inftyint_0^piexpleft[-x-(x^2+X^2-2,x,X,costheta)^{1slash2}right],x,dx,sintheta,dtheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="529" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img2814.png"
 ALT="\begin{displaymath}
E = 2\int_0^\infty \int_0^\pi \exp\left[-x-(x^2+X^2-2 x X \cos\theta)^{1/2}\right] x dx \sin\theta d\theta,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.8in{centerline{epsffile{Chapter05slashfig02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="303" BORDER="0"
 SRC="|."$dir".q|img850.png"
 ALT="\begin{figure}
\epsfysize =2.8in
\centerline{\epsffile{Chapter05/fig02.eps}}
\end{figure}">|; 

$key = q/{displaymath}{mitDelta}x,{mitDelta}pstackrel{_{{normalsize{>}}{_{{normalsize{sim}}frac{hbar}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img402.png"
 ALT="\begin{displaymath}
{\mit\Delta}x {\mit\Delta} p \stackrel {_{\normalsize &gt;}}{_{\normalsize\sim}}\frac{\hbar}{2}.
\end{displaymath}">|; 

$key = q/{displaymath}tandelta_l=frac{k,a,j_l'(k,a)-beta_{l+},j_l(k,a)}{k,a,y_l'(k,a)-beta_{l+},y_l(k,a)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2950.png"
 ALT="\begin{displaymath}
\tan \delta_l = \frac{ k a j_l'(k a) - \beta_{l+}  j_l(k a)}
{k a y_l'(k a) - \beta_{l+}  y_l(k a)}.
\end{displaymath}">|; 

$key = q/0leqsleqinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img132.png"
 ALT="$0\leq s\leq \infty$">|; 

$key = q/{displaymath}sum_ia_i^{,2},|c_i|^2-left(sum_ia_i,|c_i|^2right)^2=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img692.png"
 ALT="\begin{displaymath}
\sum_i a_i^{ 2} \vert c_i\vert^2 - \left(\sum_i a_i \vert c_i\vert^2\right)^2=0.
\end{displaymath}">|; 

$key = q/displaystylefrac{psi_{200}-psi_{210}}{sqrt{2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2281.png"
 ALT="$\displaystyle \frac{\psi_{200} -\psi_{210}}{\sqrt{2}}.$">|; 

$key = q/m_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1280.png"
 ALT="$m_p$">|; 

$key = q/{displaymath}[J^2,L^2]=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1892.png"
 ALT="\begin{displaymath}[J^2,L^2]= 0.
\end{displaymath}">|; 

$key = q/{displaymath}mbox{{boldmath{{mu{}_p=frac{g_p,e}{2,m_p},{{bf{S}_p,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2360.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\mu$}_p = \frac{g_p e}{2 m_p} {\bf S}_p,
\end{displaymath}">|; 

$key = q/{displaymath}Hequiv-frac{hbar^2}{2,m},frac{partial^2}{partialx^2}+V(x).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img558.png"
 ALT="\begin{displaymath}
H \equiv -\frac{\hbar^2}{2 m} \frac{\partial^2}{\partial x^2} + V(x).
\end{displaymath}">|; 

$key = q/{displaymath}P(uinu:u+du)=P(u),du,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img119.png"
 ALT="\begin{displaymath}
P(u\in u:u+du) = P(u) du,
\end{displaymath}">|; 

$key = q/K>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img996.png"
 ALT="$K&gt;0$">|; 

$key = q/displaystylex;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1345.png"
 ALT="$\displaystyle x$">|; 

$key = q/3P_{3slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2335.png"
 ALT="$3P_{3/2}$">|; 

$key = q/5.59;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2364.png"
 ALT="$5.59$">|; 

$key = q/L^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="$L^2$">|; 

$key = q/s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img130.png"
 ALT="$s$">|; 

$key = q/displaystylefrac{1}{2pi,hbar}int_{-infty}^{infty}int_{-infty}^{infty}int_{-inftyt),phi(p,t),p,{{rm{e}^{+{{rm{i},(p-p'),xslashhbar},dx,dp,dp',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="455" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img590.png"
 ALT="$\displaystyle \frac{1}{2\pi \hbar}\int_{-\infty}^{\infty}\int_{-\infty}^{\inft...
...phi^\ast(p',t) \phi(p,t) p {\rm e}^{+{\rm i} (p-p') x/\hbar} dx dp dp',$">|; 

$key = q/E>V(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img884.png"
 ALT="$E&gt;V(x)$">|; 

$key = q/displaystylefrac{l_x,pi}{a},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1235.png"
 ALT="$\displaystyle \frac{l_x \pi}{a},$">|; 

$key = q/-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img151.png"
 ALT="$-1$">|; 

$key = q/0<x<a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img793.png"
 ALT="$0&lt;x&lt;a$">|; 

$key = q/{displaymath}alpha^2+beta^2=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1929.png"
 ALT="\begin{displaymath}
\alpha^2 + \beta^2 = 1.
\end{displaymath}">|; 

$key = q/{displaymath}psi^{(2)}_{l,1slash2;j,m+1slash2}=alpha,psi^{(1)}_{l,1slash2;m,1slash2}+beta,psi^{(1)}_{l,1slash2;m+1,-1slash2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="369" HEIGHT="37" BORDER="0"
 SRC="|."$dir".q|img1925.png"
 ALT="\begin{displaymath}
\psi^{(2)}_{l,1/2;j,m+1/2} = \alpha \psi^{(1)}_{l,1/2;m,1/2} + \beta 
\psi^{(1)}_{l,1/2;m+1,-1/2},
\end{displaymath}">|; 

$key = q/m_j=m+m_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1927.png"
 ALT="$m_j=m+m_s$">|; 

$key = q/p^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2297.png"
 ALT="$p^2$">|; 

$key = q/{displaymath}int_{-infty}^infty{{rm{sinc}^2(x),dx=pi,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img2561.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty {\rm sinc}^2(x) dx = \pi,
\end{displaymath}">|; 

$key = q/{displaymath}langlexrangle=frac{1}{2pi,hbar}int_{-infty}^{infty}int_{-infty}^{infrac{partial}{partialp}right)phi(p,t),dx,dp,dp'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="560" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img597.png"
 ALT="\begin{displaymath}
\langle x\rangle= \frac{1}{2\pi \hbar}\int_{-\infty}^{\inft...
...bar \frac{\partial}{\partial p}\right)\phi(p,t) dx dp dp'.
\end{displaymath}">|; 

$key = q/l,m=2,0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="$l,m=2,0$">|; 

$key = q/B_0=E_0slashc;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2553.png"
 ALT="$B_0=E_0/c$">|; 

$key = q/1slash2,1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2023.png"
 ALT="$1/2, 1/2$">|; 

$key = q/psi_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img271.png"
 ALT="$\psi_2$">|; 

$key = q/{displaymath}psi_0({{bf{r})=sqrt{n},exp(,{{rm{i},k,z)=sqrt{n},exp(,{{rm{i},k,rcostheta),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2905.png"
 ALT="\begin{displaymath}
\psi_0({\bf r}) = \sqrt{n} \exp( {\rm i} k z)= \sqrt{n} \exp( {\rm i} k r\cos\theta),
\end{displaymath}">|; 

$key = q/n>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2220.png"
 ALT="$n&gt;1$">|; 

$key = q/Wslashh;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img253.png"
 ALT="$W/h$">|; 

$key = q/displaystyle[sigma_y,sigma_z];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1804.png"
 ALT="$\displaystyle [\sigma_y, \sigma_z]$">|; 

$key = q/displaystylec_-(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1843.png"
 ALT="$\displaystyle c_-(0)$">|; 

$key = q/{displaymath}psi(y)=h(y),{{rm{e}^{-y^2slash2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1008.png"
 ALT="\begin{displaymath}
\psi(y) = h(y) {\rm e}^{-y^2/2},
\end{displaymath}">|; 

$key = q/{displaymath}S_pm,chi_mp=hbar,chi_pm.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1939.png"
 ALT="\begin{displaymath}
S_\pm \chi_\mp = \hbar \chi_\pm.
\end{displaymath}">|; 

$key = q/displaystyleleftlanglepsi_0({{bf{r}_1)left|frac{a_0}{r_1}right|psi_0({{bf{r}_2)rightrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2810.png"
 ALT="$\displaystyle \left\langle \psi_0({\bf r}_1)\left\vert\frac{a_0}{r_1}\right\vert\psi_0({\bf r}_2)\right\rangle.$">|; 

$key = q/3slash2,-3slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2015.png"
 ALT="$3/2, -3/2$">|; 

$key = q/L_i,L_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1509.png"
 ALT="$L_i L_i$">|; 

$key = q/{displaymath}E=hbar,omega{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img308.png"
 ALT="\begin{displaymath}
E = \hbar \omega
\end{displaymath}">|; 

$key = q/displaystyle{{bf{L}times{{bf{L}+{{bf{S}times{{bf{S}+{{bf{L}times{{bf{S}+{{bf{S}times{{bf{L}={{bf{L}times{{bf{L}+{{bf{S}times{{bf{S};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="411" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1874.png"
 ALT="$\displaystyle {\bf L}\times{\bf L} + {\bf S}\times{\bf S}+{\bf L}\times{\bf S}+{\bf S}\times{\bf L} ={\bf L}\times{\bf L} + {\bf S}\times{\bf S}$">|; 

$key = q/E_Fproptoa^{-2}=V^{-2slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1262.png"
 ALT="$E_F\propto a^{-2}=V^{-2/3}$">|; 

$key = q/{displaymath}lambda_e=1.2times10^{-9},[E({{rm{eV})]^{-1slash2},{{rm{m},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img301.png"
 ALT="\begin{displaymath}
\lambda_e = 1.2\times 10^{-9} [E({\rm eV})]^{-1/2} {\rm m},
\end{displaymath}">|; 

$key = q/displaystylepsi^{(1)}_{pm1,pm1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1993.png"
 ALT="$\displaystyle \psi^{(1)}_{\pm 1,\pm 1/2}$">|; 

$key = q/-1slash2,-1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2067.png"
 ALT="$-1/2, -1/2$">|; 

$key = q/{displaymath}|omega+omega_{fi}|stackrel{_{{normalsize{<}}{_{{normalsize{sim}}frac{2pi}{t},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img2497.png"
 ALT="\begin{displaymath}
\vert\omega+\omega_{fi}\vert\stackrel {_{\normalsize &lt;}}{_{\normalsize\sim}}\frac{2\pi}{t},
\end{displaymath}">|; 

$key = q/3P_{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2334.png"
 ALT="$3P_{1/2}$">|; 

$key = q/|c_1|^2=x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img693.png"
 ALT="$\vert c_1\vert^2=x$">|; 

$key = q/delta_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img682.png"
 ALT="$\delta_{ij}$">|; 

$key = q/{displaymath}langlen,l,m|L^4,z-2,L^2,z,L^2+z,L^4-2,hbar^2,(L^2,z+z,L^2)|n',l',mrangle=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="514" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2205.png"
 ALT="\begin{displaymath}
\langle n,l,m\vert L^4 z-2 L^2 z L^2 + z L^4 - 2 \hbar^2 (L^2 z+z L^2) \vert n',l',m\rangle = 0.
\end{displaymath}">|; 

$key = q/{displaymath}psi=sum_{i}c_i,psi_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img676.png"
 ALT="\begin{displaymath}
\psi = \sum_{i}c_i \psi_i,
\end{displaymath}">|; 

$key = q/{mitOmega}(X);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="${\mit\Omega}(X)$">|; 

$key = q/textstyleequiv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1193.png"
 ALT="$\textstyle \equiv$">|; 

$key = q/3,{mitDelta}k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img351.png"
 ALT="$3 {\mit\Delta}k$">|; 

$key = q/{displaymath}frac{2,m}{hbar^2}frac{|V_0|}{mu,k}ll1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2902.png"
 ALT="\begin{displaymath}
\frac{2 m}{\hbar^2} \frac{\vert V_0\vert}{\mu  k} \ll 1.
\end{displaymath}">|; 

$key = q/displaystylephi_0';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img363.png"
 ALT="$\displaystyle \phi_0'$">|; 

$key = q/{displaymath}{{rm{e}^{,2,{{rm{i},varphi}=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1139.png"
 ALT="\begin{displaymath}
{\rm e}^{ 2 {\rm i} \varphi} = 1.
\end{displaymath}">|; 

$key = q/displaystyley;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1347.png"
 ALT="$\displaystyle y$">|; 

$key = q/omega_{fi}>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2514.png"
 ALT="$\omega_{fi}&gt;0$">|; 

$key = q/{displaymath}h(y)=sum_{i=0}^inftyc_i,y^i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1011.png"
 ALT="\begin{displaymath}
h(y) = \sum_{i=0}^\infty c_i y^i.
\end{displaymath}">|; 

$key = q/displaystylepsi_2-frac{e_{12}}{E_1-E_2},psi_1+cdots.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2146.png"
 ALT="$\displaystyle \psi_2 - \frac{e_{12}}{E_1-E_2} \psi_1+ \cdots.$">|; 

$key = q/displaystylefrac{Z''}{Z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1223.png"
 ALT="$\displaystyle \frac{Z''}{Z}$">|; 

$key = q/E_a=E_b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1149.png"
 ALT="$E_a=E_b$">|; 

$key = q/{displaymath}E_F=frac{pi^2,hbar^2}{2,m_e}left(frac{3,n}{pi}right)^{2slash3},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1252.png"
 ALT="\begin{displaymath}
E_F = \frac{\pi^2 \hbar^2}{2 m_e}\left(\frac{3 n}{\pi}\right)^{2/3},
\end{displaymath}">|; 

$key = q/dot{~}equivdslashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1839.png"
 ALT="$\dot{&nbsp;}\equiv d/dt$">|; 

$key = q/displaystyles_2,(s_2+1),hbar^2,chi_{s_1,s_2;s,m_s}^{(2)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2053.png"
 ALT="$\displaystyle s_2 (s_2+1) \hbar^2
 \chi_{s_1,s_2;s,m_s}^{(2)},$">|; 

$key = q/{displaymath}langlec,f(u)rangle=c,langlef(u)rangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img109.png"
 ALT="\begin{displaymath}
\langle c  f(u)\rangle = c \langle f(u)\rangle.
\end{displaymath}">|; 

$key = q/|Y_{l,m}(theta,phi)|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="$\vert Y_{l,m}(\theta ,\phi )\vert^{ 2}$">|; 

$key = q/0<x_1<x_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img887.png"
 ALT="$0&lt;x_1&lt;x_2$">|; 

$key = q/psi(x_1,x_2,ldotsx_N,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1096.png"
 ALT="$\psi(x_1,x_2,\ldots x_N,t)$">|; 

$key = q/a_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img766.png"
 ALT="$a_2$">|; 

$key = q/{{bf{x}_{nl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2258.png"
 ALT="${\bf x}_{nl}$">|; 

$key = q/{displaymath}langle{{bf{S}_pcdot{{bf{S}_erangle=frac{1}{4},hbar^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img2383.png"
 ALT="\begin{displaymath}
\langle {\bf S}_p\cdot{\bf S}_e\rangle = \frac{1}{4} \hbar^2
\end{displaymath}">|; 

$key = q/displaystyle-frac{e^2}{4piepsilon_0,m_e,c^2,r^3},{{bf{v}times{{bf{r}cdot{{bf{S};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2312.png"
 ALT="$\displaystyle - \frac{e^2}{4\pi\epsilon_0 m_e c^2 r^3} {\bf v}\times {\bf r}\cdot{\bf S}$">|; 

$key = q/{displaymath}langlepsi|H|psirangle=|c_0|^{,2},E_0+sum_{n>0}|c_n|^{,2},E_n.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2698.png"
 ALT="\begin{displaymath}
\langle \psi\vert H\vert\psi\rangle = \vert c_0\vert^{ 2} E_0 + \sum_{n&gt;0} \vert c_n\vert^{ 2} E_n.
\end{displaymath}">|; 

$key = q/r_1=R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img918.png"
 ALT="$r_1=R$">|; 

$key = q/displaystylej,(j+1),hbar^2,psi^{(2)}_{l,s;j,m_j},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1917.png"
 ALT="$\displaystyle j (j+1) \hbar^2 \psi^{(2)}_{l,s;j,m_j},$">|; 

$key = q/B=4.5times10^{10},{{rm{N},{{rm{m}^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1271.png"
 ALT="$B= 4.5\times 10^{10} {\rm N} {\rm m}^{-2}$">|; 

$key = q/{displaymath}(nabla^2+k^2),u({{bf{r})=rho({{bf{r}){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2846.png"
 ALT="\begin{displaymath}
(\nabla^2+k^2) u({\bf r}) = \rho({\bf r})
\end{displaymath}">|; 

$key = q/{displaymath}P_{irightarrowf}(t)=frac{t^2}{hbar^2}left{|V_{fi}|^{,2},{{rm{sinc}^c}^2left[(omega-omega_{fi}),tslash2right]right}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="569" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2501.png"
 ALT="\begin{displaymath}
P_{i\rightarrow f}(t) = \frac{t^2}{\hbar^2}\left\{
\vert V_{...
...} {\rm sinc}^2\left[(\omega-\omega_{fi}) t/2\right]\right\}.
\end{displaymath}">|; 

$key = q/langlef|mbox{{boldmath{{epsilon{}!cdot!{{bf{p}|irangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2681.png"
 ALT="$\langle f\vert\mbox{\boldmath$\epsilon$}\!\cdot\!{\bf p}\vert i\rangle$">|; 

$key = q/rrightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1615.png"
 ALT="$r\rightarrow\infty$">|; 

$key = q/{displaymath}|T|^{,2}=frac{|psi_2|^{,2}}{|psi_1|^{,2}}=expleft(-2,int_{x_1}^{x_2}|k(x')|,dx'right){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img897.png"
 ALT="\begin{displaymath}
\vert T\vert^{ 2} = \frac{\vert\psi_2\vert^{ 2}}{\vert\psi...
... = \exp\left(-2 \int_{x_1}^{x_2} \vert k(x')\vert dx'\right)
\end{displaymath}">|; 

$key = q/{displaymath}langleHrangle=8,E_0+langleV_{ee}rangle,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2727.png"
 ALT="\begin{displaymath}
\langle H\rangle = 8 E_0 + \langle V_{ee}\rangle,
\end{displaymath}">|; 

$key = q/{displaymath}left(frac{d}{dtheta}+l,cotthetaright)f(theta)equivfrac{1}{(sinthetafrac{d}{dtheta}left[(sintheta)^l,f(theta)right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1448.png"
 ALT="\begin{displaymath}
\left(\frac{d}{d\theta}+l \cot\theta\right)f(\theta)\equiv
...
...a)^l}\frac{d}{d\theta}\left[
(\sin\theta)^l f(\theta)\right],
\end{displaymath}">|; 

$key = q/{displaymath}pequiv-{{rm{i},hbar,frac{partial}{partialx}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img538.png"
 ALT="\begin{displaymath}
p \equiv -{\rm i} \hbar \frac{\partial}{\partial x}.
\end{displaymath}">|; 

$key = q/displaystyleV_{fi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2483.png"
 ALT="$\displaystyle V_{fi}$">|; 

$key = q/{displaymath}hbar,{{rm{e}^{,{{rm{i},phi}left(frac{partial}{partialtheta}+{{rm{i}i}right)Theta_{l,l}(theta),{{rm{e}^{,i,l,phi}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1439.png"
 ALT="\begin{displaymath}
\hbar {\rm e}^{ {\rm i} \phi}\left(\frac{\partial}{\parti...
...ial\phi}\right)\Theta_{l,l}(\theta) {\rm e}^{ i l \phi}=0.
\end{displaymath}">|; 

$key = q/phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img212.png"
 ALT="$\phi$">|; 

$key = q/|c_i|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img744.png"
 ALT="$\vert c_i\vert^{ 2}$">|; 

$key = q/x<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img815.png"
 ALT="$x&lt;0$">|; 

$key = q/a_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1041.png"
 ALT="$a_+$">|; 

$key = q/displaystyleint_{-1}^1frac{dmu}{sqrt{r_1^{,2}+r_2^{,2}-2,r_1,r_2,mu}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2741.png"
 ALT="$\displaystyle \int_{-1}^1 \frac{d\mu}{\sqrt{r_1^{ 2}+r_2^{ 2}-2 r_1 r_2 \mu}}$">|; 

$key = q/displaystylephi_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img361.png"
 ALT="$\displaystyle \phi_0$">|; 

$key = q/{displaymath}a=n,a_0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1621.png"
 ALT="\begin{displaymath}
a = n a_0,
\end{displaymath}">|; 

$key = q/psi_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img678.png"
 ALT="$\psi_i$">|; 

$key = q/{displaymath}{{bf{L}times{{bf{L}={{rm{i},hbar,{{bf{L}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1867.png"
 ALT="\begin{displaymath}
{\bf L}\times{\bf L} = {\rm i} \hbar {\bf L}.
\end{displaymath}">|; 

$key = q/f(theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1449.png"
 ALT="$f(\theta)$">|; 

$key = q/{displaymath}langlel,1slash2;j,m_j|S_z|l,1slash2;j,m_jrangle=pmfrac{m_j,hbar}{2,l+1}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2347.png"
 ALT="\begin{displaymath}
\langle l,1/2;j,m_j\vert S_z\vert l,1/2;j,m_j\rangle = \pm \frac{m_j \hbar}{2 l+1}
\end{displaymath}">|; 

$key = q/beta=m,alphaslashhbar^2,k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1063.png"
 ALT="$\beta= m \alpha/\hbar^2 k$">|; 

$key = q/p,x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img552.png"
 ALT="$p x$">|; 

$key = q/psi(x,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img147.png"
 ALT="$\psi(x,t)$">|; 

$key = q/displaystylefrac{pi}{epsilon_0,hbar^2},|mbox{{boldmath{{epsilon{}!cdot!{{bf{d}_{if}|^{,2},rho(omega_{fi}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2581.png"
 ALT="$\displaystyle \frac{\pi}{\epsilon_0 \hbar^2} \vert\mbox{\boldmath$\epsilon$}\!\cdot \!{\bf d}_{if}\vert^{ 2} \rho(\omega_{fi}),$">|; 

$key = q/psi_{210};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2269.png"
 ALT="$\psi_{210}$">|; 

$key = q/E+dE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1298.png"
 ALT="$E+dE$">|; 

$key = q/mu=costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2940.png"
 ALT="$\mu = \cos\theta$">|; 

$key = q/{displaymath}psi_{E,fermion}(x_1,x_2)=frac{1}{sqrt{2}}left[psi(x_1,E_a),psi(x_2,E_b)-psi(x_2,E_a),psi(x_1,E_b)right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="542" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1148.png"
 ALT="\begin{displaymath}
\psi_{E fermion}(x_1,x_2) = \frac{1}{\sqrt{2}}\left[\psi(x_1,E_a) \psi(x_2,E_b)-\psi(x_2,E_a) \psi(x_1,E_b)\right],
\end{displaymath}">|; 

$key = q/displaystyle{{bf{p}+q,{{bf{A},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2522.png"
 ALT="$\displaystyle {\bf p} + q { \bf A},$">|; 

$key = q/{displaymath}langlexrangle_{ab}=int_{-infty}^inftypsi^ast(x,E_a),x,psi(x,E_b),dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img1155.png"
 ALT="\begin{displaymath}
\langle x \rangle_{ab} = \int_{-\infty}^\infty \psi^\ast(x,E_a) x \psi(x,E_b) dx.
\end{displaymath}">|; 

$key = q/cos^2(thetaslash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1859.png"
 ALT="$\cos^2(\theta/2)$">|; 

$key = q/{displaymath}H=Omega,S_z,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1833.png"
 ALT="\begin{displaymath}
H = \Omega S_z,
\end{displaymath}">|; 

$key = q/{displaymath}H,psi=E,psi.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1208.png"
 ALT="\begin{displaymath}
H \psi = E \psi.
\end{displaymath}">|; 

$key = q/psi_2(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img604.png"
 ALT="$\psi_2(x)$">|; 

$key = q/{displaymath}langleprangle=int_{-infty}^{infty}int_{-infty}^{infty}phi^ast(p',t),phi(p,t),p,delta(p-p'),dp,dp'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="376" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img591.png"
 ALT="\begin{displaymath}
\langle p\rangle = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\phi^\ast(p',t) \phi(p,t) p \delta(p-p') dp dp'.
\end{displaymath}">|; 

$key = q/E_0=-e^2slash(8pi,epsilon_0,a_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2770.png"
 ALT="$E_0=-e^2/(8\pi \epsilon_0 a_0)$">|; 

$key = q/q=sqrt{2,m,(V_0-E)slashhbar^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img857.png"
 ALT="$q=\sqrt{2 m (V_0-E)/\hbar^2}$">|; 

$key = q/{displaymath}H=H_0+H_1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img2095.png"
 ALT="\begin{displaymath}
H = H_0 + H_1.
\end{displaymath}">|; 

$key = q/1,-1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2013.png"
 ALT="$1,-1/2$">|; 

$key = q/n_f<n_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1677.png"
 ALT="$n_f&lt;n_i$">|; 

$key = q/{displaymath}|T|^{,2}=expleft(-frac{4,sqrt{2}}{3},frac{m^{1slash2},W^{,3slash2}}{hbar,e,{calE}}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="269" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img908.png"
 ALT="\begin{displaymath}
\vert T\vert^{ 2} = \exp\left(-\frac{4 \sqrt{2}}{3} \frac{m^{1/2} W^{ 3/2}}{\hbar e {\cal E}}\right).
\end{displaymath}">|; 

$key = q/{displaymath}{mitOmega}({{mitSigma}}otimes{{mitSigma}})={mitOmega}({{mitSigma}}),{mitOmega}({{mitSigma}}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img85.png"
 ALT="\begin{displaymath}
{\mit\Omega}({{\mit\Sigma}}\otimes{{\mit\Sigma}}) = {\mit\Omega}({{\mit\Sigma}})  {\mit\Omega}({{\mit\Sigma}}).
\end{displaymath}">|; 

$key = q/{displaymath}d^{,2}_{if}=|langlef|e,x|irangle|^{,2}+|langlef|e,y|irangle|^{,2}+|langlef|e,z|irangle|^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="353" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img2596.png"
 ALT="\begin{displaymath}
d^{ 2}_{if} = \vert\langle f\vert e x\vert i\rangle\vert^{...
...vert^{ 2}+ \vert\langle f\vert e z\vert i\rangle\vert^{ 2}.
\end{displaymath}">|; 

$key = q/a=1.25,lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$a=1.25 \lambda $">|; 

$key = q/{displaymath}int_{-infty}^inftyint_{-infty}^inftyint_{-infty}^infty|psi(x,y,z,t)|^2,dx,dy,dz=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img1179.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty \vert\psi(x,y,z,t)\vert^2 dx dy dz = 1.
\end{displaymath}">|; 

$key = q/E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="$E$">|; 

$key = q/{displaymath}L_z=x,p_y-y,p_x,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2182.png"
 ALT="\begin{displaymath}
L_z = x p_y - y p_x,
\end{displaymath}">|; 

$key = q/{mitDelta}y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img280.png"
 ALT="${\mit\Delta}y$">|; 

$key = q/displaystyle[S_y,S_z];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1702.png"
 ALT="$\displaystyle [S_y, S_z]$">|; 

$key = q/P_n(t)=|c_n(t)|^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2399.png"
 ALT="$P_n(t)=\vert c_n(t)\vert^2$">|; 

$key = q/displaystylesqrt{frac{15}{32pi}},sin^2theta,{{rm{e}^{pm2,{{rm{i},phi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1484.png"
 ALT="$\displaystyle \sqrt{\frac{15}{32\pi}} \sin^2\theta {\rm e}^{\pm 2 {\rm i} \phi}.$">|; 

$key = q/psi_x(x,x');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img705.png"
 ALT="$\psi_x(x,x')$">|; 

$key = q/n=0,1,2,cdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1022.png"
 ALT="$n=0,1,2,\cdots$">|; 

$key = q/|phi(p')|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img732.png"
 ALT="$\vert\phi(p')\vert^{ 2}$">|; 

$key = q/k'ggk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3003.png"
 ALT="$k' \gg k$">|; 

$key = q/1slashL;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img755.png"
 ALT="$1/L$">|; 

$key = q/displaystyle[x_i,x_j];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1082.png"
 ALT="$\displaystyle [x_i,x_j]$">|; 

$key = q/chi_{s,s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1772.png"
 ALT="$\chi_{s,s}$">|; 

$key = q/chi(E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img641.png"
 ALT="$\chi(E)$">|; 

$key = q/{displaymath}langleHrangle(Z)=left[2,Z^2-frac{5}{4},Z-4,Z,(Z-2)right]E_0=left[-2,Z^2+frac{27}{4},Zright]E_0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="510" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2771.png"
 ALT="\begin{displaymath}
\langle H\rangle(Z) = \left[2 Z^2 - \frac{5}{4} Z - 4 Z (Z-2)\right] E_0
= \left[-2 Z^2+ \frac{27}{4} Z\right] E_0.
\end{displaymath}">|; 

$key = q/{displaymath}DeltaE_{j=l+1slash2}=mu_B,B,frac{2,l+2}{2,l+1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2353.png"
 ALT="\begin{displaymath}
\Delta E_{j=l+1/2} = \mu_B B \frac{2 l+2}{2 l+1}.
\end{displaymath}">|; 

$key = q/{displaymath}A,chiequivleft(array{{cc}A_{11},&A_{12}A_{21},&A_{22}array{right)left(array{{c}c_+c_-array{right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1794.png"
 ALT="\begin{displaymath}
A \chi \equiv \left(\begin{array}{cc}A_{11},&amp; A_{12}\\\\
A_{...
...rray}\right)\left(\begin{array}{c}c_+ c_-\end{array}\right).
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{l,(l+1slash2),(l+1),n^3,a_0^{,3}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1662.png"
 ALT="$\displaystyle \frac{1}{l (l+1/2) (l+1) n^3 a_0^{ 3}}.$">|; 

$key = q/{displaymath}langle+|H_1|+rangle=langle-|H_1|-rangle=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2451.png"
 ALT="\begin{displaymath}
\langle +\vert H_1\vert+\rangle = \langle -\vert H_1\vert-\rangle = 0,
\end{displaymath}">|; 

$key = q/displaystylesigma_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1809.png"
 ALT="$\displaystyle \sigma_x$">|; 

$key = q/displaystyleointY_{l,m}^{,ast},(L^2-L_z^{,2}-hbar,L_z),Y_{l,m},dOmega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1410.png"
 ALT="$\displaystyle \oint Y_{l,m}^{ \ast} (L^2 -L_z^{ 2}-\hbar L_z) Y_{l,m} d\Omega$">|; 

$key = q/displaystylepsi_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img772.png"
 ALT="$\displaystyle \psi_1$">|; 

$key = q/A_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2547.png"
 ALT="$A_0$">|; 

$key = q/{displaymath}langlei|arangle^ast=langlea|irangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2109.png"
 ALT="\begin{displaymath}
\langle i\vert a\rangle^\ast=\langle a\vert i\rangle.
\end{displaymath}">|; 

$key = q/{rmsinc}^2(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2493.png"
 ALT="${\rm sinc}^2(x)$">|; 

$key = q/{displaymath}langler^krangle=int_0^inftyr^{2+k},|R_{n,l}(r)|^{,2},dr,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img1652.png"
 ALT="\begin{displaymath}
\langle r^k\rangle = \int_0^\infty r^{2+k} \vert R_{n,l}(r)\vert^{ 2} dr,
\end{displaymath}">|; 

$key = q/{displaymath}frac{d^2R_{n,l}}{dz^2}+frac{2}{z}frac{dR_{n,l}}{dz}+left[1-frac{l,(l+1)}{z^2}right]R_{n,l}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1546.png"
 ALT="\begin{displaymath}
\frac{d^2 R_{n,l}}{dz^2} + \frac{2}{z}\frac{dR_{n,l}}{dz} + \left[1
- \frac{l (l+1)}{z^2}\right] R_{n,l} = 0.
\end{displaymath}">|; 

$key = q/H_{nm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2472.png"
 ALT="$H_{nm}$">|; 

$key = q/mgeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1464.png"
 ALT="$m\geq 0$">|; 

$key = q/displaystyleint|f(theta)|^2,dOmega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2937.png"
 ALT="$\displaystyle \int \vert f(\theta)\vert^2 d\Omega$">|; 

$key = q/{displaymath}chi_{a}^nomath_inline{dagnomath_inline{,chi_{a'}=delta_{aa'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1739.png"
 ALT="\begin{displaymath}
\chi_{a}^\dag \chi_{a'} = \delta_{aa'}.
\end{displaymath}">|; 

$key = q/y=(x-x_0)slash(2,{mitDelta}x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img338.png"
 ALT="$y=(x-x_0)/ (2 {\mit\Delta} x)$">|; 

$key = q/{displaymath}x-c,t={{rm{constant}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img240.png"
 ALT="\begin{displaymath}
x - c  t = {\rm constant}.
\end{displaymath}">|; 

$key = q/{displaymath}u=frac{1}{2}left(frac{epsilon_0,|E_0|^2}{2}+frac{|B_0|^2}{2,mu_0}right)=frac{1}{2},epsilon_0,|E_0|^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="309" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2551.png"
 ALT="\begin{displaymath}
u = \frac{1}{2}\left(\frac{\epsilon_0 \vert E_0\vert^2}{2}+...
...{2 \mu_0}\right) = \frac{1}{2} \epsilon_0 \vert E_0\vert^2,
\end{displaymath}">|; 

$key = q/psi_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1029.png"
 ALT="$\psi_n$">|; 

$key = q/-e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1579.png"
 ALT="$-e$">|; 

$key = q/psi^{(1)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1926.png"
 ALT="$\psi^{(1)}$">|; 

$key = q/{displaymath}rho(omega)=frac{hbar}{pi^2,c^3},frac{omega^3}{exp(hbar,omegaslashk_B,T)-1},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2599.png"
 ALT="\begin{displaymath}
\rho(\omega) = \frac{\hbar}{\pi^2 c^3} \frac{\omega^3}{\exp(\hbar \omega/k_B T)-1},
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter05slashfig05.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="496" HEIGHT="340" BORDER="0"
 SRC="|."$dir".q|img902.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter05/fig05.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter09slashfig02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="331" BORDER="0"
 SRC="|."$dir".q|img1650.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter09/fig02.eps}}
\end{figure}">|; 

$key = q/{displaymath}langleV_{ee}rangle=-frac{4,E_0}{pi^2},intfrac{{{rm{e}^{-2,(r_1+r_2)-2,r_1,r_2,costheta}},d^3{{bf{r}_1,d^3{{bf{r}_2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="413" HEIGHT="63" BORDER="0"
 SRC="|."$dir".q|img2731.png"
 ALT="\begin{displaymath}
\langle V_{ee}\rangle = -\frac{4 E_0}{\pi^2} \int
\frac{{\...
...2^{ 2}-2 r_1 r_2 \cos\theta}} d^3{\bf r}_1 d^3{\bf r}_2,
\end{displaymath}">|; 

$key = q/-1,1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2010.png"
 ALT="$-1, 1/2$">|; 

$key = q/E_0=13.6,{{rm{eV};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2385.png"
 ALT="$E_0=13.6 {\rm eV}$">|; 

$key = q/w_{irightarrowf}^{abs},dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2568.png"
 ALT="$w_{i\rightarrow f}^{abs} dt$">|; 

$key = q/psi(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1007.png"
 ALT="$\psi(y)$">|; 

$key = q/{displaymath}{{bf{p}!cdot!{{bf{A}={{bf{A}!cdot!{{bf{p},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2534.png"
 ALT="\begin{displaymath}
{\bf p}\!\cdot\!{\bf A} = {\bf A}\!\cdot \! {\bf p},
\end{displaymath}">|; 

$key = q/{displaymath}j_l=v,(1-|R|^{,2}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img820.png"
 ALT="\begin{displaymath}
j_l=v (1-\vert R\vert^{ 2}),
\end{displaymath}">|; 

$key = q/E_{bind}=0.13,E_0=-1.77;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2831.png"
 ALT="$E_{bind} = 0.13  E_0 = -1.77$">|; 

$key = q/{displaymath}|T|^{,2}=frac{4,k^2,q^2}{4,k^2,q^2+(k^2+q^2)^{,2},sinh^2(q,a)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img863.png"
 ALT="\begin{displaymath}
\vert T\vert^{ 2} = \frac{4 k^2 q^2}{4 k^2 q^2 + (k^2+q^2)^{ 2} \sinh^2(q a)}.
\end{displaymath}">|; 

$key = q/displaystylelangle1,0,0|z^2|1,0,0rangle=frac{1}{3},langle1,0,0|r^2|1,0,0rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2230.png"
 ALT="$\displaystyle \langle 1,0,0\vert z^2\vert 1,0,0\rangle = \frac{1}{3} \langle 1,0,0\vert r^2\vert 1,0,0\rangle,$">|; 

$key = q/displaystylempsqrt{frac{15}{8pi}},sintheta,costheta,{{rm{e}^{pm{{rm{i},phi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1482.png"
 ALT="$\displaystyle \mp\sqrt{\frac{15}{8\pi}} \sin\theta \cos\theta {\rm e}^{\pm{\rm i} \phi},$">|; 

$key = q/l=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1432.png"
 ALT="$l=1$">|; 

$key = q/displaystylepsi^{(1)}_{0,-1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1999.png"
 ALT="$\displaystyle \psi^{(1)}_{0,-1/2}$">|; 

$key = q/displaystylefrac{Y''}{Y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1221.png"
 ALT="$\displaystyle \frac{Y''}{Y}$">|; 

$key = q/|c_+|^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1817.png"
 ALT="$\vert c_+\vert^2$">|; 

$key = q/{displaymath}(H_0+V),psi=E,psi,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2842.png"
 ALT="\begin{displaymath}
(H_0+V) \psi = E \psi,
\end{displaymath}">|; 

$key = q/E=p^2slash2,m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2894.png"
 ALT="$E= p^2/2 m$">|; 

$key = q/{displaymath}I({{bf{r}_2)=intfrac{{{rm{e}^{-2,r_1}}{sqrt{r_1^{,2}+r_2^{,2}-2,r_1,r_2,costheta}},d^3{{bf{r}_1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="62" BORDER="0"
 SRC="|."$dir".q|img2733.png"
 ALT="\begin{displaymath}
I({\bf r}_2) = \int \frac{{\rm e}^{-2 r_1}}{\sqrt{r_1^{ 2}+r_2^{ 2}-2 r_1 r_2 \cos\theta}} d^3{\bf r}_1.
\end{displaymath}">|; 

$key = q/{displaymath}R_{2,1},left(sqrt{1slash3},Y_{1,0},chi_++sqrt{2slash3},Y_{1,1},chi_-right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img2086.png"
 ALT="\begin{displaymath}
R_{2,1} \left(\sqrt{1/3} Y_{1,0} \chi_+ + \sqrt{2/3} Y_{1,1} \chi_-\right).
\end{displaymath}">|; 

$key = q/{displaymath}|e_{12}|ll|E_1-E_2|.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2147.png"
 ALT="\begin{displaymath}
\vert e_{12}\vert\ll \vert E_1-E_2\vert.
\end{displaymath}">|; 

$key = q/P_1(t)=|c_1(t)|^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2427.png"
 ALT="$P_1(t)=\vert c_1(t)\vert^2$">|; 

$key = q/V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img448.png"
 ALT="$V$">|; 

$key = q/Z,e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1687.png"
 ALT="$Z e$">|; 

$key = q/{displaymath}beta_{l-}=left.frac{1}{(u_lslashr)}frac{d(u_lslashr)}{dr}right|_{r=a}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2957.png"
 ALT="\begin{displaymath}
\beta_{l-} = \left.\frac{1}{(u_l/r)} \frac{d(u_l/r)}{dr}\right\vert _{r=a}.
\end{displaymath}">|; 

$key = q/displaystyleH;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2523.png"
 ALT="$\displaystyle H$">|; 

$key = q/{displaymath}psi=sum_nc_n,psi_n,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2693.png"
 ALT="\begin{displaymath}
\psi = \sum_n c_n \psi_n,
\end{displaymath}">|; 

$key = q/displaystylefrac{2}{(3,a_0)^{3slash2}}left(1-frac{2,r}{3,a_0}+frac{2,r^2}{27,a_0^{,2}}right)expleft(-frac{r}{3,a_0}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1645.png"
 ALT="$\displaystyle \frac{2}{(3 a_0)^{3/2}}\left(1-
\frac{2 r}{3 a_0} + \frac{2 r^2}{27 a_0^{ 2}}\right)\exp\left(-\frac{r}{3 a_0}\right),$">|; 

$key = q/displaystylepsi_2({{bf{r}_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2719.png"
 ALT="$\displaystyle \psi_2({\bf r}_2)$">|; 

$key = q/beta=2,(k-k_0),{mitDelta}x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img340.png"
 ALT="$\beta = 2 (k-k_0) {\mit\Delta}x$">|; 

$key = q/{displaymath}Phi_m(phi)sim{{rm{e}^{,{{rm{i},m,phi}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1397.png"
 ALT="\begin{displaymath}
\Phi_m(\phi)\sim {\rm e}^{ {\rm i} m \phi}.
\end{displaymath}">|; 

$key = q/M_odot=2times10^{30},{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1291.png"
 ALT="$M_\odot = 2\times 10^{30} {\rm kg}$">|; 

$key = q/{displaymath}Phi_m(phi)=frac{{{rm{e}^{,{{rm{i},m,phi}}{sqrt{2pi}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1403.png"
 ALT="\begin{displaymath}
\Phi_m(\phi) = \frac{{\rm e}^{ {\rm i} m \phi}}{\sqrt{2\pi}}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter09slashfig01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="376" HEIGHT="322" BORDER="0"
 SRC="|."$dir".q|img1544.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter09/fig01.eps}}
\end{figure}">|; 

$key = q/|V_{fi}|^2=|V_{if}^nomath_inline{dagnomath_inline{|^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2517.png"
 ALT="$\vert V_{fi}\vert^2=\vert V_{if}^\dag \vert^2$">|; 

$key = q/P_ypropto{mitDelta}y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img285.png"
 ALT="$P_y\propto {\mit\Delta}y$">|; 

$key = q/{displaymath}psi(x)=psi_0,expleft(int_0^x{{rm{i},k(x'),dx'right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img875.png"
 ALT="\begin{displaymath}
\psi(x) = \psi_0 \exp\left(\int_0^x{\rm i} k(x') dx'\right),
\end{displaymath}">|; 

$key = q/langlen,l,m|y|n',l',m'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2639.png"
 ALT="$\langle n,l,m\vert y\vert n',l',m'\rangle$">|; 

$key = q/+A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img158.png"
 ALT="$+A$">|; 

$key = q/1.4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1295.png"
 ALT="$1.4$">|; 

$key = q/{displaymath}psi_1=psi_2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img272.png"
 ALT="\begin{displaymath}
\psi_1=\psi_2,
\end{displaymath}">|; 

$key = q/{displaymath}int_{-infty}^inftypsi_a^ast,(A,psi_a),dx=int_{-infty}^infty(A,psi_a)^ast,psi_a,dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img656.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \psi_a^\ast (A \psi_a) dx = \int_{-\infty}^\infty
(A \psi_a)^\ast \psi_a dx,
\end{displaymath}">|; 

$key = q/E_{nlm}=E_{n'l'm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2219.png"
 ALT="$E_{nlm}=E_{n'l'm}$">|; 

$key = q/displaystyle{{rm{i},frac{2^7}{3^5},a_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2664.png"
 ALT="$\displaystyle {\rm i} \frac{2^7}{3^5} a_0,$">|; 

$key = q/l=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1431.png"
 ALT="$l=0$">|; 

$key = q/mbox{{boldmath{{mu{}_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2361.png"
 ALT="$\mbox{\boldmath$\mu$}_p$">|; 

$key = q/Vrightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2844.png"
 ALT="$V\rightarrow 0$">|; 

$key = q/displaystylechi^{(1)}_{1slash2,1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2082.png"
 ALT="$\displaystyle \chi^{(1)}_{1/2,1/2},$">|; 

$key = q/2P_{1slash2}rightarrow1S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2680.png"
 ALT="$2P_{1/2}\rightarrow 1S$">|; 

$key = q/e^2rightarrow2,e^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2716.png"
 ALT="$e^2\rightarrow 2 e^2$">|; 

$key = q/x_1+dx_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1075.png"
 ALT="$x_1+dx_1$">|; 

$key = q/2P_{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2331.png"
 ALT="$2P_{1/2}$">|; 

$key = q/S_e^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2378.png"
 ALT="$S_e^{ 2}$">|; 

$key = q/x=x_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img886.png"
 ALT="$x=x_1$">|; 

$key = q/r=a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1540.png"
 ALT="$r=a$">|; 

$key = q/{displaymath}Y_{l,l-1}simL_-,Y_{l,l}sim{{rm{e}^{-{{rm{i},phi}left(-frac{partial}phi}right)(sintheta)^l,{{rm{e}^{,{{rm{i},l,phi},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="442" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1446.png"
 ALT="\begin{displaymath}
Y_{l,l-1} \sim L_- Y_{l,l} \sim {\rm e}^{-{\rm i} \phi}\le...
...tial\phi}\right) (\sin\theta)^l {\rm e}^{ {\rm i} l \phi},
\end{displaymath}">|; 

$key = q/{displaymath}psi(t)=frac{1}{sqrt{2pi,hbar}}int_{-infty}^{infty}chi(E),{{rm{e}^{-{{rm{i},E,tslashhbar},dE.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img640.png"
 ALT="\begin{displaymath}
\psi(t)= \frac{1}{\sqrt{2\pi \hbar}}\int_{-\infty}^{\infty}\chi(E) {\rm e}^{-{\rm i} E t/\hbar} dE.
\end{displaymath}">|; 

$key = q/b_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img771.png"
 ALT="$b_2$">|; 

$key = q/j_l(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="$j_l(z)$">|; 

$key = q/c(x')=psi(x');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img716.png"
 ALT="$c(x')=\psi(x')$">|; 

$key = q/displaystyleDeltaE_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2283.png"
 ALT="$\displaystyle \Delta E_1$">|; 

$key = q/{displaymath}Y_{l,m}(theta,phi)=Theta_{l,m}(theta),Phi_m(phi).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1391.png"
 ALT="\begin{displaymath}
Y_{l,m}(\theta,\phi) = \Theta_{l,m}(\theta) \Phi_m(\phi).
\end{displaymath}">|; 

$key = q/{displaymath}psi(x,t)=int_{-infty}^{infty}bar{psi}(k),{{rm{e}^{,{{rm{i},phi(k)},dk,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img354.png"
 ALT="\begin{displaymath}
\psi(x,t) = \int_{-\infty}^{\infty} \bar{\psi}(k) {\rm e}^{ {\rm i} \phi(k)} dk,
\end{displaymath}">|; 

$key = q/q,asim1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img870.png"
 ALT="$q a\sim 1$">|; 

$key = q/k,all1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2970.png"
 ALT="$k a\ll 1$">|; 

$key = q/O(x,p,E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img562.png"
 ALT="$O(x,p,E)$">|; 

$key = q/{displaymath}(A,B-B,A),psi_i=(A,B-B,a_i),psi_i=(A-a_i),B,psi_i=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="441" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img699.png"
 ALT="\begin{displaymath}
(A B-B A) \psi_i = (A B-B a_i) \psi_i = (A-a_i) B \psi_i = 0,
\end{displaymath}">|; 

$key = q/{displaymath}H_1=frac{e^2}{8pi,epsilon_0,m_e^{,2},c^2,r^3},{{bf{L}cdot{{bf{S}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2315.png"
 ALT="\begin{displaymath}
H_1 = \frac{e^2}{8\pi \epsilon_0 m_e^{ 2} c^2 r^3} {\bf L}\cdot {\bf S}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{dlangleHrangle}{dZ}=left[-4,Z+frac{27}{4}right]E_0=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img2772.png"
 ALT="\begin{displaymath}
\frac{d\langle H\rangle}{dZ} = \left[-4 Z+ \frac{27}{4}\right] E_0 = 0.
\end{displaymath}">|; 

$key = q/{displaymath}{{rm{Average~Age}simeqP_{18}times18+P_{19}times19+P_{20}times20+cdots.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="433" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img94.png"
 ALT="\begin{displaymath}
{\rm Average&nbsp;Age} \simeq P_{18}\times 18 + P_{19}\times 19 + P_{20}\times 20
+\cdots.
\end{displaymath}">|; 

$key = q/displaystyle(0,,0,,k),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2587.png"
 ALT="$\displaystyle (0, 0, k),$">|; 

$key = q/P(u);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img120.png"
 ALT="$P(u)$">|; 

$key = q/y>sqrt{lambda};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="$y&gt;\sqrt {\lambda }$">|; 

$key = q/x_1=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img903.png"
 ALT="$x_1=0$">|; 

$key = q/-1,0,1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1973.png"
 ALT="$-1,0,1$">|; 

$key = q/Dggd;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img263.png"
 ALT="$D\gg d$">|; 

$key = q/{{bf{n}cdot{{bf{r}=d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="${\bf n}\cdot {\bf r} = d$">|; 

$key = q/omega_{21}pm2,gamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2442.png"
 ALT="$\omega_{21}\pm 2 \gamma$">|; 

$key = q/{displaymath}p_i=-{{rm{i},hbar,frac{partial}{partialx_i}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1080.png"
 ALT="\begin{displaymath}
p_i = - {\rm i} \hbar \frac{\partial}{\partial x_i}.
\end{displaymath}">|; 

$key = q/p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img257.png"
 ALT="$p$">|; 

$key = q/{displaymath}psi_X(X,t)=psi_{0},{{rm{e}^{,{{rm{i},(P',Xslashhbar-E',tslashhbar)},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1125.png"
 ALT="\begin{displaymath}
\psi_X(X,t) = \psi_{0} {\rm e}^{ {\rm i} (P' X/\hbar - E' t/\hbar)},
\end{displaymath}">|; 

$key = q/p=-{{rm{i},hbar,partialslashpartialx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img564.png"
 ALT="$p=-{\rm i} \hbar \partial/\partial x$">|; 

$key = q/{displaymath}psi_{nlm}^{(1)'}=psi_{nlm}^{(1)}+sum_{n'neqn,l'=lpm1}frac{e_{n'l'nl}}{E_n-E_{n'}},psi_{n'l'm}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2262.png"
 ALT="\begin{displaymath}
\psi_{nlm}^{(1)'} = \psi_{nlm}^{(1)} + \sum_{n'\neq n,l'=l\pm 1}
\frac{e_{n'l'nl}}{E_n-E_{n'}} \psi_{n'l'm}.
\end{displaymath}">|; 

$key = q/displaystyleq^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img955.png"
 ALT="$\displaystyle q^2$">|; 

$key = q/i,j=1,2,3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1870.png"
 ALT="$i,j=1,2,3$">|; 

$key = q/{displaymath}tandelta_l=frac{-(k,a)^{2,l+1}}{(2,l+1),[(2,l-1)!!]^{,2}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2977.png"
 ALT="\begin{displaymath}
\tan\delta_l = \frac{-(k a)^{2 l+1}}{(2 l+1)  [(2 l-1)!!]^{ 2}}.
\end{displaymath}">|; 

$key = q/tany;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img982.png"
 ALT="$\tan y$">|; 

$key = q/{displaymath}H_{nm}(t)=langlen|H_1(t)|mrangle,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2407.png"
 ALT="\begin{displaymath}
H_{nm}(t) = \langle n\vert H_1(t)\vert m\rangle,
\end{displaymath}">|; 

$key = q/beta=Zslasha_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1690.png"
 ALT="$\beta = Z/a_0$">|; 

$key = q/displaystylesqrt{n},psi_{n-1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1040.png"
 ALT="$\displaystyle \sqrt{n} \psi_{n-1}.$">|; 

$key = q/{displaymath}w_{irightarrowf}^{spn}=frac{N_f}{N_i},w^{abs}_{frightarrowi}-w_{irightarrowf}^{stm}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2607.png"
 ALT="\begin{displaymath}
w_{i\rightarrow f}^{spn} = \frac{N_f}{N_i} w^{abs}_{f\rightarrow i}-
w_{i\rightarrow f}^{stm}.
\end{displaymath}">|; 

$key = q/{displaymath}w_{irightarrowf}simalpha^5,omega_{if}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2687.png"
 ALT="\begin{displaymath}
w_{i\rightarrow f} \sim \alpha^5 \omega_{if}
\end{displaymath}">|; 

$key = q/{displaymath}frac{X''}{X}+frac{Y''}{Y}+frac{Z''}{Z}=-frac{2,m}{hbar^2},E,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1217.png"
 ALT="\begin{displaymath}
\frac{X''}{X} + \frac{Y''}{Y} + \frac{Z''}{Z} = -\frac{2 m}{\hbar^2} E,
\end{displaymath}">|; 

$key = q/m'=m-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2645.png"
 ALT="$m'=m-1$">|; 

$key = q/displaystylelangleA^2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img690.png"
 ALT="$\displaystyle \langle A^2\rangle$">|; 

$key = q/displaystyleexp(-{{rm{i},k,a),frac{sin[k,(r-a)]}{k,r}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2966.png"
 ALT="$\displaystyle \exp(-{\rm i}  k a)  \frac{ \sin[k (r-a)]}{k r}.$">|; 

$key = q/chi_a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1738.png"
 ALT="$\chi_a$">|; 

$key = q/displaystyle2,{{rm{i},hbar,left(L^2,L_y,x-L_x,y+{{rm{i},hbar,zright);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2195.png"
 ALT="$\displaystyle 2 {\rm i} \hbar \left(L^2, L_y x-L_x y + {\rm i} \hbar z\right)$">|; 

$key = q/displaystylesum_i|c_i|^2,a_i^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img691.png"
 ALT="$\displaystyle \sum_i \vert c_i\vert^2 a_i^{ 2}.$">|; 

$key = q/psi_a^ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img665.png"
 ALT="$\psi_a^\ast$">|; 

$key = q/{displaymath}|T|^{,2}=frac{4,k^2,q^2}{4,k^2,q^2+(k^2-q^2)^{,2},sin^2(q,a)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="294" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img847.png"
 ALT="\begin{displaymath}
\vert T\vert^{ 2} = \frac{4 k^2 q^2}{4 k^2 q^2 + (k^2-q^2)^{ 2} \sin^2(q a)}.
\end{displaymath}">|; 

$key = q/psi^{(1)}_{0,pm1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1975.png"
 ALT="$\psi^{(1)}_{0,\pm 1/2}$">|; 

$key = q/{bfn}cdot{bfr}=d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img175.png"
 ALT="${\bf n}\cdot{\bf r} = d$">|; 

$key = q/displaystyle-hbar^2,left{4,(L_x,x+L_y,y+L_z,z),L_z-4,(L_x^{,2}+L_y^{,2}+L_z^{,2}),zright.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="432" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2199.png"
 ALT="$\displaystyle -\hbar^2 \left\{4 (L_x x+ L_y y+L_z z) L_z
-4 (L_x^{ 2}+L_y^{ 2}+L_z^{ 2}) z\right.$">|; 

$key = q/{{bf{A}({bf{r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2525.png"
 ALT="${\bf A}(\bf r)$">|; 

$key = q/j(b,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img484.png"
 ALT="$j(b,t)$">|; 

$key = q/displaystyle{{rm{i},hbar,{{bf{L}+{{rm{i},hbar,{{bf{S};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1875.png"
 ALT="$\displaystyle {\rm i} \hbar {\bf L} + {\rm i} \hbar {\bf S}$">|; 

$key = q/displaystylepsi(x,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img566.png"
 ALT="$\displaystyle \psi(x,t)$">|; 

$key = q/displaystylelanglef|V^nomath_inline{dagnomath_inline{|irangle=langlei|V|frangle^ast.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2486.png"
 ALT="$\displaystyle \langle f\vert V^\dag \vert i\rangle = \langle i\vert V\vert f\rangle^\ast.$">|; 

$key = q/displaystyleL^2,psi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1534.png"
 ALT="$\displaystyle L^2 \psi$">|; 

$key = q/{displaymath}psi(0)=sum_mc_m,psi_m,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2389.png"
 ALT="\begin{displaymath}
\psi(0) = \sum_m c_m \psi_m,
\end{displaymath}">|; 

$key = q/1slashk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2988.png"
 ALT="$1/k$">|; 

$key = q/r=a,z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1586.png"
 ALT="$r = a z$">|; 

$key = q/{displaymath}sigma_E,sigma_tgeqfrac{hbar}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img634.png"
 ALT="\begin{displaymath}
\sigma_E \sigma_t\geq \frac{\hbar}{2}.
\end{displaymath}">|; 

$key = q/bar{E}proptoM^{4slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1292.png"
 ALT="$\bar{E}\propto M^{4/3}$">|; 

$key = q/2S_{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2330.png"
 ALT="$2S_{1/2}$">|; 

$key = q/{displaymath}H=frac{p_x^{,2}}{2,m}+V(x,t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1204.png"
 ALT="\begin{displaymath}
H = \frac{p_x^{ 2}}{2 m} + V(x,t),
\end{displaymath}">|; 

$key = q/psi_{21-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2271.png"
 ALT="$\psi_{21-1}$">|; 

$key = q/displaystylelangle1|H_1|1rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2130.png"
 ALT="$\displaystyle \langle 1\vert H_1\vert 1\rangle,$">|; 

$key = q/displaystylechi_-,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2455.png"
 ALT="$\displaystyle \chi_-,$">|; 

$key = q/z=k,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1563.png"
 ALT="$z=k a$">|; 

$key = q/displaystyleS_+,chi_{s,m_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1933.png"
 ALT="$\displaystyle S_+ \chi_{s,m_s}$">|; 

$key = q/displaystyle(x-m),alpha-[l,(l+1)-m,(m+1)]^{1slash2},beta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1946.png"
 ALT="$\displaystyle (x - m) \alpha - [l (l+1)-m (m+1)]^{1/2} \beta$">|; 

$key = q/{displaymath}|omega-omega_{fi}|stackrel{_{{normalsize{<}}{_{{normalsize{sim}}frac{2pi}{t},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img2498.png"
 ALT="\begin{displaymath}
\vert\omega-\omega_{fi}\vert\stackrel {_{\normalsize &lt;}}{_{\normalsize\sim}}\frac{2\pi}{t},
\end{displaymath}">|; 

$key = q/F_+simeq-1.13;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2829.png"
 ALT="$F_+\simeq -1.13$">|; 

$key = q/nabla!cdot!{{bf{A}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2535.png"
 ALT="$\nabla\!\cdot\!{\bf A} = 0$">|; 

$key = q/A_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2924.png"
 ALT="$A_l$">|; 

$key = q/E_{bind}=-2.8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2833.png"
 ALT="$E_{bind}=-2.8$">|; 

$key = q/H_1=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2394.png"
 ALT="$H_1=0$">|; 

$key = q/omega(k);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img397.png"
 ALT="$\omega(k)$">|; 

$key = q/displaystyle{{rm{i},hbar,(x,p_y-y,p_x)={{rm{i},hbar,L_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1317.png"
 ALT="$\displaystyle {\rm i} \hbar (x p_y-y p_x) = {\rm i} \hbar L_z,$">|; 

$key = q/displaystyle[l,(l+1)-m,(m-1)]^{1slash2},hbar,Y_{l,m-1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1390.png"
 ALT="$\displaystyle [l (l+1)-m (m-1)]^{1/2} \hbar Y_{l,m-1}.$">|; 

$key = q/(1slash2),[x,p+(x,p)^nomath_inline{dagnomath_inline{];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img554.png"
 ALT="$(1/2) [x p + (x p)^\dag ]$">|; 

$key = q/O;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img524.png"
 ALT="$O$">|; 

$key = q/gamma,tll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2507.png"
 ALT="$\gamma t\ll 1$">|; 

$key = q/displaystylea_+,psi_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1037.png"
 ALT="$\displaystyle a_+ \psi_n$">|; 

$key = q/k'equivdkslashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img879.png"
 ALT="$k'\equiv dk/dx$">|; 

$key = q/{displaymath}H_0,chi_pm=mpfrac{g,e,hbar,B_0}{4,m},chi_pm.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2449.png"
 ALT="\begin{displaymath}
H_0 \chi_\pm = \mp \frac{g e \hbar B_0}{4 m} \chi_\pm.
\end{displaymath}">|; 

$key = q/{displaymath}z=int_{-infty}^inftypsi^ast,A,B,psi,dx-langleArangle,langleBrangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img618.png"
 ALT="\begin{displaymath}
z =\int_{-\infty}^\infty \psi^\ast A B \psi dx -\langle A\rangle \langle B\rangle.
\end{displaymath}">|; 

$key = q/m_s'=m_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2653.png"
 ALT="$m_s'=m_s$">|; 

$key = q/p^2slash(2,m);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img439.png"
 ALT="$p^2/(2 m)$">|; 

$key = q/(k,r)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2686.png"
 ALT="$(k r)^2$">|; 

$key = q/displaystylefrac{pi}{epsilon_0,hbar^2},|mbox{{boldmath{{epsilon{}!cdot!{{bf{d}_{if}|^{,2},rho(omega_{if}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2583.png"
 ALT="$\displaystyle \frac{\pi}{\epsilon_0 \hbar^2} \vert\mbox{\boldmath$\epsilon$}\!\cdot\!{\bf d}_{if}\vert^{ 2} \rho(\omega_{if}),$">|; 

$key = q/{displaymath}langlei|H_0+H_1|Erangle=E,langlei|Erangle,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2125.png"
 ALT="\begin{displaymath}
\langle i\vert H_0 + H_1\vert E\rangle = E \langle i\vert E\rangle,
\end{displaymath}">|; 

$key = q/V(x)-E=W-e,{calE},x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img901.png"
 ALT="$V(x)-E=W-e {\cal E} x$">|; 

$key = q/X_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img78.png"
 ALT="$X_i$">|; 

$key = q/{displaymath}Y_{l,l-1}(theta,phi)simfrac{{{rm{e}^{,{{rm{i},(l-1),phi}}{(sintheta{sintheta}frac{d}{dtheta}right)(sintheta)^{2,l}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="347" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1450.png"
 ALT="\begin{displaymath}
Y_{l,l-1}(\theta,\phi)\sim \frac{{\rm e}^{ {\rm i} (l-1) ...
...ac{1}{\sin\theta}\frac{d}{d\theta}\right)
(\sin\theta)^{2 l}.
\end{displaymath}">|; 

$key = q/displaystyle[L_y^{,2},L_x]+[L_z^{,2},L_x];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1328.png"
 ALT="$\displaystyle [L_y^{ 2},L_x] + [L_z^{ 2},L_x]$">|; 

$key = q/tilde{psi}_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2703.png"
 ALT="$\tilde{\psi}_0$">|; 

$key = q/2,hbar^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2381.png"
 ALT="$2 \hbar^2$">|; 

$key = q/{displaymath}E=(n+1slash2),hbar,omega,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1021.png"
 ALT="\begin{displaymath}
E = (n+1/2) \hbar \omega,
\end{displaymath}">|; 

$key = q/{displaymath}int_{-infty}^{infty}delta(x-x'),delta(x-x''),dx=delta(x'-x'').{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img712.png"
 ALT="\begin{displaymath}
\int_{-\infty}^{\infty} \delta(x-x') \delta(x-x'') dx = \delta(x'-x'').
\end{displaymath}">|; 

$key = q/s_1=s_2=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2058.png"
 ALT="$s_1=s_2=1/2$">|; 

$key = q/{displaymath}sigmasim{mitDelta}v,tsimfrac{hbar,t}{m,{mitDelta}x},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img409.png"
 ALT="\begin{displaymath}
\sigma \sim {\mit\Delta}v t \sim \frac{\hbar t}{m {\mit\Delta}x},
\end{displaymath}">|; 

$key = q/nu={mitDelta}Eslashh;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1673.png"
 ALT="$\nu={\mit\Delta}E/h$">|; 

$key = q/displaystylel';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2649.png"
 ALT="$\displaystyle l'$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter09slashfig03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="373" HEIGHT="331" BORDER="0"
 SRC="|."$dir".q|img1651.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter09/fig03.eps}}
\end{figure}">|; 

$key = q/displaystyleA_0,mbox{{boldmath{{epsilon{},cos!left({{bf{k}!cdot!{{bf{r}-omegatright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2539.png"
 ALT="$\displaystyle A_0  \mbox{\boldmath$\epsilon$} \cos\!\left
({\bf k}\!\cdot\!{\bf r}
- \omega t\right),$">|; 

$key = q/{displaymath}psi(x,t)=psi_0,{{rm{e}^{,{{rm{i},(k,x-omega,t)},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img217.png"
 ALT="\begin{displaymath}
\psi(x,t) = \psi_0 {\rm e}^{ {\rm i} (k x-\omega t)},
\end{displaymath}">|; 

$key = q/x_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img265.png"
 ALT="$x_1$">|; 

$key = q/i=1,2,3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1496.png"
 ALT="$i=1,2,3$">|; 

$key = q/displaystylephi_0''&=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img365.png"
 ALT="$\displaystyle \phi_0''&amp;=$">|; 

$key = q/psi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img320.png"
 ALT="$\psi$">|; 

$key = q/displaystylecos(alphaslash2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1842.png"
 ALT="$\displaystyle \cos(\alpha/2),$">|; 

$key = q/displaystyleleftlanglefrac{1}{r}rightrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1657.png"
 ALT="$\displaystyle \left\langle \frac{1}{r}\right\rangle$">|; 

$key = q/displaystyley_1(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1557.png"
 ALT="$\displaystyle y_1(z)$">|; 

$key = q/V(x)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1054.png"
 ALT="$V(x)=0$">|; 

$key = q/{displaymath}Tsimeqfrac{p^2}{2,m}-frac{p^4}{8,m^3,c^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img2291.png"
 ALT="\begin{displaymath}
T \simeq \frac{p^2}{2 m} - \frac{p^4}{8 m^3 c^2}.
\end{displaymath}">|; 

$key = q/{displaymath}s,(s+1)geqm_s,(m_s+1).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1754.png"
 ALT="\begin{displaymath}
s (s+1) \geq m_s (m_s+1).
\end{displaymath}">|; 

$key = q/displaystyle2,alpha,({mitDelta}k)^{,2},t.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img378.png"
 ALT="$\displaystyle 2 \alpha ({\mit\Delta}k)^{ 2} t.$">|; 

$key = q/displaystylelanglef|V|irangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2484.png"
 ALT="$\displaystyle \langle f\vert V\vert i\rangle,$">|; 

$key = q/{displaymath}B=frac{5}{3},P=frac{pi^3,hbar^2}{9,m}left(frac{3,n}{pi}right)^{5slash3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1268.png"
 ALT="\begin{displaymath}
B = \frac{5}{3} P = \frac{\pi^3 \hbar^2}{9 m}\left(\frac{3 n}{\pi}\right)^{5/3}.
\end{displaymath}">|; 

$key = q/P(X);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="$P(X)$">|; 

$key = q/displaystyledelta_{mm'}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1395.png"
 ALT="$\displaystyle \delta_{mm'}.$">|; 

$key = q/{displaymath}frac{Deltalambda}{lambda}simfrac{DeltaE_{2P}}{hbar,omega}sim4times10^{-8}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2676.png"
 ALT="\begin{displaymath}
\frac{\Delta\lambda}{\lambda} \sim \frac{\Delta E_{2P}}{\hbar \omega}\sim 4 \times 10^{-8}.
\end{displaymath}">|; 

$key = q/H_1neq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2395.png"
 ALT="$H_1\neq 0$">|; 

$key = q/displaystylez,p_x-x,p_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1307.png"
 ALT="$\displaystyle z p_x - x p_z,$">|; 

$key = q/{{rm{sinc}(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="${\rm sinc}(x)$">|; 

$key = q/E_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2243.png"
 ALT="$E_n$">|; 

$key = q/displaystyle-frac{cosz}{z},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1556.png"
 ALT="$\displaystyle - \frac{\cos z}{z},$">|; 

$key = q/{displaymath}epsilon_{ijk},epsilon_{ilm}equivdelta_{jl},delta_{km}-delta_{jm},delta_{kl}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1512.png"
 ALT="\begin{displaymath}
\epsilon_{ijk} \epsilon_{ilm}\equiv \delta_{jl} \delta_{km}-\delta_{jm} \delta_{kl}.
\end{displaymath}">|; 

$key = q/{displaymath}P_n(t)=|langlepsi_n|psirangle|^2=left|c_n,expleft(-{{rm{i},E_n,tslashhbarright)right|^2=|c_n|^2=P_n(0),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2392.png"
 ALT="\begin{displaymath}
P_n(t) = \vert\langle \psi_n\vert\psi\rangle\vert^2 = \left\...
...,E_n t/\hbar\right)\right\vert^2 = \vert c_n\vert^2 = P_n(0),
\end{displaymath}">|; 

$key = q/d^{,2}_{if};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2595.png"
 ALT="$d^{ 2}_{if}$">|; 

$key = q/{displaymath}psi(x,t)=bar{psi},{{rm{e}^{,{{rm{i},(k,x-omega,t)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img227.png"
 ALT="\begin{displaymath}
\psi (x,t) = \bar{\psi} {\rm e}^{ {\rm i} (k x-\omega t)}.
\end{displaymath}">|; 

$key = q/displaystylefrac{sinz}{z^2}-frac{cosz}{z},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1554.png"
 ALT="$\displaystyle \frac{\sin z}{z^2} - \frac{\cos z}{z},$">|; 

$key = q/displaystyle-hbar^2,left{4,(L_x,x+L_y,y+L_z,z),L_z-2,(L^2,z+z,L^2)right}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="411" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2201.png"
 ALT="$\displaystyle -\hbar^2 \left\{4 (L_x x+ L_y y+L_z z) L_z-2 (L^2 z+z L^2)\right\}.$">|; 

$key = q/displaystyleA+B,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img839.png"
 ALT="$\displaystyle A+B,$">|; 

$key = q/displaystyle2,{{rm{i},sigma_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1803.png"
 ALT="$\displaystyle 2 {\rm i} \sigma_z,$">|; 

$key = q/displaystyleleft.+2,(L^2,z-z,L^2)right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2200.png"
 ALT="$\displaystyle \left.+2 (L^2 z-z L^2)\right\}$">|; 

$key = q/{displaymath}H,psi=E,psi{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2713.png"
 ALT="\begin{displaymath}
H \psi = E \psi
\end{displaymath}">|; 

$key = q/displaystyle{{bf{E};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2527.png"
 ALT="$\displaystyle {\bf E}$">|; 

$key = q/{displaymath}langlexrangle=frac{1}{2pi,hbar}int_{-infty}^{infty}phi^ast(p)left({{i},hbar,frac{partial}{partialp}right)phi(p),dp.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img598.png"
 ALT="\begin{displaymath}
\langle x\rangle= \frac{1}{2\pi \hbar}\int_{-\infty}^{\inft...
...{\rm i} \hbar \frac{\partial}{\partial p}\right)\phi(p) dp.
\end{displaymath}">|; 

$key = q/2P_{3slash2}rightarrow1S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2679.png"
 ALT="$2P_{3/2}\rightarrow 1S$">|; 

$key = q/{displaymath}E_{nl}'=E_n+e_{nlnl}+sum_{n',l'=lpm1}frac{|e_{n'l'nl}|^2}{E_n-E_{n'}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img2236.png"
 ALT="\begin{displaymath}
E_{nl}' = E_n + e_{nlnl} + \sum_{n',l'=l\pm 1}\frac{\vert e_{n'l'nl}\vert^2}{E_n-E_{n'}},
\end{displaymath}">|; 

$key = q/{displaymath}{{rm{i},hbar,frac{partialpsi}{partialt}=H,psi.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img561.png"
 ALT="\begin{displaymath}
{\rm i} \hbar \frac{\partial\psi}{\partial t} = H \psi.
\end{displaymath}">|; 

$key = q/E_i>E_f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2600.png"
 ALT="$E_i&gt; E_f$">|; 

$key = q/Gamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3031.png"
 ALT="$\Gamma$">|; 

$key = q/{displaymath}langlef|mbox{{boldmath{{epsilon{}!cdot!{{bf{p},exp(,{{rm{i},{{bf{k}ldmath{{epsilon{}}!cdot!langlef|{{bf{p}|irangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2577.png"
 ALT="\begin{displaymath}
\langle f\vert\mbox{\boldmath$\epsilon$}\!\cdot\!{\bf p} \e...
...dmath$\epsilon$}}\!\cdot\!\langle f\vert{\bf p}\vert i\rangle.
\end{displaymath}">|; 

$key = q/psi(x,E_b);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1161.png"
 ALT="$\psi(x,E_b)$">|; 

$key = q/displaystyle-frac{g,e,B_1}{4,m}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2459.png"
 ALT="$\displaystyle -\frac{g e B_1}{4 m}.$">|; 

$key = q/{displaymath}int_{-infty}^inftypsi_1^ast,psi_2,dx=c,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img762.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \psi_1^\ast \psi_2 dx = c,
\end{displaymath}">|; 

$key = q/{displaymath}{{rm{sinc},xequivfrac{sin,x}{x}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img2491.png"
 ALT="\begin{displaymath}
{\rm sinc}  x\equiv \frac{\sin x}{x}.
\end{displaymath}">|; 

$key = q/psi_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img270.png"
 ALT="$\psi_1$">|; 

$key = q/displaystyle[L^2,L_x];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1327.png"
 ALT="$\displaystyle [L^2,L_x]$">|; 

$key = q/{displaymath}k^2=frac{2,m,|E|}{hbar^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img966.png"
 ALT="\begin{displaymath}
k^2 = \frac{2 m \vert E\vert}{\hbar^2}.
\end{displaymath}">|; 

$key = q/n!!=n,(n-2),(n-4)cdots1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2976.png"
 ALT="$n!! = n (n-2) (n-4)\cdots 1$">|; 

$key = q/{displaymath}int_{-infty}^infty|psi(x,t)|^2,dx=1{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img1070.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \vert\psi(x,t)\vert^2 dx = 1
\end{displaymath}">|; 

$key = q/displaystylek,T,{{rm{e}^{,{{rm{i},k,a}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img845.png"
 ALT="$\displaystyle k T {\rm e}^{ {\rm i} k a}.$">|; 

$key = q/m'=m+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2647.png"
 ALT="$m'=m+1$">|; 

$key = q/chi_pm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1921.png"
 ALT="$\chi_\pm$">|; 

$key = q/{displaymath}1+3+5+cdots+2,(n-1)+1=n^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1667.png"
 ALT="\begin{displaymath}
1 + 3 + 5 + \cdots +2 (n-1)+1 = n^2.
\end{displaymath}">|; 

$key = q/k_B,TllE_F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1258.png"
 ALT="$k_B T\ll E_F$">|; 

$key = q/{displaymath}Y_{l,l}(theta,phi)=Theta_{l,l}(theta),{{rm{e}^{,{{rm{i},l,phi}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1438.png"
 ALT="\begin{displaymath}
Y_{l,l}(\theta,\phi) = \Theta_{l,l}(\theta) {\rm e}^{ {\rm i} l \phi}
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{L}={{bf{r}times{{bf{p}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1303.png"
 ALT="\begin{displaymath}
{\bf L} = {\bf r}\times {\bf p}.
\end{displaymath}">|; 

$key = q/{displaymath}m_{s,max}-m_{s,min}=2,s=k,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1774.png"
 ALT="\begin{displaymath}
m_{s max}-m_{s min} = 2 s = k,
\end{displaymath}">|; 

$key = q/displaystylesin(alphaslash2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1844.png"
 ALT="$\displaystyle \sin(\alpha/2).$">|; 

$key = q/displaystyleR_{1,0}(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1638.png"
 ALT="$\displaystyle R_{1,0}(r)$">|; 

$key = q/V_0ggE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img871.png"
 ALT="$V_0\gg E$">|; 

$key = q/displaystyle-frac{2}{X},{{rm{e}^{-X}int_0^Xleft[{{rm{e}^{-2,x},(1+X+x)-(1+X-x)right]x,dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="421" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2797.png"
 ALT="$\displaystyle - \frac{2}{X} {\rm e}^{-X}\int_0^X \left[{\rm e}^{-2 x} (1+X+x)-
(1+X-x)\right]x dx$">|; 

$key = q/{displaymath}|T|^{,2}=expleft(-frac{2,sqrt{2,m}}{hbar}int_{0}^{Wslashe,{calE}}sqrt{W-e,{calE},x},dxright).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="378" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img906.png"
 ALT="\begin{displaymath}
\vert T\vert^{ 2} = \exp\left(-\frac{2 \sqrt{2 m}}{\hbar}\int_{0}^{W/e {\cal E}}
\sqrt{W-e {\cal E} x } dx\right).
\end{displaymath}">|; 

$key = q/{calO}(epsilon^2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2163.png"
 ALT="${\cal O}(\epsilon^2)$">|; 

$key = q/m+1,-1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1964.png"
 ALT="$m+1, -1/2$">|; 

$key = q/displaystylefrac{{{rm{e}^{-{{rm{i},p,x_0slashhbar}}{(2pi,sigma_p^{,2})^{1slash4}},{{rm{e}^{-(p-p_0)^{,2}slash4,sigma_p^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img630.png"
 ALT="$\displaystyle \frac{{\rm e}^{-{\rm i} p x_0/\hbar}}{(2\pi \sigma_p^{ 2})^{1/4}} {\rm e}^{-(p-p_0)^{ 2}/4 \sigma_p^{ 2}},$">|; 

$key = q/P(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img140.png"
 ALT="$P(t)$">|; 

$key = q/{displaymath}L_-,L_+,Y_{l,m}=c^+_{l,m},c^-_{l,m+1},Y_{l,m}=[l,(l+1)-m,(m+1)],hbar^2,Y_{l,m},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="487" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img1382.png"
 ALT="\begin{displaymath}
L_- L_+ Y_{l,m} = c^+_{l,m} c^-_{l,m+1} Y_{l,m} =
[l (l+1)-m (m+1)] \hbar^2 Y_{l,m},
\end{displaymath}">|; 

$key = q/{displaymath}ointY^{,ast}_{l',m'}(theta,phi),Y_{l,m}(theta,phi),dOmega=delta_{ll'},delta_{mm'},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1370.png"
 ALT="\begin{displaymath}
\oint Y^{ \ast}_{l',m'}(\theta,\phi) Y_{l,m}(\theta,\phi) d\Omega = \delta_{ll'} \delta_{mm'},
\end{displaymath}">|; 

$key = q/V(-x)=V(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img958.png"
 ALT="$V(-x)=V(x)$">|; 

$key = q/T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img159.png"
 ALT="$T$">|; 

$key = q/v=pslashm=hbar,kslashm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img821.png"
 ALT="$v = p/m=\hbar k/m$">|; 

$key = q/displaystylefrac{2}{a_0^{,3slash2}},expleft(-frac{r}{a_0}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1639.png"
 ALT="$\displaystyle \frac{2}{a_0^{ 3/2}} \exp\left(-\frac{r}{a_0}\right),$">|; 

$key = q/sigma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img388.png"
 ALT="$\sigma$">|; 

$key = q/s=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2064.png"
 ALT="$s=0$">|; 

$key = q/mbox{{boldmath{{mu{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1823.png"
 ALT="$\mbox{\boldmath$\mu$}$">|; 

$key = q/{displaymath}J^2=({{bf{L}+{{bf{S})cdot({{bf{L}+{{bf{S})=L^2+S^2+2,{{bf{L}cdot{{bf{S},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1889.png"
 ALT="\begin{displaymath}
J^2 = ({\bf L}+{\bf S})\cdot({\bf L}+{\bf S}) = L^2+ S^2 + 2 {\bf L}\cdot{\bf S},
\end{displaymath}">|; 

$key = q/{displaymath}I(y)propto|psi(y,t)|^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img275.png"
 ALT="\begin{displaymath}
I(y) \propto \vert\psi(y,t)\vert^{ 2}.
\end{displaymath}">|; 

$key = q/displaystyleDeltaE_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2285.png"
 ALT="$\displaystyle \Delta E_2$">|; 

$key = q/displaystylec_f(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2488.png"
 ALT="$\displaystyle c_f(t)$">|; 

$key = q/{displaymath}a_pm=frac{1}{sqrt{2}}left(mpfrac{d}{dy}+yright).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1033.png"
 ALT="\begin{displaymath}
a_\pm = \frac{1}{\sqrt{2}}\left(\mp \frac{d}{dy}+y\right).
\end{displaymath}">|; 

$key = q/{displaymath}y=sqrt{frac{m,omega}{hbar}},x,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img999.png"
 ALT="\begin{displaymath}
y = \sqrt{\frac{m \omega}{\hbar}} x,
\end{displaymath}">|; 

$key = q/displaystyleleft(frac{l+m+1}{2,l+1}right)^{1slash2}!psi^{(2)}_{l+1slash2,m+1slas{l-m}{2,l+1}right)^{1slash2}!psi^{(2)}_{l-1slash2,m+1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="451" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1960.png"
 ALT="$\displaystyle \left(\frac{l+m+1}{2 l+1}\right)^{1/2}\!\psi^{(2)}_{l+1/2,m+1/2} + \left(\frac{l-m}{2 l+1}\right)^{1/2}\!\psi^{(2)}_{l-1/2,m+1/2},$">|; 

$key = q/{displaymath}chiequivleft(array{{c}c_+c_-array{right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1789.png"
 ALT="\begin{displaymath}
\chi \equiv \left(\begin{array}{c}c_+ c_-\end{array}\right).
\end{displaymath}">|; 

$key = q/durightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img118.png"
 ALT="$du\rightarrow 0$">|; 

$key = q/{displaymath}E_{nl}'=E_n+lambda_{nl}+sum_{n'neqn,l'=lpm1}frac{|e_{n'l'nl}|^2}{E_n-E_{n'}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img2261.png"
 ALT="\begin{displaymath}
E_{nl}' = E_n +\lambda_{nl}+\sum_{n'\neq n,l'=l\pm 1}\frac{\vert e_{n'l'nl}\vert^2}{E_n-E_{n'}},
\end{displaymath}">|; 

$key = q/omega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2621.png"
 ALT="$\omega_0$">|; 

$key = q/{{rm{arg}(z)=tan^{-1}(yslashx);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img206.png"
 ALT="${\rm arg}(z)=\tan^{-1}(y/x)$">|; 

$key = q/{displaymath}psi_E=sum_klanglek|Erangle,psi_k,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2151.png"
 ALT="\begin{displaymath}
\psi_E = \sum_k \langle k\vert E\rangle \psi_k,
\end{displaymath}">|; 

$key = q/j_1(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="$j_1(z)$">|; 

$key = q/Dgglambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img273.png"
 ALT="$D\gg \lambda$">|; 

$key = q/{displaymath}|psi|rightarrow0mbox{hspace{0.5cm}ashspace{0.5cm}}|x|rightarrowinfty.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img479.png"
 ALT="\begin{displaymath}
\vert\psi\vert\rightarrow 0\mbox{\hspace{0.5cm}as\hspace{0.5cm}}\vert x\vert\rightarrow\infty.
\end{displaymath}">|; 

$key = q/1S_{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2329.png"
 ALT="$1S_{1/2}$">|; 

$key = q/xneq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img575.png"
 ALT="$x\neq 0$">|; 

$key = q/B_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1864.png"
 ALT="$B_0$">|; 

$key = q/{displaymath}H(x_1,x_2,ldots,x_N,t)=sum_{i=1,N}H_i(x_i,t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1093.png"
 ALT="\begin{displaymath}
H(x_1,x_2,\ldots, x_N,t)=\sum_{i=1,N} H_i(x_i,t),
\end{displaymath}">|; 

$key = q/-infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img192.png"
 ALT="$-\infty$">|; 

$key = q/displaystyleexp(-{{rm{i},k,a)frac{[cos(k,a),sin(k,r)-sin(k,a),cos(k,r)]}{k,r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="390" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2965.png"
 ALT="$\displaystyle \exp(-{\rm i}  k a) \frac{[\cos (k a)  \sin (k r)
-\sin (k a)  \cos (k r)]}{k r}$">|; 

$key = q/{displaymath}alpha=2,e^2sum_{n>1}frac{|langle1,0,0|z|n,1,0rangle|^2}{E_{n00}-E_{100}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img2221.png"
 ALT="\begin{displaymath}
\alpha = 2 e^2\sum_{n&gt;1}\frac{\vert\langle 1,0,0\vert z\vert n,1,0\rangle\vert^2}{E_{n00}-E_{100}}.
\end{displaymath}">|; 

$key = q/nsim8.6times10^{28},{{rm{m}^{-3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1269.png"
 ALT="$n\sim 8.6\times 10^{28} {\rm m}^{-3}$">|; 

$key = q/{displaymath}int_{-infty}^infty|psi_i(x_i,t)|^2,dx_i=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img1101.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \vert\psi_i(x_i,t)\vert^2 dx_i = 1.
\end{displaymath}">|; 

$key = q/{displaymath}psi(r,theta,phi)=R_{n,l}(r),Y_{l,m}(theta,phi),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1541.png"
 ALT="\begin{displaymath}
\psi(r,\theta,\phi) = R_{n,l}(r) Y_{l,m}(\theta,\phi),
\end{displaymath}">|; 

$key = q/{displaymath}c=nu,lambda,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img234.png"
 ALT="\begin{displaymath}
c = \nu \lambda,
\end{displaymath}">|; 

$key = q/displaystyle28.9,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img949.png"
 ALT="$\displaystyle 28.9,$">|; 

$key = q/{displaymath}left(-frac{hbar^2}{2,m_e},nabla^2-frac{e^2}{4pi,epsilon_0,r_{1,2}}rt)psi_0({{bf{r}_{1,2})=E_0,psi_0({{bf{r}_{1,2}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="377" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2801.png"
 ALT="\begin{displaymath}
\left(-\frac{\hbar^2}{2 m_e} \nabla^2 - \frac{e^2}{4\pi \...
...,2}}\right)\psi_0({\bf r}_{1,2}) = E_0 \psi_0({\bf r}_{1,2}),
\end{displaymath}">|; 

$key = q/{displaymath}langle(x_1-x_2)^2rangle_{fermion}=langle(x_1-x_2)^2rangle_{dist}+2,|langlexrangle_{ab}|^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="390" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1158.png"
 ALT="\begin{displaymath}
\langle (x_1-x_2)^2\rangle_{fermion} = \langle (x_1-x_2)^2\rangle_{dist} +
2 \vert\langle x\rangle_{ab}\vert^2,
\end{displaymath}">|; 

$key = q/{displaymath}sigma^2_x=frac{2,sigma^2}{sqrt{pi}},int_{-infty}^{infty}y^2,{{rm{e}^{-y^2},dy.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img500.png"
 ALT="\begin{displaymath}
\sigma^2_x =\frac{2 \sigma^2}{\sqrt{\pi}} \int_{-\infty}^{\infty} y^2 {\rm e}^{-y^2} dy.
\end{displaymath}">|; 

$key = q/displaystylesqrt{frac{1}{3}},psi^{(1)}_{0,1slash2}-sqrt{frac{2}{3}},psi^{(1)}_{1,-1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1988.png"
 ALT="$\displaystyle \sqrt{\frac{1}{3}} \psi^{(1)}_{0,1/2} -
\sqrt{ \frac{2}{3}} \psi^{(1)}_{1,-1/2},$">|; 

$key = q/{displaymath}chi=c_+,chi_++c_-,chi_-=left(array{{c}c_+c_-array{right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1815.png"
 ALT="\begin{displaymath}
\chi = c_+ \chi_++c_- \chi_- = \left(\begin{array}{c}c_+ c_-\end{array}\right).
\end{displaymath}">|; 

$key = q/{displaymath}sigma_{{rm{total}simeqsum_{l=0}^{k,a}frac{2pi}{k^2},(2,l+1)simeq2pi,a^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img2986.png"
 ALT="\begin{displaymath}
\sigma_{\rm total} \simeq \sum_{l=0}^{k a} \frac{2\pi}{k^2}  (2 l+1) \simeq
2\pi  a^2.
\end{displaymath}">|; 

$key = q/{displaymath}phi(p)=frac{{{rm{e}^{-{{rm{i},p,x_0slashhbar}}{sqrt{2pi,hbar}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img583.png"
 ALT="\begin{displaymath}
\phi(p) = \frac{{\rm e}^{-{\rm i} p x_0/\hbar}}{\sqrt{2\pi \hbar}}.
\end{displaymath}">|; 

$key = q/1slash6+1slash6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img75.png"
 ALT="$1/6+1/6$">|; 

$key = q/F_pm<-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2824.png"
 ALT="$F_\pm &lt; -1$">|; 

$key = q/displaystylefrac{partial}{partialx_1}=frac{m_1}{m_1+m_2}frac{partial}{partialX}+frac{partial}{partialx'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1115.png"
 ALT="$\displaystyle \frac{\partial}{\partial x_1} = \frac{m_1}{m_1+m_2}\frac{\partial}{\partial X} + \frac{\partial}{\partial x'},$">|; 

$key = q/psi_b(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1168.png"
 ALT="$\psi_b(x)$">|; 

$key = q/{displaymath}langleOrangle=int_{-infty}^{infty}psi^ast(x,t),O(x),psi(x,t),dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img540.png"
 ALT="\begin{displaymath}
\langle O \rangle = \int_{-\infty}^{\infty}\psi^\ast(x,t) O(x) \psi(x,t) dx.
\end{displaymath}">|; 

$key = q/h,nu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img251.png"
 ALT="$h \nu$">|; 

$key = q/sin^2delta_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2985.png"
 ALT="$\sin^2\delta_l$">|; 

$key = q/a=x+{{rm{i},p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img764.png"
 ALT="$a = x + {\rm i} p$">|; 

$key = q/{displaymath}-frac{hbar^2}{2,m},frac{partial^2}{partialx^2}+V(x)={{rm{i},hbar,frac{partial}{partialt},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img559.png"
 ALT="\begin{displaymath}
-\frac{\hbar^2}{2 m} \frac{\partial^2}{\partial x^2} + V(x) = {\rm i} \hbar \frac{\partial}{\partial t},
\end{displaymath}">|; 

$key = q/displaystyleleft(frac{l-m}{2,l+1}right)^{1slash2}!psi^{(2)}_{l+1slash2,m+1slash2+m+1}{2,l+1}right)^{1slash2}!psi^{(2)}_{l-1slash2,m+1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="451" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1962.png"
 ALT="$\displaystyle \left(\frac{l-m}{2 l+1}\right)^{1/2}\!\psi^{(2)}_{l+1/2,m+1/2} - \left(\frac{l+m+1}{2 l+1}\right)^{1/2}\!
\psi^{(2)}_{l-1/2,m+1/2}.$">|; 

$key = q/{displaymath}langle1|H_1|1rangle=langle2|H_1|2rangle=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2410.png"
 ALT="\begin{displaymath}
\langle 1\vert H_1\vert 1\rangle = \langle 2\vert H_1\vert 2\rangle = 0.
\end{displaymath}">|; 

$key = q/pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3014.png"
 ALT="$\pi/2$">|; 

$key = q/{displaymath}langleS_xrangle=frac{hbar}{2},sinalpha,cos(Omega,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1854.png"
 ALT="\begin{displaymath}
\langle S_x\rangle = \frac{\hbar}{2} \sin\alpha \cos(\Omega t).
\end{displaymath}">|; 

$key = q/m_{s_2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2037.png"
 ALT="$m_{s_2}$">|; 

$key = q/langle(x_1-x_2)^2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1170.png"
 ALT="$\langle (x_1-x_2)^2\rangle$">|; 

$key = q/{displaymath}H_1=-frac{g,e,B_1}{2,m},left[cos(omega,t),S_x+sin(omega,t),S_yright].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="327" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2448.png"
 ALT="\begin{displaymath}
H_1 = -\frac{g e B_1}{2 m} \left[\cos(\omega t) S_x+ \sin(\omega t) S_y\right].
\end{displaymath}">|; 

$key = q/displaystylepsi_1+frac{e_{12}^ast}{E_1-E_2},psi_2+cdots,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2144.png"
 ALT="$\displaystyle \psi_1+ \frac{e_{12}^\ast}{E_1-E_2} \psi_2+ \cdots,$">|; 

$key = q/displaystyley_l(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1549.png"
 ALT="$\displaystyle y_l(z)$">|; 

$key = q/|psi(x_1,x_2,t)|^2,dx_1,dx_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1135.png"
 ALT="$\vert\psi(x_1,x_2,t)\vert^2 dx_1 dx_2$">|; 

$key = q/m_eslashm_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2365.png"
 ALT="$m_e/m_p$">|; 

$key = q/z=y-y_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img343.png"
 ALT="$z=y-y_0$">|; 

$key = q/chi^{(2)}_{s_1,s_2;s,m_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2084.png"
 ALT="$\chi^{(2)}_{s_1,s_2;s,m_s}$">|; 

$key = q/a';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img660.png"
 ALT="$a'$">|; 

$key = q/{displaymath}[S^2,S_x]=[S^2,S_y]=[S^2,S_z]=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1707.png"
 ALT="\begin{displaymath}[S^2, S_x]= [S^2, S_y] = [S^2,S_z] = 0.
\end{displaymath}">|; 

$key = q/displaystylelangle1|H_1|2rangle=langle2|H_1|1rangle^ast.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2134.png"
 ALT="$\displaystyle \langle 1\vert H_1\vert 2\rangle = \langle 2\vert H_1\vert 1\rangle^\ast.$">|; 

$key = q/{displaymath}int_{-infty}^infty|c(p')|^{,2},dp'=int_{-infty}^{infty}|phi(p')|^{,2},dp'=int_{-infty}^infty|psi(x')|^{,2},dx'=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="459" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img733.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \vert c(p')\vert^{ 2} dp'= \int_{-\i...
...,dp' = \int_{-\infty}^\infty \vert\psi(x')\vert^{ 2} dx' =1.
\end{displaymath}">|; 

$key = q/{displaymath}langlem|nrangle=delta_{nm},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2149.png"
 ALT="\begin{displaymath}
\langle m\vert n\rangle = \delta_{nm},
\end{displaymath}">|; 

$key = q/displaystyle{{rm{i},hbar,S_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1701.png"
 ALT="$\displaystyle {\rm i} \hbar S_z,$">|; 

$key = q/{displaymath}langleUrangle(Z)=-4,Z,(Z-2),E_0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2769.png"
 ALT="\begin{displaymath}
\langle U\rangle(Z) = -4 Z (Z-2) E_0,
\end{displaymath}">|; 

$key = q/f({{bf{k},{{bf{k}');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2872.png"
 ALT="$f({\bf k},{\bf k}')$">|; 

$key = q/{displaymath}P_{irightarrowf}^{abs}(t)=int_{-infty}^inftyfrac{t^2,e^2}{2,epsilon{{rm{sinc}^2[(omega-omega_{fi}),tslash2],domega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="633" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2558.png"
 ALT="\begin{displaymath}
P_{i\rightarrow f}^{abs}(t) = \int_{-\infty}^\infty\frac{t^2...
...rho(\omega) {\rm sinc}^2[(\omega-\omega_{fi}) t/2] d\omega.
\end{displaymath}">|; 

$key = q/sigma_A^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img609.png"
 ALT="$\sigma_A^{ 2}$">|; 

$key = q/displaystylefrac{{{rm{e}^{+{{rm{i},E,t_0slashhbar}}{(2pi,sigma_E^{,2})^{1slash4}},{{rm{e}^{-(E-E_0)^{,2}slash4,sigma_E^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img646.png"
 ALT="$\displaystyle \frac{{\rm e}^{+{\rm i} E t_0/\hbar}}{(2\pi \sigma_E^{ 2})^{1/4}} {\rm e}^{-(E-E_0)^{ 2}/4 \sigma_E^{ 2}},$">|; 

$key = q/displaystyleA,psi_a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img661.png"
 ALT="$\displaystyle A \psi_a$">|; 

$key = q/C^{12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1277.png"
 ALT="$C^{12}$">|; 

$key = q/a_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1589.png"
 ALT="$a_0$">|; 

$key = q/l=l'=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2210.png"
 ALT="$l=l'=0$">|; 

$key = q/{displaymath}frac{dsigma}{dOmega}simeqleft(frac{Z,Z',e^2}{16pi,epsilon_0,E}right)^2frac{1}{sin^4(thetaslash2)},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img2893.png"
 ALT="\begin{displaymath}
\frac{d\sigma}{d\Omega} \simeq\left(\frac{Z  Z'  e^2}{16\pi \epsilon_0 E}\right)^2
\frac{1}{\sin^4(\theta/2)},
\end{displaymath}">|; 

$key = q/P(u_1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img98.png"
 ALT="$P(u_1)$">|; 

$key = q/omega_n=E_nslashhbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2465.png"
 ALT="$\omega_n=E_n/\hbar$">|; 

$key = q/{displaymath}c_n(0)=delta_{ni}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2471.png"
 ALT="\begin{displaymath}
c_n(0) = \delta_{ni}.
\end{displaymath}">|; 

$key = q/{displaymath}k^2=frac{2,m,E}{hbar^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img796.png"
 ALT="\begin{displaymath}
k^2 = \frac{2 m E}{\hbar^2}.
\end{displaymath}">|; 

$key = q/{displaymath}sigma_{{rm{total}=frac{4pi}{k^2}sum_l(2,l+1),sin^2delta_l,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2941.png"
 ALT="\begin{displaymath}
\sigma_{\rm total} = \frac{4\pi}{k^2} \sum_l (2 l+1) \sin^2\delta_l,
\end{displaymath}">|; 

$key = q/+hbarslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1818.png"
 ALT="$+\hbar/2$">|; 

$key = q/{displaymath}v=frac{p}{m},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img298.png"
 ALT="\begin{displaymath}
v = \frac{p}{m},
\end{displaymath}">|; 

$key = q/displaystylee_{11};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2129.png"
 ALT="$\displaystyle e_{11}$">|; 

$key = q/{displaymath}frac{d^2psi}{dx^2}=-q^2,psi,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img831.png"
 ALT="\begin{displaymath}
\frac{d^2 \psi}{d x^2} = - q^2 \psi,
\end{displaymath}">|; 

$key = q/m'=m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2267.png"
 ALT="$m'=m$">|; 

$key = q/m<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1468.png"
 ALT="$m&lt;0$">|; 

$key = q/a_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2916.png"
 ALT="$a_l$">|; 

$key = q/V_0=(4slash3),E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="$V_0 = (4/3) E$">|; 

$key = q/q,agg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img872.png"
 ALT="$q a\gg 1$">|; 

$key = q/p_j,p_i=p_i,p_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1521.png"
 ALT="$p_j p_i=p_i p_j$">|; 

$key = q/{displaymath}{calR}_l(r)=frac{u_l(r)}{r},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2954.png"
 ALT="\begin{displaymath}
{\cal R}_l (r) = \frac{u_l(r)}{r},
\end{displaymath}">|; 

$key = q/{displaymath}J=2int_0^inftyint_0^piexpleft[-x-(x^2+X^2-2,x,X,costheta)^{1slash2}right],x^2,dx,sintheta,dtheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="534" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img2789.png"
 ALT="\begin{displaymath}
J = 2\int_0^\infty \int_0^\pi \exp\left[-x-(x^2+X^2-2 x X \cos\theta)^{1/2}\right] x^2 dx \sin\theta d\theta,
\end{displaymath}">|; 

$key = q/R_{n,l}(r)simZ(rslasha),exp(-rslasha)slash(rslasha)rightarrowexp(rslasha)slash(rslasha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="439" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1614.png"
 ALT="$R_{n,l}(r)\sim Z(r/a) \exp(-r/a)/(r/a)\rightarrow \exp(r/a)/(r/a)$">|; 

$key = q/chi^nomath_inline{dagnomath_inline{,chi';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1731.png"
 ALT="$\chi^\dag \chi'$">|; 

$key = q/{displaymath}P_{irightarrowf}^{stm}(t)=frac{t^2}{hbar^2},|V_{if}^nomath_inline{drm{sinc}^2left[(omega-omega_{if}),tslash2right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2515.png"
 ALT="\begin{displaymath}
P_{i\rightarrow f}^{stm}(t) = \frac{t^2}{\hbar^2} 
\vert V_...
...ert^{ 2} {\rm sinc}^2\left[(\omega-\omega_{if}) t/2\right],
\end{displaymath}">|; 

$key = q/displaystylepsi^{(1)}_{m+1,-1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1961.png"
 ALT="$\displaystyle \psi^{(1)}_{m+1,-1/2}$">|; 

$key = q/langleL_xrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1486.png"
 ALT="$\langle L_x\rangle$">|; 

$key = q/|{{bf{p}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2892.png"
 ALT="$\vert{\bf p}\vert$">|; 

$key = q/displaystyles,(s+1),hbar^2,psi^{(1)}_{l,s;m,m_s},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1902.png"
 ALT="$\displaystyle s (s+1) \hbar^2 \psi^{(1)}_{l,s;m,m_s},$">|; 

$key = q/{displaymath}|psi(x,0)|^{,2}proptoexpleft[-frac{(x-x_0)^{,2}}{2,({mitDelta}x)^{,2}}right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img329.png"
 ALT="\begin{displaymath}
\vert\psi(x,0)\vert^{ 2} \propto \exp\left[- \frac{(x-x_0)^{ 2}}{2 ({\mit\Delta}x)^{ 2}}\right].
\end{displaymath}">|; 

$key = q/displaystyleint_0^{2pi}Phi^{,ast}_{m'}(phi),Phi_{m}(phi),dphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1394.png"
 ALT="$\displaystyle \int_0^{2\pi}\Phi^{ \ast}_{m'}(\phi) \Phi_{m}(\phi) d\phi$">|; 

$key = q/a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$a$">|; 

$key = q/delta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2978.png"
 ALT="$\delta_0$">|; 

$key = q/displaystylefrac{omega_{fi}^{,3},d_{if}^{,2}}{3pi,epsilon_0,hbar,c^3},frac{1}{exp(hbar,omega_{fi}slashk_B,T)-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2612.png"
 ALT="$\displaystyle \frac{\omega_{fi}^{ 3} d_{if}^{ 2}}{3\pi \epsilon_0 \hbar c^3} \frac{1}
{\exp(\hbar \omega_{fi}/k_B T)-1},$">|; 

$key = q/{mitOmega}({mitSigma});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="${\mit\Omega}({\mit\Sigma})$">|; 

$key = q/{displaymath}|T|^{,2}simeqexpleft(-2,sqrt{2},betaleft[frac{pi}{2}sqrt{frac{E_c}{E}}-2right]right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img927.png"
 ALT="\begin{displaymath}
\vert T\vert^{ 2} \simeq \exp\left(-2 \sqrt{2} \beta\left[\frac{\pi}{2}\sqrt{\frac{E_c}{E}}-2\right]\right).
\end{displaymath}">|; 

$key = q/u=costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1458.png"
 ALT="$u=\cos\theta$">|; 

$key = q/displaystylea_-,psi_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1039.png"
 ALT="$\displaystyle a_- \psi_n$">|; 

$key = q/displaystyle{{bf{A};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2538.png"
 ALT="$\displaystyle {\bf A}$">|; 

$key = q/{displaymath}omega_{nm}=frac{E_n-E_m}{hbar}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2408.png"
 ALT="\begin{displaymath}
\omega_{nm} = \frac{E_n-E_m}{\hbar}.
\end{displaymath}">|; 

$key = q/displaystylem,frac{dlanglexrangle}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img514.png"
 ALT="$\displaystyle m \frac{d\langle x\rangle}{dt}$">|; 

$key = q/{displaymath}P_l(costheta)=sqrt{frac{4pi}{2,l+1}},Y_{l,0}(theta,varphi).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2910.png"
 ALT="\begin{displaymath}
P_l(\cos\theta) = \sqrt{\frac{4\pi}{2 l+1}}  Y_{l,0}(\theta,\varphi).
\end{displaymath}">|; 

$key = q/dpsislashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img837.png"
 ALT="$d\psi/d x$">|; 

$key = q/displaystyle-{{rm{i},hbar,frac{partial}{partialx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1194.png"
 ALT="$\displaystyle -{\rm i} \hbar \frac{\partial}{\partial x},$">|; 

$key = q/{displaymath}x,x^2=x^2,x.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img531.png"
 ALT="\begin{displaymath}
x x^2 = x^2 x.
\end{displaymath}">|; 

$key = q/displaystyleL_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1304.png"
 ALT="$\displaystyle L_x$">|; 

$key = q/{mitDeltak}=1slash(2,{mitDelta}x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img379.png"
 ALT="${\mit\Delta k}=1/(2 {\mit\Delta}x)$">|; 

$key = q/{displaymath}langlen,l,m|[L_z,x_+]-hbar,x_+|n',l',m'rangle=hbar,(m-m'-1),langlen,l,m|x_+|n',l',m'rangle=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="582" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2642.png"
 ALT="\begin{displaymath}
\langle n,l,m\vert[L_z,x_+]-\hbar x_+\vert n',l',m'\rangle...
...,(m-m'-1) 
\langle n,l,m\vert x_+\vert n',l',m'\rangle = 0,
\end{displaymath}">|; 

$key = q/{displaymath}leftlangle|mbox{{boldmath{{epsilon{}!cdot!{{bf{d}_{if}|^{,2}rightraif},frac{intintsin^2theta,cos^2phi,dOmega}{4pi},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2593.png"
 ALT="\begin{displaymath}
\left\langle \vert\mbox{\boldmath$\epsilon$}\!\cdot\!{\bf d}...
...{if} 
\frac{\int\int\sin^2\theta \cos^2\phi d\Omega}{4\pi},
\end{displaymath}">|; 

$key = q/displaystylefrac{3}{20}left(frac{9pi}{8}right)^{2slash3}frac{hbar^2}{m_e}left(frac{M}{m_p}right)^{5slash3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1283.png"
 ALT="$\displaystyle \frac{3}{20}\left(\frac{9\pi}{8}\right)^{2/3}\frac{\hbar^2}{m_e}
\left(\frac{M}{m_p}\right)^{5/3},$">|; 

$key = q/theta=theta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2736.png"
 ALT="$\theta=\theta_1$">|; 

$key = q/{displaymath}|bar{psi}(k)|^{,2}proptoexpleft[-frac{(k-k_0)^{,2}}{2,({mitDelta}k)^{,2}}right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img349.png"
 ALT="\begin{displaymath}
\vert\bar{\psi}(k)\vert^{ 2} \propto \exp\left[- \frac{(k-k_0)^{ 2}}{2 ({\mit\Delta}k)^{ 2}}\right].
\end{displaymath}">|; 

$key = q/phi_2(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img769.png"
 ALT="$\phi_2(x)$">|; 

$key = q/1,-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2070.png"
 ALT="$1, -1$">|; 

$key = q/{displaymath}|T|^{,2}=expleft(-2,sqrt{2},betaint_{1}^{E_cslashE}left[frac{1}{y}-frac{E}{E_c}right]^{1slash2}dyright),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="372" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img921.png"
 ALT="\begin{displaymath}
\vert T\vert^{ 2} = \exp\left(-2 \sqrt{2} \beta \int_{1}^{E_c/E}\left[\frac{1}{y}-\frac{E}{E_c}\right]^{1/2} dy\right),
\end{displaymath}">|; 

$key = q/{displaymath}[J^2,S_z]neq0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1895.png"
 ALT="\begin{displaymath}[J^2,S_z]\neq 0.
\end{displaymath}">|; 

$key = q/{displaymath}frac{psi({{bf{r})-psi_0({{bf{r})}{sqrt{n}}=frac{exp(,{{rm{i},k,r)}{r},f(theta).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2933.png"
 ALT="\begin{displaymath}
\frac{\psi({\bf r} )- \psi_0({\bf r}) }{ \sqrt{n}} =
\frac{\exp( {\rm i} k r)}{r} 
f(\theta).
\end{displaymath}">|; 

$key = q/|T|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img824.png"
 ALT="$\vert T\vert^{ 2}$">|; 

$key = q/{displaymath}alpha<frac{16}{3},4piepsilon_0,a_0^{,3}simeq5.3,,4piepsilon_0,a_0^{,3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img2233.png"
 ALT="\begin{displaymath}
\alpha &lt; \frac{16}{3} 4\pi\epsilon_0 a_0^{ 3}\simeq 5.3  4\pi\epsilon_0 a_0^{ 3}.
\end{displaymath}">|; 

$key = q/{displaymath}Hequiv{{rm{i},hbar,frac{partial}{partialt}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img560.png"
 ALT="\begin{displaymath}
H \equiv {\rm i} \hbar \frac{\partial}{\partial t}.
\end{displaymath}">|; 

$key = q/{displaymath}chi^nomath_inline{dagnomath_inline{,chi=(c_+^ast,c_-^ast)left(arrayht)=c_+^ast,c_++c_-^ast,c_-=|c_+|^2+|c_-|^2geq0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="478" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1791.png"
 ALT="\begin{displaymath}
\chi^\dag \chi = (c_+^\ast, c_-^\ast)\left(\begin{array}{c}...
...+ + c_-^\ast c_- = \vert c_+\vert^2 + \vert c_-\vert^2\geq 0.
\end{displaymath}">|; 

$key = q/displaystyleS^2-S_z^{,2}+hbar,S_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1712.png"
 ALT="$\displaystyle S^2-S_z^{ 2}+\hbar S_z,$">|; 

$key = q/chi_{s,m_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1932.png"
 ALT="$\chi_{s,m_s}$">|; 

$key = q/displaystylem,hbar,Y_{l,m}(theta,phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1363.png"
 ALT="$\displaystyle m \hbar Y_{l,m}(\theta,\phi),$">|; 

$key = q/{displaymath}frac{1}{lambda}=Rleft(frac{1}{n_f^{,2}}-frac{1}{n_i^{,2}}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1675.png"
 ALT="\begin{displaymath}
\frac{1}{\lambda} = R\left(\frac{1}{n_f^{ 2}}-\frac{1}{n_i^{ 2}}\right),
\end{displaymath}">|; 

$key = q/lambda_{nl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2257.png"
 ALT="$\lambda_{nl}$">|; 

$key = q/E_{total}<E_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2823.png"
 ALT="$E_{total}&lt; E_0$">|; 

$key = q/{displaymath}E=frac{p^2}{2,m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img299.png"
 ALT="\begin{displaymath}
E = \frac{p^2}{2 m}.
\end{displaymath}">|; 

$key = q/A^2=beta^3slashpi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1689.png"
 ALT="$A^2=\beta^3/\pi$">|; 

$key = q/S_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1748.png"
 ALT="$S_+$">|; 

$key = q/displaystyle({{bf{L}+{{bf{S})times({{bf{L}+{{bf{S});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1873.png"
 ALT="$\displaystyle ({\bf L}+{\bf S})\times({\bf L}+{\bf S})$">|; 

$key = q/{displaymath}qequiv|{{bf{k}-{{bf{k}'|=2,k,sin(thetaslash2),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2880.png"
 ALT="\begin{displaymath}
q \equiv \vert{\bf k} - {\bf k}'\vert = 2  k  \sin (\theta/2),
\end{displaymath}">|; 

$key = q/psi^{(2)}_{3slash2,pm3slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1980.png"
 ALT="$\psi^{(2)}_{3/2,\pm 3/2}$">|; 

$key = q/chi=S_+,chi_{s,m_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1751.png"
 ALT="$\chi=S_+ \chi_{s,m_s}$">|; 

$key = q/displaystyle+hbar,S_-.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1718.png"
 ALT="$\displaystyle +\hbar S_-.$">|; 

$key = q/{{bf{r}=(x,,y,,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img179.png"
 ALT="${\bf r}=(x, y, z)$">|; 

$key = q/-1,0,+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1433.png"
 ALT="$-1, 0, +1$">|; 

$key = q/n=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1569.png"
 ALT="$n=3$">|; 

$key = q/{displaymath}delta_l(E_0)=frac{pi}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img3021.png"
 ALT="\begin{displaymath}
\delta_l(E_0) = \frac{\pi}{2}.
\end{displaymath}">|; 

$key = q/displaystylel,(l+1),hbar^2,psi^{(2)}_{l,s;j,m_j},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1913.png"
 ALT="$\displaystyle l (l+1) \hbar^2 \psi^{(2)}_{l,s;j,m_j},$">|; 

$key = q/S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="$S$">|; 

$key = q/{displaymath}|psi(x)|^{,2}=|psi_0|^{,2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img883.png"
 ALT="\begin{displaymath}
\vert\psi(x)\vert^{ 2} = \vert\psi_0\vert^{ 2},
\end{displaymath}">|; 

$key = q/{displaymath}H=frac{left({{bf{p}-e,{{bf{A}right)^2}{2,m_e}+e,phi+V_0(r),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2531.png"
 ALT="\begin{displaymath}
H = \frac{\left({\bf p} - e  {\bf A}\right)^2 }{2 m_e}+ e  \phi + V_0(r),
\end{displaymath}">|; 

$key = q/omega_{fi}<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2513.png"
 ALT="$\omega_{fi}&lt;0$">|; 

$key = q/{{bf{S}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2028.png"
 ALT="${\bf S}_2$">|; 

$key = q/langlen,l,m|z|n',l',mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2209.png"
 ALT="$\langle n,l,m\vert z\vert n',l',m\rangle$">|; 

$key = q/{displaymath}sigma_u=left[,leftlangle({mitDelta}u)^2rightrangle,right]^{1slash2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img115.png"
 ALT="\begin{displaymath}
\sigma_u = \left[ \left\langle({\mit\Delta} u)^2\right\rangle \right]^{1/2},
\end{displaymath}">|; 

$key = q/{displaymath}E_n=(n+1slash2),hbar,omega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1028.png"
 ALT="\begin{displaymath}
E_n = (n+1/2) \hbar \omega.
\end{displaymath}">|; 

$key = q/{displaymath}langlef(u)+g(u)rangle=langlef(u)rangle+langleg(u)rangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img107.png"
 ALT="\begin{displaymath}
\langle f(u)+g(u)\rangle= \langle f(u)\rangle+\langle g(u)\rangle.
\end{displaymath}">|; 

$key = q/displaystylepsi(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img643.png"
 ALT="$\displaystyle \psi(t)$">|; 

$key = q/{displaymath}dN=-N,nu,|T|^2,dt.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img935.png"
 ALT="\begin{displaymath}
dN = - N \nu \vert T\vert^2 dt.
\end{displaymath}">|; 

$key = q/{displaymath}k',asimeqsqrt{frac{2,m,|V_0|,a^2}{hbar^2}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img3009.png"
 ALT="\begin{displaymath}
k'  a \simeq \sqrt{\frac{2  m  \vert V_0\vert  a^2}{\hbar^2}}.
\end{displaymath}">|; 

$key = q/{displaymath}langlepsi|H|psirangle=E_0+sum_{n>0}|c_n|^{,2},(E_n-E_0).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2700.png"
 ALT="\begin{displaymath}
\langle \psi\vert H\vert\psi\rangle = E_0 + \sum_{n&gt;0} \vert c_n\vert^{ 2} (E_n-E_0).
\end{displaymath}">|; 

$key = q/A=I^{-1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2783.png"
 ALT="$A = I^{-1/2}$">|; 

$key = q/{displaymath}[x_i,p_j]={{rm{i},hbar,delta_{ij}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1516.png"
 ALT="\begin{displaymath}[x_i,p_j]= {\rm i} \hbar \delta_{ij}.
\end{displaymath}">|; 

$key = q/{displaymath}chi^nomath_inline{dagnomath_inline{,chigeq0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1729.png"
 ALT="\begin{displaymath}
\chi^\dag \chi\geq 0
\end{displaymath}">|; 

$key = q/H_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2096.png"
 ALT="$H_0$">|; 

$key = q/{displaymath}Upropto|psi|^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img245.png"
 ALT="\begin{displaymath}
U \propto \vert\psi\vert^{ 2}.
\end{displaymath}">|; 

$key = q/s_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2034.png"
 ALT="$s_1$">|; 

$key = q/{displaymath}delta_{ij}=left{array{{rcl}1&mbox{hspace{1cm}}&mbox{if{i{and{j{thes&&mbox{if{i{and{j{different}[0.5ex]array{right..{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img1513.png"
 ALT="\begin{displaymath}
\delta_{ij} = \left\{
\begin{array}{rcl}
1 &amp;\mbox{\hspace{1c...
...box{if $i$ and $j$ different} [0.5ex]
\end{array}\right. .
\end{displaymath}">|; 

$key = q/langlen,l,m|x_+|n',l',m'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2644.png"
 ALT="$\langle n,l,m\vert x_+\vert n',l',m'\rangle$">|; 

$key = q/sigma_x,sigma_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img751.png"
 ALT="$\sigma_x \sigma_p$">|; 

$key = q/nu,|T|^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img931.png"
 ALT="$\nu \vert T\vert^2$">|; 

$key = q/x_i+dx_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1099.png"
 ALT="$x_i+dx_i$">|; 

$key = q/{displaymath}{{rm{i},hbar,frac{partialpsi}{partialt}=H,psi,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1203.png"
 ALT="\begin{displaymath}
{\rm i} \hbar \frac{\partial\psi}{\partial t} = H \psi,
\end{displaymath}">|; 

$key = q/displaystyleomega(k_0),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img368.png"
 ALT="$\displaystyle \omega(k_0),$">|; 

$key = q/{{bf{S}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2027.png"
 ALT="${\bf S}_1$">|; 

$key = q/langlex-langlexranglerangle=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img521.png"
 ALT="$\langle x-\langle x\rangle\rangle = 0$">|; 

$key = q/u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img95.png"
 ALT="$u$">|; 

$key = q/{displaymath}V(r)=frac{Z_1,Z_2,e^2}{4pi,epsilon_0,r}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img914.png"
 ALT="\begin{displaymath}
V(r) = \frac{Z_1 Z_2 e^2}{4\pi \epsilon_0 r}
\end{displaymath}">|; 

$key = q/{displaymath}int_{-infty}^inftypsi_n,psi_m,dx=delta_{nm}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img1030.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \psi_n \psi_m dx = \delta_{nm}.
\end{displaymath}">|; 

$key = q/O^ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img543.png"
 ALT="$O^\ast$">|; 

$key = q/displaystyleleft(array{{cc}E_1-E+e_{11}<col_mark>e_{12}<row_mark>[0.5ex]e_{12}^alearray{right)=left(array{{c}0<row_mark>[0.5ex]0array{right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="446" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2128.png"
 ALT="$\displaystyle \left(\begin{array}{cc}E_1-E+e_{11}&amp; e_{12}  [0.5ex]
e_{12}^\as...
...gle\end{array}\right)=\left(
\begin{array}{c} 0  [0.5ex]
0\end{array}\right),$">|; 

$key = q/{displaymath}c_i=int_{-infty}^inftyint_{-infty}^inftyint_{-infty}^inftypsi_i^ast,psi_a,dx,dy,dz.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img2105.png"
 ALT="\begin{displaymath}
c_i = \int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty\psi_i^\ast \psi_a dx dy dz.
\end{displaymath}">|; 

$key = q/chi_{s,-s+2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1771.png"
 ALT="$\chi_{s,-s+2}$">|; 

$key = q/{displaymath}H_0=-frac{g,e,B_0}{2,m},S_z,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2447.png"
 ALT="\begin{displaymath}
H_0 = - \frac{g e B_0}{2 m} S_z,
\end{displaymath}">|; 

$key = q/H(x',X)=H_{x'}(x')+H_X(X)];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1121.png"
 ALT="$H(x',X) = H_{x'}(x') + H_X(X)]$">|; 

$key = q/displaystyle-k_x^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1220.png"
 ALT="$\displaystyle - k_x^{ 2},$">|; 

$key = q/{{bf{E};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2306.png"
 ALT="${\bf E}$">|; 

$key = q/{displaymath}langle{{bf{S}_pcdot{{bf{S}_erangle=-frac{3}{4},hbar^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img2382.png"
 ALT="\begin{displaymath}
\langle {\bf S}_p\cdot{\bf S}_e\rangle = - \frac{3}{4} \hbar^2
\end{displaymath}">|; 

$key = q/{displaymath}h(y)simCsum_{j}frac{y^{2,j}}{j!}simC,{{rm{e}^{,y^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1016.png"
 ALT="\begin{displaymath}
h(y) \sim C \sum_{j}\frac{y^{2 j}}{j!}\sim C {\rm e}^{ y^2}.
\end{displaymath}">|; 

$key = q/k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img145.png"
 ALT="$k$">|; 

$key = q/P';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1126.png"
 ALT="$P'$">|; 

$key = q/{displaymath}H=H_0+H_1(t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2543.png"
 ALT="\begin{displaymath}
H = H_0 + H_1(t),
\end{displaymath}">|; 

$key = q/sum_n|c_n|^2=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2470.png"
 ALT="$\sum_n\vert c_n\vert^2=1$">|; 

$key = q/P(XmidY);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.png"
 ALT="$P(X\mid Y)$">|; 

$key = q/psi_X=psi_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1129.png"
 ALT="$\psi_X=\psi_0$">|; 

$key = q/V(x)-E=W;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img900.png"
 ALT="$V(x)-E=W$">|; 

$key = q/displaystylep_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1192.png"
 ALT="$\displaystyle p_x$">|; 

$key = q/+alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img424.png"
 ALT="$+\alpha$">|; 

$key = q/x>aslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img967.png"
 ALT="$x&gt;a/2$">|; 

$key = q/{displaymath}H_1=frac{e,B}{2,m_e},(L_z+2,S_z).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2341.png"
 ALT="\begin{displaymath}
H_1 = \frac{e B}{2 m_e} (L_z+ 2 S_z).
\end{displaymath}">|; 

$key = q/x=x_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img892.png"
 ALT="$x=x_2$">|; 

$key = q/displaystyle-nablaphi-frac{partial{{bf{A}}{partialt},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2528.png"
 ALT="$\displaystyle - \nabla\phi - \frac{\partial {\bf A}}{\partial t},$">|; 

$key = q/{displaymath}psi({{bf{r}_1,{{bf{r}_2)=frac{Z^3}{pi,a_0^{,3}},expleft(-frac{Z,[r_1+r_2]}{a_0}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2754.png"
 ALT="\begin{displaymath}
\psi({\bf r}_1, {\bf r}_2) =
\frac{Z^3}{\pi a_0^{ 3}} \exp\left(- \frac{Z [r_1+r_2]}{a_0}\right),
\end{displaymath}">|; 

$key = q/{displaymath}psi_p(x,p')=frac{{{rm{e}^{+{{rm{i},p',xslashhbar}}{(2pi,hbar)^{1slash2}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img727.png"
 ALT="\begin{displaymath}
\psi_p(x,p') =\frac{ {\rm e}^{+{\rm i} p' x/\hbar}}{(2\pi \hbar)^{1/2}}.
\end{displaymath}">|; 

$key = q/epsilon_{112}=epsilon_{131}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1507.png"
 ALT="$\epsilon_{112}=\epsilon_{131}=0$">|; 

$key = q/displaystyle{{bf{J}times{{bf{J};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1872.png"
 ALT="$\displaystyle {\bf J}\times{\bf J}$">|; 

$key = q/{displaymath}P(XmidY)=P(X)+P(Y).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img73.png"
 ALT="\begin{displaymath}
P(X\mid Y) = P(X) + P(Y).
\end{displaymath}">|; 

$key = q/|z|=sqrt{x^2+y^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img203.png"
 ALT="$\vert z\vert=\sqrt{x^2+y^2}$">|; 

$key = q/displaystylechi_+,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2454.png"
 ALT="$\displaystyle \chi_+,$">|; 

$key = q/H={{rm{i},hbar,partialslashpartialt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img565.png"
 ALT="$H={\rm i} \hbar \partial/\partial t$">|; 

$key = q/displaystylefrac{1}{sqrt{4pi}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1474.png"
 ALT="$\displaystyle \frac{1}{\sqrt{4\pi}},$">|; 

$key = q/displaystylefrac{omega_{if}^{,3},d_{if}^{,2}}{3pi,epsilon_0,hbar,c^3},frac{1}{exp(hbar,omega_{if}slashk_B,T)-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2613.png"
 ALT="$\displaystyle \frac{\omega_{if}^{ 3} d_{if}^{ 2}}{3\pi \epsilon_0 \hbar c^3} \frac{1}
{\exp(\hbar \omega_{if}/k_B T)-1},$">|; 

$key = q/{displaymath}{mitDelta}x,{mitDelta}kstackrel{_{{normalsize{>}}{_{{normalsize{sim}}frac{1}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img400.png"
 ALT="\begin{displaymath}
{\mit\Delta}x {\mit\Delta} k\stackrel {_{\normalsize &gt;}}{_{\normalsize\sim}}\frac{1}{2}.
\end{displaymath}">|; 

$key = q/displaystyle+e^2,|{{bf{E}|^2sum_{n',l',m'neqn,l,m}frac{|langlen,l,m|z|n',l',m'rangle|^2}{E_{nlm}-E_{n'l'm'}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="340" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2179.png"
 ALT="$\displaystyle + e^2 \vert{\bf E}\vert^2\sum_{n',l',m'\neq n,l,m}\frac{\vert\langle n,l,m\vert z\vert n',l',m'\rangle\vert^2}{E_{nlm}-E_{n'l'm'}}.$">|; 

$key = q/c_{l+1},z^{l+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1619.png"
 ALT="$c_{l+1} z^{l+1}$">|; 

$key = q/{displaymath}E_{n00}=-frac{e^2}{8pi,epsilon_0,a_0,n^2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2223.png"
 ALT="\begin{displaymath}
E_{n00} = -\frac{e^2}{8\pi \epsilon_0 a_0 n^2},
\end{displaymath}">|; 

$key = q/{displaymath}langleprangle=m,frac{dlanglexrangle}{dt}=-{{rm{i},hbarint_{-infty}^{infty}psi^ast,frac{partialpsi}{partialx},dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img508.png"
 ALT="\begin{displaymath}
\langle p\rangle = m \frac{d\langle x\rangle}{dt}= - {\rm i...
...nfty}^{\infty}
\psi^\ast \frac{\partial\psi}{\partial x} dx.
\end{displaymath}">|; 

$key = q/{displaymath}q^2=frac{2,m,(E-V_0)}{hbar^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img832.png"
 ALT="\begin{displaymath}
q^2 = \frac{2 m (E-V_0)}{\hbar^2}.
\end{displaymath}">|; 

$key = q/{displaymath}langleV_{ee}rangle=-frac{16,E_0}{pi}int_0^infty{{rm{e}^{-2,r_2},I(r_2),r_2^{,2},dr_2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2751.png"
 ALT="\begin{displaymath}
\langle V_{ee}\rangle = -\frac{16 E_0}{\pi}\int_0^\infty
{\rm e}^{-2 r_2} I(r_2) r_2^{ 2} dr_2,
\end{displaymath}">|; 

$key = q/langleOrangle=langleOrangle^ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img541.png"
 ALT="$\langle O\rangle = \langle O \rangle^\ast$">|; 

$key = q/displaystyle-hbar,S_+,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1716.png"
 ALT="$\displaystyle - \hbar S_+,$">|; 

$key = q/{displaymath}psi(x,t)=psi_0,{{rm{e}^{,{{rm{i},(k,x-omega,t)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img225.png"
 ALT="\begin{displaymath}
\psi(x,t) = \psi_0 {\rm e}^{ {\rm i} (k x-\omega t)}.
\end{displaymath}">|; 

$key = q/displaystylesqrt{frac{2}{a}}sin(k_z,z),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1233.png"
 ALT="$\displaystyle \sqrt{\frac{2}{a}}\sin (k_z z),$">|; 

$key = q/{displaymath}bar{psi}(k)proptoint_{-infty}^{infty}psi(x,0),{{rm{e}^{-{{rm{i},k,x},dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img336.png"
 ALT="\begin{displaymath}
\bar{\psi}(k)\propto \int_{-\infty}^{\infty} \psi(x,0) {\rm e}^{-{\rm i} k x} dx.
\end{displaymath}">|; 

$key = q/displaystyleointY_{l,m}^{,ast},(L_+)^nomath_inline{dagnomath_inline{,(L_+,Y_{l,m}),dOmega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1408.png"
 ALT="$\displaystyle \oint Y_{l,m}^{ \ast} (L_+)^\dag (L_+ Y_{l,m}) d\Omega$">|; 

$key = q/displaystyleS_1^{,2},chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2039.png"
 ALT="$\displaystyle S_1^{ 2}  \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}$">|; 

$key = q/displaystylefrac{exp(,{{rm{i},delta_0),sin(k,r+delta_0)}{k,r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2994.png"
 ALT="$\displaystyle \frac{ \exp( {\rm i}  \delta_0)  \sin(k r+\delta_0)}{k r},$">|; 

$key = q/p_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1189.png"
 ALT="$p_x$">|; 

$key = q/displaystyle(L_z+S_z),psi^{(1)}_{l,s;m,m_s}=(m+m_s),hbar,psi^{(1)}_{l,s;m,m_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="334" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1908.png"
 ALT="$\displaystyle (L_z+S_z) \psi^{(1)}_{l,s;m,m_s}=
(m+m_s) \hbar \psi^{(1)}_{l,s;m,m_s}$">|; 

$key = q/{displaymath}J^2=L^2+S^2+2,{{bf{L}cdot{{bf{S},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2317.png"
 ALT="\begin{displaymath}
J^2 = L^2+S^2+ 2 {\bf L}\cdot{\bf S},
\end{displaymath}">|; 

$key = q/{displaymath}P_{irightarrowf}^{stm}(t)=1-expleft(-w_{irightarrowf}^{stm},tright),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img2574.png"
 ALT="\begin{displaymath}
P_{i\rightarrow f}^{stm}(t) = 1 - \exp\left(-w_{i\rightarrow f}^{stm} t\right),
\end{displaymath}">|; 

$key = q/S_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="$S_z$">|; 

$key = q/psi_a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img650.png"
 ALT="$\psi_a$">|; 

$key = q/V(x,t)equivV(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1103.png"
 ALT="$V(x,t)\equiv V(x)$">|; 

$key = q/{displaymath}int_{-infty}^{infty}|psi(x,t)|^{,2},dx=int_{-infty}^{infty}|phi(p,t)|^{,2},dp.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img601.png"
 ALT="\begin{displaymath}
\int_{-\infty}^{\infty} \vert\psi(x,t)\vert^{ 2} dx =\int_{-\infty}^{\infty}\vert\phi(p,t)\vert^{ 2} dp.
\end{displaymath}">|; 

$key = q/{displaymath}E=sum_{i=1,N}E_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1108.png"
 ALT="\begin{displaymath}
E = \sum_{i=1,N} E_i.
\end{displaymath}">|; 

$key = q/B_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2444.png"
 ALT="$B_1$">|; 

$key = q/p^2slash(2,m)+V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img440.png"
 ALT="$p^2/(2 m) + V$">|; 

$key = q/lambdaproptoE^{-1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img303.png"
 ALT="$\lambda\propto E^{-1/2}$">|; 

$key = q/a=0.5,lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$a=0.5 \lambda $">|; 

$key = q/{displaymath}bar{psi}(k)proptoexpleft[-{{rm{i},k,x_0-frac{(k-k_0)^{,2}}{4,({mitDelta}k)^2}right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img344.png"
 ALT="\begin{displaymath}
\bar{\psi}(k) \propto \exp\left[-{\rm i} k x_0 - \frac{(k-k_0)^{ 2}}{4 ({\mit\Delta}k)^2}\right],
\end{displaymath}">|; 

$key = q/x-c,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img239.png"
 ALT="$x-c t$">|; 

$key = q/displaystyle-frac{2}{X},{{rm{e}^{-X}int_0^Xleft[{{rm{e}^{-2,x},(1+X+x)-(1+X-x)right]dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="407" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2815.png"
 ALT="$\displaystyle - \frac{2}{X} {\rm e}^{-X}\int_0^X \left[{\rm e}^{-2 x} (1+X+x)-
(1+X-x)\right]dx$">|; 

$key = q/{displaymath}H_0=frac{p^2}{2,m}equiv-frac{hbar^2}{2,m},nabla^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2836.png"
 ALT="\begin{displaymath}
H_0 = \frac{p^2}{2 m} \equiv - \frac{\hbar^2}{2 m} \nabla^2
\end{displaymath}">|; 

$key = q/Y_{l,l-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1452.png"
 ALT="$Y_{l,l-2}$">|; 

$key = q/psi^{(1)}_{-1,pm1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1976.png"
 ALT="$\psi^{(1)}_{-1,\pm 1/2}$">|; 

$key = q/omega_{if}=-omega_{fi}>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2516.png"
 ALT="$\omega_{if}=-\omega_{fi}&gt;0$">|; 

$key = q/displaystyleleftlanglepsi_0({{bf{r}_1)left|frac{a_0}{r_2}right|psi_0({{bf{r}_1)rightrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2808.png"
 ALT="$\displaystyle \left\langle \psi_0({\bf r}_1)\left\vert\frac{a_0}{r_2}\right\vert\psi_0({\bf r}_1)\right\rangle,$">|; 

$key = q/displaystylefrac{(r_1+r_2)-|r_1-r_2|}{r_1,r_2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2743.png"
 ALT="$\displaystyle \frac{(r_1+r_2) - \vert r_1-r_2\vert}{r_1 r_2}$">|; 

$key = q/E_{200}=-e^2slash(32pi,epsilon_0,a_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2266.png"
 ALT="$E_{200}
= -e^2/(32\pi \epsilon_0 a_0)$">|; 

$key = q/{displaymath}langleHrangle(Z)=2,E_0(Z)+langleV_{ee}rangle(Z)+langleUrangle(Z),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="322" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2761.png"
 ALT="\begin{displaymath}
\langle H\rangle (Z)= 2 E_0(Z) + \langle V_{ee}\rangle(Z) + \langle U\rangle(Z),
\end{displaymath}">|; 

$key = q/displaystyleA,psi_{a'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img663.png"
 ALT="$\displaystyle A \psi_{a'}$">|; 

$key = q/{displaymath}mbox{{boldmath{{mu{}_e=-frac{e}{m_e},{{bf{S}_e.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img2369.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\mu$}_e = - \frac{e}{m_e} {\bf S}_e.
\end{displaymath}">|; 

$key = q/y=j,pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img993.png"
 ALT="$y = j \pi$">|; 

$key = q/{displaymath}V(x)=left{array{{lll}infty&mbox{hspace{1cm}}&xleq0-V_0&&0<x<L0&&xgeqLarray{right.,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="76" BORDER="0"
 SRC="|."$dir".q|img1065.png"
 ALT="\begin{displaymath}
V(x) = \left\{\begin{array}{lll}
\infty&amp;\mbox{\hspace{1cm}}&amp;x\leq 0\\\\
-V_0&amp;&amp;0&lt;x&lt;L\\\\
0 &amp;&amp;x\geq L
\end{array}\right.,
\end{displaymath}">|; 

$key = q/{displaymath}{mitOmega}(XotimesY)={mitOmega}(X),{mitOmega}(Y).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img86.png"
 ALT="\begin{displaymath}
{\mit\Omega}(X\otimes Y) = {\mit\Omega}(X)  {\mit\Omega}(Y).
\end{displaymath}">|; 

$key = q/zequiv(x,,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img201.png"
 ALT="$z\equiv (x, y)$">|; 

$key = q/{displaymath}x,(x+1)=l,(l+1),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1949.png"
 ALT="\begin{displaymath}
x (x+1) = l (l+1),
\end{displaymath}">|; 

$key = q/{displaymath}-frac{hbar^2}{2,m}left(frac{d^2}{dr^2}+frac{2}{r}frac{d}{dr}-frac{l,(l+1)}{r^2}right)R_{n,l}+V,R_{n,l}=E,R_{n,l}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="438" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1536.png"
 ALT="\begin{displaymath}
-\frac{\hbar^2}{2 m}\left(\frac{d^2}{d r^2}
+ \frac{2}{r}\f...
...- \frac{l (l+1)}{r^2}\right)R_{n,l}
+V R_{n,l} = E R_{n,l}.
\end{displaymath}">|; 

$key = q/P_1(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2439.png"
 ALT="$P_1(t)$">|; 

$key = q/{displaymath}E=E_1+E_2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img2715.png"
 ALT="\begin{displaymath}
E = E_1 + E_2.
\end{displaymath}">|; 

$key = q/displaystyleH,psi_pm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2803.png"
 ALT="$\displaystyle H \psi_\pm$">|; 

$key = q/displaystylelangleurangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img124.png"
 ALT="$\displaystyle \langle u\rangle$">|; 

$key = q/+l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1437.png"
 ALT="$+l$">|; 

$key = q/displaystylepsi_1({{bf{r}_1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2717.png"
 ALT="$\displaystyle \psi_1({\bf r}_1)$">|; 

$key = q/psi_{n',l',m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2231.png"
 ALT="$\psi_{n',l',m'}$">|; 

$key = q/displaystyle{calO}(epsilon);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2161.png"
 ALT="$\displaystyle {\cal O}(\epsilon)$">|; 

$key = q/{{rm{i},y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img194.png"
 ALT="${\rm i} y$">|; 

$key = q/displaystyleL^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1517.png"
 ALT="$\displaystyle L^2$">|; 

$key = q/{displaymath}P(X)=lim_{{mitOmega}({mitSigma})rightarrowinfty}frac{{mitOmega}(X)}{{mitOmega}({mitSigma})},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="\begin{displaymath}
P(X) = \lim_{{\mit\Omega}({\mit\Sigma})\rightarrow\infty}\frac{{\mit\Omega}(X)}{{\mit\Omega}({\mit\Sigma})},
\end{displaymath}">|; 

$key = q/nusimeqvslashR;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img929.png"
 ALT="$\nu\simeq v/R$">|; 

$key = q/displaystyle{{rm{i},hbar,{{bf{J}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1876.png"
 ALT="$\displaystyle {\rm i} \hbar {\bf J}.$">|; 

$key = q/omega_{21};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2433.png"
 ALT="$\omega_{21}$">|; 

$key = q/nu=omegaslash2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img233.png"
 ALT="$\nu=\omega/2\pi$">|; 

$key = q/chi^nomath_inline{dagnomath_inline{,chi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1728.png"
 ALT="$\chi^\dag \chi$">|; 

$key = q/1,0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="$1,0$">|; 

$key = q/6000,AA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img745.png"
 ALT="$6000 \AA$">|; 

$key = q/j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img164.png"
 ALT="$j$">|; 

$key = q/{displaymath}x_pm=x+{{rm{i},y.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2640.png"
 ALT="\begin{displaymath}
x_\pm = x + {\rm i}  y.
\end{displaymath}">|; 

$key = q/{displaymath}Z(z)=sum_kc_k,z^k.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1595.png"
 ALT="\begin{displaymath}
Z(z) = \sum_k c_k z^k.
\end{displaymath}">|; 

$key = q/rrightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1601.png"
 ALT="$r\rightarrow 0$">|; 

$key = q/{displaymath}log_{10}[tau({{rm{yr})]=-C_1-C_2,Z_1^{,2slash3}+C_3,frac{Z_1}{sqrt{E({{rm{MeV})}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="377" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img940.png"
 ALT="\begin{displaymath}
\log_{10}[\tau ({\rm yr})] = -C_1 - C_2 Z_1^{ 2/3} + C_3 \frac{Z_1}{\sqrt{E({\rm MeV})}},
\end{displaymath}">|; 

$key = q/delta_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2928.png"
 ALT="$\delta_l$">|; 

$key = q/{displaymath}E_1=E_2=4,E_0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2722.png"
 ALT="\begin{displaymath}
E_1=E_2 = 4 E_0,
\end{displaymath}">|; 

$key = q/{displaymath}Omega=frac{g,e,B}{2,m_e}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1834.png"
 ALT="\begin{displaymath}
\Omega = \frac{g e B}{2 m_e}.
\end{displaymath}">|; 

$key = q/u_2,cdots,u_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img97.png"
 ALT="$u_2, \cdots, u_M$">|; 

$key = q/lambda^{-1}=nuslashc;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1674.png"
 ALT="$\lambda^{-1}=\nu/c$">|; 

$key = q/k(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img876.png"
 ALT="$k(x)$">|; 

$key = q/{displaymath}H,psi_i=E_i,psi_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2099.png"
 ALT="\begin{displaymath}
H \psi_i = E_i \psi_i.
\end{displaymath}">|; 

$key = q/{displaymath}(d_x)_{n,n'}=e,sqrt{frac{hbar,n}{2,m_e,omega_0}},delta_{n,n'+1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2629.png"
 ALT="\begin{displaymath}
(d_x)_{n,n'} = e \sqrt{\frac{\hbar n}{2 m_e \omega_0}} \delta_{n,n'+1}.
\end{displaymath}">|; 

$key = q/{displaymath}(H_0+H_1),psi_E=E,psi_E.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2123.png"
 ALT="\begin{displaymath}
(H_0+H_1) \psi_E = E \psi_E.
\end{displaymath}">|; 

$key = q/displaystyleint_{-infty}^{infty}psi^{ast}left(-{{rm{i},hbar,frac{partial}{partialx}right)psi,dx.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img536.png"
 ALT="$\displaystyle \int_{-\infty}^{\infty}\psi^{\ast}\left(-{\rm i} \hbar 
\frac{\partial}{\partial x}\right)\psi dx.$">|; 

$key = q/{displaymath}chi=c_+,chi_++c_-,chi_-.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1788.png"
 ALT="\begin{displaymath}
\chi = c_+ \chi_+ + c_- \chi_-.
\end{displaymath}">|; 

$key = q/{displaymath}sigma_A^{,2}=langleA^2rangle-langleArangle^2=a^2-a^2=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img653.png"
 ALT="\begin{displaymath}
\sigma_A^{ 2} = \langle A^2\rangle - \langle A\rangle^2 = a^2-a^2 = 0.
\end{displaymath}">|; 

$key = q/n=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1024.png"
 ALT="$n=0$">|; 

$key = q/{displaymath}langle-|H_1|+rangle=langle+|H_1|-rangle^ast=-frac{g,e,B_1}{4,m},exp(,{{rm{i},omega,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="364" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2452.png"
 ALT="\begin{displaymath}
\langle -\vert H_1\vert+\rangle = \langle +\vert H_1\vert-\r...
...e^\ast = - \frac{g e B_1}{4 m} \exp( {\rm i} \omega t).
\end{displaymath}">|; 

$key = q/{displaymath}R=frac{-E_0}{h,c}=frac{m_e,e^4}{(4pi)^3,epsilon_0^{,2},hbar^3,c}=1.097times10^7,{{rm{m^{-1}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1676.png"
 ALT="\begin{displaymath}
R = \frac{-E_0}{h c} =\frac{m_e e^4}{(4\pi)^3 \epsilon_0^{ 2} \hbar^3 c} = 1.097\times 10^7 {\rm m^{-1}}.
\end{displaymath}">|; 

$key = q/psi^{ast}slash({{rm{i},hbar);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img471.png"
 ALT="$\psi^{\ast}/({\rm i} \hbar)$">|; 

$key = q/omeganeqomega_{21};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2437.png"
 ALT="$\omega\neq \omega_{21}$">|; 

$key = q/displaystylegamma,expleft[+{{rm{i},(omega-omega_{21}),tright],c_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2414.png"
 ALT="$\displaystyle \gamma \exp\left[+{\rm i} (\omega-\omega_{21}) t\right] c_2,$">|; 

$key = q/{displaymath}U=-frac{3}{5}frac{G,M^2}{R},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1274.png"
 ALT="\begin{displaymath}
U = -\frac{3}{5}\frac{G M^2}{R},
\end{displaymath}">|; 

$key = q/{displaymath}P_{irightarrowf}(t)ll1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2480.png"
 ALT="\begin{displaymath}
P_{i\rightarrow f}(t)\ll 1.
\end{displaymath}">|; 

$key = q/murightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2888.png"
 ALT="$\mu \rightarrow 0$">|; 

$key = q/chi_{s,-s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1769.png"
 ALT="$\chi_{s,-s}$">|; 

$key = q/tgg2pislash|omega_{fi}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2499.png"
 ALT="$t\gg 2\pi/\vert\omega_{fi}\vert$">|; 

$key = q/{scriptstylesqrt{(l+m+1)slash(2,l+1)}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1967.png"
 ALT="${\scriptstyle\sqrt{(l+m+1)/(2 l+1)}}$">|; 

$key = q/{displaymath}H_1(t)=V,exp(,{{rm{i},omega,t)+V^nomath_inline{dagnomath_inline{,exp(-{{rm{i},omega,t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2481.png"
 ALT="\begin{displaymath}
H_1(t) = V \exp( {\rm i} \omega t) + V^\dag \exp(-{\rm i} \omega t),
\end{displaymath}">|; 

$key = q/dslashdtintP(x,t),dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img761.png"
 ALT="$d/dt \int P(x,t) dx$">|; 

$key = q/displaystyleL_z,psi^{(1)}_{l,s;m,m_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1903.png"
 ALT="$\displaystyle L_z \psi^{(1)}_{l,s;m,m_s}$">|; 

$key = q/displaystylec^+_{l,m-1},c^-_{l,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1386.png"
 ALT="$\displaystyle c^+_{l,m-1} c^-_{l,m}$">|; 

$key = q/G;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1275.png"
 ALT="$G$">|; 

$key = q/dV=r^2,dr,d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1634.png"
 ALT="$dV = r^2 dr d{\mit\Omega}$">|; 

$key = q/displaystyleY_{1,pm1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1477.png"
 ALT="$\displaystyle Y_{1,\pm1}$">|; 

$key = q/|psi(y,t)|^{,2},dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img294.png"
 ALT="$\vert\psi(y,t)\vert^{ 2} dy$">|; 

$key = q/0leqxleqa;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img830.png"
 ALT="$0\leq x \leq a$">|; 

$key = q/{displaymath}langle1,0,0|r^2|1,0,0rangle=3,a_0^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2232.png"
 ALT="\begin{displaymath}
\langle 1,0,0\vert r^2\vert 1,0,0\rangle = 3 a_0^{ 2}.
\end{displaymath}">|; 

$key = q/{displaymath}(l+l'+2),(l+l'),(l-l'+1),(l-l'-1),langlen,l,m|z|n',l',mrangle=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="508" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2208.png"
 ALT="\begin{displaymath}
(l+l'+2) (l+l') (l-l'+1) (l-l'-1) \langle n,l,m\vert z\vert n',l',m\rangle = 0.
\end{displaymath}">|; 

$key = q/{displaymath}q=frac{2,j,pi}{a}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img994.png"
 ALT="\begin{displaymath}
q = \frac{2 j \pi}{a}.
\end{displaymath}">|; 

$key = q/x_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1072.png"
 ALT="$x_i$">|; 

$key = q/2a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1052.png"
 ALT="$2a$">|; 

$key = q/displaystyle[S_x,S_y];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1700.png"
 ALT="$\displaystyle [S_x, S_y]$">|; 

$key = q/displaystylefrac{a_0}{2},[3,n^2-l,(l+1)],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1656.png"
 ALT="$\displaystyle \frac{a_0}{2} [3 n^2-l (l+1)],$">|; 

$key = q/cotdelta_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3022.png"
 ALT="$\cot \delta_l$">|; 

$key = q/{displaymath}(E_n-E+e_{nn})-sum_{kneqn}frac{|e_{nk}|^2}{E_k-E_n}simeq0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img2168.png"
 ALT="\begin{displaymath}
(E_n-E+e_{nn})-\sum_{k\neq n}\frac{\vert e_{nk}\vert^2}{E_k-E_n} \simeq 0.
\end{displaymath}">|; 

$key = q/{{rm{Li}^{++};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1686.png"
 ALT="${\rm Li}^{++}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter05slashfig07.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="337" HEIGHT="327" BORDER="0"
 SRC="|."$dir".q|img983.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter05/fig07.eps}}
\end{figure}">|; 

$key = q/{displaymath}S_i=frac{hbar}{2},sigma_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1800.png"
 ALT="\begin{displaymath}
S_i = \frac{\hbar}{2} \sigma_i.
\end{displaymath}">|; 

$key = q/displaystylebar{psi}(k,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img568.png"
 ALT="$\displaystyle \bar{\psi}(k,t)$">|; 

$key = q/{displaymath}chi=left(array{{c}c_+(t)c_-(t)array{right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1836.png"
 ALT="\begin{displaymath}
\chi = \left(\begin{array}{c}c_+(t) c_-(t)\end{array}\right),
\end{displaymath}">|; 

$key = q/w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img137.png"
 ALT="$w$">|; 

$key = q/displaystylej_l(k,r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2971.png"
 ALT="$\displaystyle j_l(k r)$">|; 

$key = q/+Z,e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2758.png"
 ALT="$+Z e$">|; 

$key = q/{displaymath}left(frac{partial^2}{partialx^2}+frac{partial^2}{partialy^2}+frac{p}{partialz^2}right)psi=-frac{2,m}{hbar^2},E,psi,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1212.png"
 ALT="\begin{displaymath}
\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\p...
...^2}{\partial z^2}\right)\psi = -\frac{2 m}{\hbar^2} E \psi,
\end{displaymath}">|; 

$key = q/k',a=pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3015.png"
 ALT="$k' a = \pi/2$">|; 

$key = q/s=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2063.png"
 ALT="$s=1$">|; 

$key = q/y_l(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1559.png"
 ALT="$y_l(z)$">|; 

$key = q/{displaymath}M=2,N,m_p,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1279.png"
 ALT="\begin{displaymath}
M = 2 N m_p,
\end{displaymath}">|; 

$key = q/displaystylephi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2537.png"
 ALT="$\displaystyle \phi$">|; 

$key = q/beta=beta_1slash(1+{{rm{i},beta_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img382.png"
 ALT="$\beta=\beta_1/(1+{\rm i} \beta_2)$">|; 

$key = q/z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img197.png"
 ALT="$z$">|; 

$key = q/psi_i(x_i,t)=psi_{E_i}(x_i),exp(-{{rm{i},E_i,tslashhbar);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1105.png"
 ALT="$\psi_i(x_i,t) = \psi_{E_i}(x_i) \exp(-{\rm i}  E_i t/\hbar)$">|; 

$key = q/N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img128.png"
 ALT="$N$">|; 

$key = q/{displaymath}beta=left(frac{Z_1,Z_2,e^2,m,R}{4pi,epsilon_0,hbar^2}right)^{1slash2}=0.74,Z_1^{2slash3}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img922.png"
 ALT="\begin{displaymath}
\beta = \left(\frac{Z_1 Z_2 e^2 m R}{4\pi \epsilon_0 \hbar^2}\right)^{1/2} = 0.74 Z_1^{2/3}
\end{displaymath}">|; 

$key = q/V({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2837.png"
 ALT="$V({\bf r})$">|; 

$key = q/|psi(x')|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img719.png"
 ALT="$\vert\psi(x')\vert^{ 2}$">|; 

$key = q/displaystylesintheta,sinphi,frac{partial}{partialr}+frac{costheta,sinphi}{r},fraialtheta}+frac{cosphi}{r,sintheta},frac{partial}{partialphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="357" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1354.png"
 ALT="$\displaystyle \sin\theta \sin\phi \frac{\partial}{\partial r} + \frac{\cos\th...
...partial\theta} + \frac{\cos\phi}{r \sin\theta} \frac{\partial}{\partial\phi},$">|; 

$key = q/displaystyleleft.leftlanglesum_nc_n,psi_nright|Hleft|sum_m,c_m,psi_mrightrangleright.=sum_{n,m}c_n^{,ast},c_m,langlepsi_n|H|psi_mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="400" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2696.png"
 ALT="$\displaystyle \left.\left\langle \sum_n c_n \psi_n\right\vert
H\left\vert\sum_...
...\right. = \sum_{n,m} c_n^{ \ast} c_m \langle \psi_n\vert H\vert\psi_m\rangle$">|; 

$key = q/{displaymath}H,psi_i=E_i,psi_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img741.png"
 ALT="\begin{displaymath}
H \psi_i = E_i \psi_i,
\end{displaymath}">|; 

$key = q/{displaymath}H_0,psi_{nlm}=E_n,psi_{nlm},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2235.png"
 ALT="\begin{displaymath}
H_0 \psi_{nlm} = E_n \psi_{nlm},
\end{displaymath}">|; 

$key = q/z_{n,l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1564.png"
 ALT="$z_{n,l}$">|; 

$key = q/{displaymath}Y_{l,-l}(theta,phi)sim(sintheta)^l,{{rm{e}^{-{{rm{i},l,phi}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1443.png"
 ALT="\begin{displaymath}
Y_{l,-l}(\theta,\phi) \sim (\sin\theta)^l {\rm e}^{-{\rm i} l \phi}.
\end{displaymath}">|; 

$key = q/displaystylelangle2|H_1|2rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2132.png"
 ALT="$\displaystyle \langle 2\vert H_1\vert 2\rangle,$">|; 

$key = q/{displaymath}A_{ij}=int_{-infty}^inftypsi_i^ast,A,psi_j,dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img2112.png"
 ALT="\begin{displaymath}
A_{ij} = \int_{-\infty}^\infty\psi_i^\ast A \psi_j dx,
\end{displaymath}">|; 

$key = q/3slash2,3slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2025.png"
 ALT="$3/2, 3/2$">|; 

$key = q/j(a,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img483.png"
 ALT="$j(a,t)$">|; 

$key = q/{displaymath}langle2,0,0|z|2,1,0rangle=langle2,1,0|z|2,0,0rangle=3,a_0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2273.png"
 ALT="\begin{displaymath}
\langle 2,0,0\vert z\vert 2,1,0\rangle = \langle 2,1,0\vert z\vert 2,0,0\rangle = 3 a_0.
\end{displaymath}">|; 

$key = q/omegaslashc=k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img261.png"
 ALT="$\omega/c = k$">|; 

$key = q/{displaymath}L^2=x_i,p_j,x_i,p_j-x_i,p_j,x_j,p_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1514.png"
 ALT="\begin{displaymath}
L^2 = x_i p_j x_i p_j - x_i p_j x_j p_i.
\end{displaymath}">|; 

$key = q/displaystyleA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1282.png"
 ALT="$\displaystyle A$">|; 

$key = q/{displaymath}A_{ij}=langlepsi_i|A|psi_jrangleequivlanglei|A|jrangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2115.png"
 ALT="\begin{displaymath}
A_{ij}=\langle \psi_i\vert A\vert\psi_j\rangle \equiv \langle i\vert A\vert j\rangle.
\end{displaymath}">|; 

$key = q/|bar{psi}(k)|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img356.png"
 ALT="$\vert\bar{\psi}(k)\vert$">|; 

$key = q/displaystyle[s,(s+1)-m_s,(m_s+1)]^{1slash2},hbar,chi_{s,m_s+1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1934.png"
 ALT="$\displaystyle [s (s+1)-m_s (m_s+1)]^{1/2} \hbar \chi_{s,m_s+1},$">|; 

$key = q/{displaymath}I({{bf{r}_2)=2piint_0^inftyint_0^pifrac{{{rm{e}^{-2,r_1}}{sqrt{r_1^,costheta_1}},r_1^{,2},dr_1,sintheta_1,dtheta_1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="485" HEIGHT="62" BORDER="0"
 SRC="|."$dir".q|img2738.png"
 ALT="\begin{displaymath}
I({\bf r}_2) = 2\pi\int_0^\infty\int_0^\pi
\frac{{\rm e}^{-2...
..._2 \cos\theta_1}} 
r_1^{ 2} dr_1 \sin\theta_1 d\theta_1.
\end{displaymath}">|; 

$key = q/{displaymath}a_0=frac{4pi,epsilon_0,hbar^2}{m_e,e^2}=5.3times10^{-11},{{rm{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1623.png"
 ALT="\begin{displaymath}
a_0 = \frac{4\pi \epsilon_0 \hbar^2}{m_e e^2} = 5.3\times 10^{-11} {\rm m}.
\end{displaymath}">|; 

$key = q/V(x)=-alpha,delta(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1060.png"
 ALT="$V(x) = -\alpha \delta (x)$">|; 

$key = q/(1slash2),hbar,omega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2635.png"
 ALT="$(1/2) \hbar \omega_0$">|; 

$key = q/displaystylefrac{l_z,pi}{a}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1239.png"
 ALT="$\displaystyle \frac{l_z \pi}{a}.$">|; 

$key = q/J_z=L_z+S_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2343.png"
 ALT="$J_z=L_z+S_z$">|; 

$key = q/displaystylefrac{e^2}{16pi,epsilon_0,m_e^{,2},c^2}leftlangle1,1slash2;j,m_jleft|frac{J^2-L^2-S^2}{r^3}right|l,1slash2;j,m_jrightrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="445" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2321.png"
 ALT="$\displaystyle \frac{e^2}{16\pi \epsilon_0 m_e^{ 2} c^2}\left\langle
1,1/2;j,m_j\left\vert\frac{J^2-L^2-S^2}{r^3}\right\vert l,1/2;j,m_j\right\rangle$">|; 

$key = q/displaystyle[l,(l+1)+3slash4-m-1],hbar^2,Y_{l,m+1},chi_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1944.png"
 ALT="$\displaystyle [l (l+1)+3/4-m-1] \hbar^2 Y_{l,m+1} \chi_-$">|; 

$key = q/displaystyleS^{2},chi_{s_1,s_2;s,m_s}^{(2)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2054.png"
 ALT="$\displaystyle S^{2}  \chi_{s_1,s_2;s,m_s}^{(2)}$">|; 

$key = q/g=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1830.png"
 ALT="$g=2$">|; 

$key = q/displaystyle-frac{dV(langlexrangle)}{dlanglexrangle}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img523.png"
 ALT="$\displaystyle -\frac{dV(\langle x\rangle)}{d\langle x\rangle}.$">|; 

$key = q/{displaymath}tandelta_0=frac{sin(k,a)slashk,a}{-cos(k,a)slashka}=-tan(k,a),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2962.png"
 ALT="\begin{displaymath}
\tan\delta_0 = \frac{\sin (k a)/k a}{-\cos (k a)/ka} = -\tan (k a),
\end{displaymath}">|; 

$key = q/v_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img408.png"
 ALT="$v_0$">|; 

$key = q/{displaymath}langleprangle=int_{-infty}^{infty}phi^ast(p,t),p,phi(p,t),dp=int_{-infty}^{infty}p,|phi|^{,2},dp.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="362" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img592.png"
 ALT="\begin{displaymath}
\langle p\rangle = \int_{-\infty}^{\infty}\phi^\ast(p,t) p\...
...p,t) dp = \int_{-\infty}^{\infty}p \vert\phi\vert^{ 2} dp.
\end{displaymath}">|; 

$key = q/sigma_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1801.png"
 ALT="$\sigma_i$">|; 

$key = q/displaystylem,hbar,psi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1533.png"
 ALT="$\displaystyle m \hbar \psi,$">|; 

$key = q/P_{irightarrowf}ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2508.png"
 ALT="$P_{i\rightarrow f}\ll 1$">|; 

$key = q/hbar^2,k^2slash(2,m);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img438.png"
 ALT="$\hbar^2 k^2/(2 m)$">|; 

$key = q/psi(-x)=psi(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img959.png"
 ALT="$\psi(-x)=\psi(x)$">|; 

$key = q/l+1slash2,m+1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1966.png"
 ALT="$l+1/2, m+1/2$">|; 

$key = q/{displaymath}P(x,t)=|psi(x,t)|^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1174.png"
 ALT="\begin{displaymath}
P(x,t) = \vert\psi(x,t)\vert^2.
\end{displaymath}">|; 

$key = q/{displaymath}DeltaE_{nlm}=E_n,frac{alpha^2}{n^2}left(frac{n}{l+1slash2}-frac{3}{4}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2302.png"
 ALT="\begin{displaymath}
\Delta E_{nlm} = E_n \frac{\alpha^2}{n^2}\left(\frac{n}{l+1/2}-\frac{3}{4}\right),
\end{displaymath}">|; 

$key = q/{displaymath}DeltaE_{l,1slash2;j,m_j}=mu_B,B,m_jleft[1pmfrac{1}{2,l+1}right]{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2348.png"
 ALT="\begin{displaymath}
\Delta E_{l,1/2;j,m_j} = \mu_B B m_j\left[1\pm \frac{1}{2 l+1}\right]
\end{displaymath}">|; 

$key = q/{displaymath}langlen,l'^{(1)},m|n,l^{(1)},mrangle=delta_{ll'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2249.png"
 ALT="\begin{displaymath}
\langle n,l'^{(1)},m\vert n,l^{(1)},m\rangle = \delta_{ll'}.
\end{displaymath}">|; 

$key = q/{displaymath}Y_{l,l-2}sim{{rm{e}^{-{{rm{i},(l-2),phi}left[frac{d}{dtheta}+(l-1),{sintheta}frac{d}{dtheta}right)(sintheta)^{2,l}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="540" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1454.png"
 ALT="\begin{displaymath}
Y_{l,l-2}\sim {\rm e}^{-{\rm i} (l-2) \phi}\left[\frac{d}{...
...ac{1}{\sin\theta}\frac{d}{d\theta}\right)
(\sin\theta)^{2 l}.
\end{displaymath}">|; 

$key = q/{displaymath}[A,B]equivA,B-B,A.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img621.png"
 ALT="\begin{displaymath}[A,B]\equiv A B-B A.
\end{displaymath}">|; 

$key = q/{{rm{He}^+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1685.png"
 ALT="${\rm He}^+$">|; 

$key = q/{displaymath}psi(x,y,z,t)=psi(x,y,z),{{rm{e}^{-{{rm{i},E,tslashhbar},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1209.png"
 ALT="\begin{displaymath}
\psi(x,y,z,t) = \psi(x,y,z) {\rm e}^{-{\rm i} E t/\hbar},
\end{displaymath}">|; 

$key = q/{displaymath}sum_{i=1}^{M}P(X_i)=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img81.png"
 ALT="\begin{displaymath}
\sum_{i=1}^{M} P(X_i) =1.
\end{displaymath}">|; 

$key = q/{displaymath}psi_0=A,{{rm{e}^{,{{rm{i},varphi},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img219.png"
 ALT="\begin{displaymath}
\psi_0 = A {\rm e}^{ {\rm i} \varphi},
\end{displaymath}">|; 

$key = q/{displaymath}P(XmidY)=lim_{{mitOmega}({mitSigma})rightarrowinfty}frac{{mitOmega}(XmidY)}{{mitOmega}({mitSigma})},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img70.png"
 ALT="\begin{displaymath}
P(X\mid Y) =\lim_{{\mit\Omega}({\mit\Sigma})\rightarrow\infty}
\frac{
{\mit\Omega}(X \mid Y)}{{\mit\Omega}({\mit\Sigma})},
\end{displaymath}">|; 

$key = q/displaystyle-frac{2}{X}int_X^infty{{rm{e}^{-2,x}left[{{rm{e}^{-X},(1+X+x)-{{rm{e}^X,(1-X+x)right]x,dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="452" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2798.png"
 ALT="$\displaystyle -\frac{2}{X}\int_X^\infty {\rm e}^{-2 x}\left[{\rm e}^{-X} (1+X+x)-
{\rm e}^X (1-X+x)\right] x dx,$">|; 

$key = q/{{rm{sinc}^2[(omega-omega_{fi}),tslash2];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2559.png"
 ALT="${\rm sinc}^2[(\omega-\omega_{fi}) t/2]$">|; 

$key = q/{displaymath}langleArangle=a,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img686.png"
 ALT="\begin{displaymath}
\langle A\rangle= a,
\end{displaymath}">|; 

$key = q/displaystyleleft{array{{lcl}2slashr_1<col_mark><mbox_mark>{hspace{1cm}}<col_markr_2<col_mark><col_mark><mbox_mark>{for{r_1<r_2{}array{right.,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2744.png"
 ALT="$\displaystyle \left\{\begin{array}{lcl}2/r_1&amp;\mbox{\hspace{1cm}}&amp;\mbox{for
$r_1&gt;r_2$}\\\\
2/r_2&amp;&amp;\mbox{for $r_1&lt;r_2$}\end{array}\right.,$">|; 

$key = q/{displaymath}langlei|A|jrangle^ast=langlej|A^nomath_inline{dagnomath_inline{|irangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2117.png"
 ALT="\begin{displaymath}
\langle i\vert A\vert j\rangle^\ast=\langle j\vert A^\dag \vert i\rangle.
\end{displaymath}">|; 

$key = q/{displaymath}lambda=frac{c}{sqrt{nu^2-nu_0^{,2}}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img452.png"
 ALT="\begin{displaymath}
\lambda = \frac{c}{\sqrt{\nu^2 - \nu_0^{ 2}}},
\end{displaymath}">|; 

$key = q/displaystylempsqrt{frac{3}{8pi}},sintheta,{{rm{e}^{pm{{rm{i},phi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1478.png"
 ALT="$\displaystyle \mp \sqrt{\frac{3}{8\pi}} \sin\theta {\rm e}^{\pm{\rm i} \phi},$">|; 

$key = q/displaystyleint_{-pi}^piTheta^{,ast}_{l',m'}(theta),Theta_{l,m}(theta),sintheta,dtheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1392.png"
 ALT="$\displaystyle \int_{-\pi}^\pi \Theta^{ \ast}_{l',m'}(\theta) \Theta_{l,m}(\theta) \sin\theta d\theta$">|; 

$key = q/displaystylesqrt{frac{2}{a}}sin(k_x,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1229.png"
 ALT="$\displaystyle \sqrt{\frac{2}{a}}\sin (k_x x),$">|; 

$key = q/{displaymath}beta_{l+}=beta_{l-}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2958.png"
 ALT="\begin{displaymath}
\beta_{l+} = \beta_{l-}.
\end{displaymath}">|; 

$key = q/displaystyle-frac{1}{2,m_e,c^2}left(E_n^{,2}-2,E_n,langlen,l,m|V|n,l,mrangle+langlen,l,m|V^2|n,l,mrangleright);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="500" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2299.png"
 ALT="$\displaystyle -\frac{1}{2 m_e c^2}\left(E_n^{ 2} - 2 E_n \langle
n,l,m\vert V\vert n,l,m\rangle + \langle n,l,m\vert V^2\vert n,l,m\rangle\right)$">|; 

$key = q/phi(p);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img631.png"
 ALT="$\phi(p)$">|; 

$key = q/g=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1827.png"
 ALT="$g=1$">|; 

$key = q/{displaymath}|psi(x_1,x_2,t)|^2=|psi(x_2,x_1,t)|^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1136.png"
 ALT="\begin{displaymath}
\vert\psi(x_1,x_2,t)\vert^2 = \vert\psi(x_2,x_1,t)\vert^2,
\end{displaymath}">|; 

$key = q/ineqj;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1166.png"
 ALT="$i\neq j$">|; 

$key = q/x=(j-varphislash2pi),lambda+v,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img166.png"
 ALT="$x=(j-\varphi/2\pi) \lambda+ v t$">|; 

$key = q/{displaymath}-lleqmleql.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1418.png"
 ALT="\begin{displaymath}
-l \leq m \leq l.
\end{displaymath}">|; 

$key = q/P'=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1128.png"
 ALT="$P'=0$">|; 

$key = q/(A,B)^nomath_inline{dagnomath_inline{=B^nomath_inline{dagnomath_inline{,A^nomath_inline{dagnomath_inline{;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img550.png"
 ALT="$(A B)^\dag = B^\dag A^\dag $">|; 

$key = q/{displaymath}E_n=(n+1slash2),hbar,omega_0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2620.png"
 ALT="\begin{displaymath}
E_n = (n+1/2) \hbar \omega_0,
\end{displaymath}">|; 

$key = q/displaystyleE_2';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2141.png"
 ALT="$\displaystyle E_2'$">|; 

$key = q/2,n^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1668.png"
 ALT="$2 n^2$">|; 

$key = q/V(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img436.png"
 ALT="$V(x)$">|; 

$key = q/{displaymath}(a-a'),int_{-infty}^inftypsi_a^ast,psi_{a'},dx=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img670.png"
 ALT="\begin{displaymath}
(a-a')  \int_{-\infty}^\infty\psi_a^\ast \psi_{a'} dx = 0.
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{n^2,a_0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1658.png"
 ALT="$\displaystyle \frac{1}{n^2 a_0},$">|; 

$key = q/{calR}_0(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2999.png"
 ALT="${\cal R}_0(r)$">|; 

$key = q/V_0>E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img865.png"
 ALT="$V_0&gt;E$">|; 

$key = q/{{bf{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img180.png"
 ALT="${\bf n}$">|; 

$key = q/displaystylefrac{2,m,E}{hbar^2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img954.png"
 ALT="$\displaystyle \frac{2 m E}{\hbar^2},$">|; 

$key = q/displaystyleS^2,psi^{(1)}_{l,s;m,m_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1901.png"
 ALT="$\displaystyle S^2 \psi^{(1)}_{l,s;m,m_s}$">|; 

$key = q/X(0)=X(a)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1214.png"
 ALT="$X(0)=X(a)=0$">|; 

$key = q/1-P_{irightarrowf}^{abs}(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2567.png"
 ALT="$1-P_{i\rightarrow f}^{abs}(t)$">|; 

$key = q/{displaymath}langleV_{ee}rangle=-frac{4,E_0}{pi^2},intfrac{{{rm{e}^{-2,(hat{r}_1-hat{{bf{r}_2|},d^3hat{{bf{r}_1,d^3hat{{bf{r}_2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2729.png"
 ALT="\begin{displaymath}
\langle V_{ee}\rangle = -\frac{4 E_0}{\pi^2} \int
\frac{{\...
... r}_1-\hat{\bf r}_2\vert} d^3\hat{\bf r}_1 d^3\hat{\bf r}_2,
\end{displaymath}">|; 

$key = q/L=(S,P,D,F,cdots);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2327.png"
 ALT="$L=(S,P,D,F,\cdots)$">|; 

$key = q/psi_a(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1167.png"
 ALT="$\psi_a(x)$">|; 

$key = q/displaystylelanglepsi|H|psirangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2695.png"
 ALT="$\displaystyle \langle\psi\vert H\vert\psi\rangle$">|; 

$key = q/{displaymath}psi_E(x,t,E_i)=psi_i(x),{{rm{e}^{-{{rm{i},E_i,tslashhbar},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img736.png"
 ALT="\begin{displaymath}
\psi_E(x,t,E_i) = \psi_i(x) {\rm e}^{-{\rm i} E_i t/\hbar},
\end{displaymath}">|; 

$key = q/displaystylefrac{3}{5},G,M^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1285.png"
 ALT="$\displaystyle \frac{3}{5} G M^2.$">|; 

$key = q/{displaymath}[H,t]={{rm{i},hbar,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img633.png"
 ALT="\begin{displaymath}[H,t]= {\rm i} \hbar,
\end{displaymath}">|; 

$key = q/displaystyle1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img121.png"
 ALT="$\displaystyle 1$">|; 

$key = q/|x|leqaslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img969.png"
 ALT="$\vert x\vert\leq a/2$">|; 

$key = q/{displaymath}J_z=L_z+S_z,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1896.png"
 ALT="\begin{displaymath}
J_z = L_z+S_z,
\end{displaymath}">|; 

$key = q/pm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2349.png"
 ALT="$\pm$">|; 

$key = q/V({{bf{r})=V(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2903.png"
 ALT="$V({\bf r}) = V(r)$">|; 

$key = q/displaystylepsi_0({{bf{r}_2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2720.png"
 ALT="$\displaystyle \psi_0({\bf r}_2),$">|; 

$key = q/displaystyleS_-,S_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1713.png"
 ALT="$\displaystyle S_- S_+$">|; 

$key = q/sigma_E,sigma_t=hbarslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img647.png"
 ALT="$\sigma_E \sigma_t=\hbar/2$">|; 

$key = q/displaystylepsi_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img774.png"
 ALT="$\displaystyle \psi_2$">|; 

$key = q/{displaymath}int_{-infty}^{infty}{{rm{e}^{-y^2},dy=sqrt{pi},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img466.png"
 ALT="\begin{displaymath}
\int_{-\infty}^{\infty}{\rm e}^{-y^2} dy = \sqrt{\pi},
\end{displaymath}">|; 

$key = q/{displaymath}bar{E}=E_Fleft.int_0^{l_F}l^2,4pi,l^2,dlrightslashfrac{4}{3},pi,l_F^{,5}=frac{3}{5},E_F,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="318" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1254.png"
 ALT="\begin{displaymath}
\bar{E} = E_F\left.\int_0^{l_F}l^2 4\pi l^2 dl\right/\frac{4}{3} \pi l_F^{ 5}=
\frac{3}{5} E_F,
\end{displaymath}">|; 

$key = q/E'=P'^{,2}slash2,M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1127.png"
 ALT="$E' = P'^{ 2}/2 M$">|; 

$key = q/{displaymath}|psi_0|^{,2}sqrt{2},sigma,int_{-infty}^{infty}{{rm{e}^{-y^2},dy=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img465.png"
 ALT="\begin{displaymath}
\vert\psi_0\vert^{ 2}\sqrt{2} \sigma \int_{-\infty}^{\infty}{\rm e}^{-y^2} dy=1.
\end{displaymath}">|; 

$key = q/10^{-9};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img304.png"
 ALT="$10^{-9}$">|; 

$key = q/{displaymath}psi({{bf{r})=sqrt{n},sum_{l=0}^infty{{rm{i}^{,l},(2,l+1),{calR}_l(r),P_l(costheta),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img2953.png"
 ALT="\begin{displaymath}
\psi({\bf r}) = \sqrt{n} \sum_{l=0}^\infty
{\rm i}^{ l}  (2 l+1) {\cal R}_l(r) P_l(\cos\theta),
\end{displaymath}">|; 

$key = q/langle1slashrrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2766.png"
 ALT="$\langle 1/r\rangle$">|; 

$key = q/langle(x-langlexrangle)^{,2}rangle=sigma_x^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img522.png"
 ALT="$\langle (x-\langle x\rangle)^{ 2}\rangle = \sigma_x^{ 2}$">|; 

$key = q/{displaymath}{{bf{U},{{bf{x}=lambda,{{bf{x},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2253.png"
 ALT="\begin{displaymath}
{\bf U} {\bf x} = \lambda  {\bf x},
\end{displaymath}">|; 

$key = q/m';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2636.png"
 ALT="$m'$">|; 

$key = q/displaystyleL^2,psi^{(1)}_{l,s;m,m_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1899.png"
 ALT="$\displaystyle L^2 \psi^{(1)}_{l,s;m,m_s}$">|; 

$key = q/{displaymath}V_0-E=frac{hbar^2kappa^2}{2,m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2998.png"
 ALT="\begin{displaymath}
V_0 - E = \frac{\hbar^2 \kappa^2}{2 m}.
\end{displaymath}">|; 

$key = q/a_i,b_iequiva_1,b_1+a_2,b_2+a_3,b_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1508.png"
 ALT="$a_i b_i\equiv a_1 b_1+a_2 b_2+a_3 b_3$">|; 

$key = q/{displaymath}bar{psi}(k)propto{{rm{e}^{-{{rm{i},(k-k_0),x_0}int_{-infty}^{infty}x_0)-frac{(x-x_0)^2}{4,({mitDelta}x)^2}right]dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="525" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img337.png"
 ALT="\begin{displaymath}
\bar{\psi}(k) \propto
{\rm e}^{-{\rm i} (k-k_0) x_0}\int_{...
...0) (x-x_0) - \frac{(x-x_0)^2}{4 ({\mit\Delta}x)^2}\right]dx.
\end{displaymath}">|; 

$key = q/A,chi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1733.png"
 ALT="$A \chi$">|; 

$key = q/mathop{mathgroupsymoperatorstan}nolimitsy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$\mathop {\mathgroup \symoperators tan}\nolimits y$">|; 

$key = q/R=1.06times10^{-10};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2832.png"
 ALT="$R=1.06\times 10^{-10}$">|; 

$key = q/-78.98;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2775.png"
 ALT="$-78.98$">|; 

$key = q/{displaymath}langleS_zrangle=frac{hbar}{2},cosalpha{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1852.png"
 ALT="\begin{displaymath}
\langle S_z \rangle = \frac{\hbar}{2} \cos\alpha
\end{displaymath}">|; 

$key = q/Z(0)=Z(a)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1216.png"
 ALT="$Z(0)=Z(a)= 0$">|; 

$key = q/displaystylesum_{n>1}|langle1,0,0|z|n,1,0rangle|^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2227.png"
 ALT="$\displaystyle \sum_{n&gt;1}\vert\langle 1,0,0\vert z\vert n,1,0\rangle\vert^2$">|; 

$key = q/displaystylephi(k_0)=k_0,x-omega_0,t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img362.png"
 ALT="$\displaystyle \phi(k_0) = k_0 x-\omega_0 t,$">|; 

$key = q/c=3times10^8,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img236.png"
 ALT="$c=3\times 10^8 {\rm m/s}$">|; 

$key = q/|psi|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img498.png"
 ALT="$\vert\psi\vert^{ 2}$">|; 

$key = q/{displaymath}L^4,z-2,L^2,z,L^2+z,L^4-2,hbar^2,(L^2,z+z,L^2)=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="379" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2204.png"
 ALT="\begin{displaymath}
L^4 z-2 L^2 z L^2 + z L^4 - 2 \hbar^2 (L^2 z+z L^2) = 0.
\end{displaymath}">|; 

$key = q/{displaymath}(a_1-a_2)^2,x,(1-x)=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img696.png"
 ALT="\begin{displaymath}
(a_1-a_2)^2 x (1-x) = 0.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{n}cdot{{bf{r}=(j-varphislash2pi),lambda+v,t,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img185.png"
 ALT="\begin{displaymath}
{\bf n}\cdot{\bf r} = (j-\varphi/2\pi) \lambda + v t,
\end{displaymath}">|; 

$key = q/displaystylel,(l+1),hbar^2,psi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1535.png"
 ALT="$\displaystyle l (l+1) \hbar^2 \psi.$">|; 

$key = q/R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img818.png"
 ALT="$R$">|; 

$key = q/displaystyleE_1,psi_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2120.png"
 ALT="$\displaystyle E_1 \psi_1,$">|; 

$key = q/displaystyle{{bf{k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2586.png"
 ALT="$\displaystyle {\bf k}$">|; 

$key = q/{displaymath}E_f-E_i=hbar,omega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2512.png"
 ALT="\begin{displaymath}
E_f - E_i = \hbar \omega.
\end{displaymath}">|; 

$key = q/W;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img250.png"
 ALT="$W$">|; 

$key = q/int_{-infty}^{infty}dxslash(x^2+a^2)=pislasha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img752.png"
 ALT="$\int_{-\infty}^{\infty} dx/(x^2+a^2) = \pi/a$">|; 

$key = q/{displaymath}psi(x_2,x_1,t)=-psi(x_1,x_2,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1143.png"
 ALT="\begin{displaymath}
\psi(x_2,x_1,t) = -\psi(x_1,x_2,t).
\end{displaymath}">|; 

$key = q/psi^{(2)}_{l,s;j,m_j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1911.png"
 ALT="$\psi^{(2)}_{l,s;j,m_j}$">|; 

$key = q/P_l(costheta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2909.png"
 ALT="$P_l(\cos\theta)$">|; 

$key = q/{{bf{v}'=hbar,{{bf{k}'slashm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2866.png"
 ALT="${\bf v}' = \hbar {\bf k}'/m$">|; 

$key = q/displaystylesqrt{frac{3}{4pi}},costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1476.png"
 ALT="$\displaystyle \sqrt{\frac{3}{4\pi}} \cos\theta,$">|; 

$key = q/displaystyleS_{1z},chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2043.png"
 ALT="$\displaystyle S_{1z}  \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}$">|; 

$key = q/{displaymath}q^2=4,k^2,sin^2(thetaslash2)=2,k^2,(1-costheta).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2887.png"
 ALT="\begin{displaymath}
q^2 = 4 k^2  \sin^2(\theta/2) = 2 k^2  (1-\cos\theta).
\end{displaymath}">|; 

$key = q/c,chi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1720.png"
 ALT="$c \chi$">|; 

$key = q/{displaymath}DeltaE=frac{8}{3},g_p,frac{m_e}{m_p},alpha^2,E_0=5.88times10^{-6},{{rm{eV},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="314" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img2384.png"
 ALT="\begin{displaymath}
\Delta E = \frac{8}{3} g_p \frac{m_e}{m_p} \alpha^2 E_0 = 5.88\times 10^{-6} {\rm eV},
\end{displaymath}">|; 

$key = q/i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img79.png"
 ALT="$i$">|; 

$key = q/k=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2260.png"
 ALT="$k=1$">|; 

$key = q/I(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img283.png"
 ALT="$I(y)$">|; 

$key = q/{displaymath}{{bf{j}=frac{hbar}{m},{{rm{Im}(psi^ast,nablapsi).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2862.png"
 ALT="\begin{displaymath}
{\bf j} = \frac{\hbar}{m} {\rm Im}(\psi^\ast \nabla\psi).
\end{displaymath}">|; 

$key = q/m_s=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2065.png"
 ALT="$m_s=-1$">|; 

$key = q/|bar{psi}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img232.png"
 ALT="$\vert\bar{\psi}\vert$">|; 

$key = q/displaystyle1.73.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img946.png"
 ALT="$\displaystyle 1.73.$">|; 

$key = q/|{{bf{E}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2171.png"
 ALT="$\vert{\bf E}\vert$">|; 

$key = q/psi^{(1)}_{1,pm1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1974.png"
 ALT="$\psi^{(1)}_{1,\pm 1/2}$">|; 

$key = q/langleL_x^{,2}rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1487.png"
 ALT="$\langle L_x^{ 2}\rangle$">|; 

$key = q/{displaymath}l_F=left(frac{3,N}{pi}right)^{1slash3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1249.png"
 ALT="\begin{displaymath}
l_F = \left(\frac{3 N}{\pi}\right)^{1/3}.
\end{displaymath}">|; 

$key = q/{displaymath}q^2=frac{2,m,(V_0+E)}{hbar^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img971.png"
 ALT="\begin{displaymath}
q^2 = \frac{2 m (V_0+E)}{\hbar^2}.
\end{displaymath}">|; 

$key = q/displaystylek_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1234.png"
 ALT="$\displaystyle k_x$">|; 

$key = q/{displaymath}tanleft(sqrt{2,m,(V_0+E)},,Lslashhbarright)=-sqrt{(V_0+E)slash(-E)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="396" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1066.png"
 ALT="\begin{displaymath}
\tan\left(\sqrt{2 m (V_0+E)}  L/\hbar\right) = - \sqrt{(V_0+E)/(-E)}.
\end{displaymath}">|; 

$key = q/displaystyleC_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img945.png"
 ALT="$\displaystyle C_3$">|; 

$key = q/{displaymath}H,psi_n=(n+1slash2),hbar,omega,psi_n=E_n,psi_n{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1044.png"
 ALT="\begin{displaymath}
H \psi_n = (n+1/2) \hbar \omega \psi_n = E_n \psi_n
\end{displaymath}">|; 

$key = q/tau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img937.png"
 ALT="$\tau$">|; 

$key = q/{displaymath}H=-frac{hbar^2}{2,m_e},nabla^2-frac{e^2}{4pi,epsilon_0}left(frac{1}{r_1}+frac{1}{r_2}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2800.png"
 ALT="\begin{displaymath}
H = -\frac{\hbar^2}{2 m_e} \nabla^2 - \frac{e^2}{4\pi \epsilon_0}\left(\frac{1}{r_1}+\frac{1}{r_2}\right).
\end{displaymath}">|; 

$key = q/{displaymath}sqrt{frac{2,m,|V_0|,a^2}{hbar^2}}=frac{pi}{2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img3020.png"
 ALT="\begin{displaymath}
\sqrt{\frac{2 m \vert V_0 \vert a^2}{\hbar^2} } = \frac{\pi}{2}
\end{displaymath}">|; 

$key = q/{displaymath}[L_z,x_pm]=pm,hbar,x_pm.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2641.png"
 ALT="\begin{displaymath}[L_z, x_\pm]= \pm \hbar x_\pm.
\end{displaymath}">|; 

$key = q/displaystyleY_{1,0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1475.png"
 ALT="$\displaystyle Y_{1,0}$">|; 

$key = q/displaystyleS_z,chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2047.png"
 ALT="$\displaystyle S_z  \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}$">|; 

$key = q/{displaymath}s,(s+1)geqm_s,(m_s-1).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1756.png"
 ALT="\begin{displaymath}
s (s+1)\geq m_s (m_s-1).
\end{displaymath}">|; 

$key = q/{displaymath}int_{-infty}^{infty}f(x),delta(x),dx=f(0),int_{-infty}^{infty}delta(x),dx=f(0),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img580.png"
 ALT="\begin{displaymath}
\int_{-\infty}^{\infty}f(x) \delta(x) dx = f(0) \int_{-\infty}^{\infty}\delta(x) dx=f(0),
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{d}_{if}=langlef|e,{{bf{r}|irangle{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2584.png"
 ALT="\begin{displaymath}
{\bf d}_{if} = \langle f\vert e {\bf r}\vert i\rangle
\end{displaymath}">|; 

$key = q/2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="$2$">|; 

$key = q/E=E_0slashn^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2668.png"
 ALT="$E = E_0/n^2$">|; 

$key = q/{displaymath}[L_+,L_-]=2,hbar,L_z.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1340.png"
 ALT="\begin{displaymath}[L_+, L_-]= 2 \hbar L_z.
\end{displaymath}">|; 

$key = q/y=(2,j-1),pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img986.png"
 ALT="$y = (2 j-1) \pi/2$">|; 

$key = q/{displaymath}{{rm{i},hbar,frac{partialpsi}{partialt}=-frac{hbar^2}{2,m}frac{partial^2psi}{partialx^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img435.png"
 ALT="\begin{displaymath}
{\rm i} \hbar \frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2 m}\frac{\partial^2\psi}{\partial x^2}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{dsigma}{dOmega}simeqleft(frac{2,m,Z,Z',e^2}{4pi,epsilon_0,hbar^2}right)^2frac{1}{16,k^4,sin^4(thetaslash2)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img2890.png"
 ALT="\begin{displaymath}
\frac{d\sigma}{d\Omega} \simeq \left(\frac{2 m  Z  Z'  e...
..._0 \hbar^2}\right)^2
\frac{1}{ 16  k^4  \sin^4( \theta/2)}.
\end{displaymath}">|; 

$key = q/{displaymath}sigma_x,sigma_pgeqfrac{hbar}{2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img625.png"
 ALT="\begin{displaymath}
\sigma_x \sigma_p\geq \frac{\hbar}{2},
\end{displaymath}">|; 

$key = q/E_0=-13.6,{{rm{eV};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2723.png"
 ALT="$E_0=-13.6 {\rm eV}$">|; 

$key = q/{displaymath}w_{irightarrowf}^{spn}=frac{omega_{if}^{,3},d_{if}^{,2}}{3pi,epsilon_0,hbar,c^3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2611.png"
 ALT="\begin{displaymath}
w_{i\rightarrow f}^{spn} = \frac{\omega_{if}^{ 3} d_{if}^{ 2}}{3\pi \epsilon_0 \hbar c^3}.
\end{displaymath}">|; 

$key = q/x=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img576.png"
 ALT="$x=0$">|; 

$key = q/{displaymath}frac{d^2psi(x)}{dx^2}={{rm{i},k'(x),psi(x)-k^2(x),psi(x),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img878.png"
 ALT="\begin{displaymath}
\frac{d^2\psi(x)}{dx^2} = {\rm i} k'(x) \psi(x) - k^2(x) \psi(x),
\end{displaymath}">|; 

$key = q/displaystylecotdelta_l(E_0)+left(frac{dcotdelta_l}{dE}right)_{E=E_0}(E-E_0)+cdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3024.png"
 ALT="$\displaystyle \cot \delta_l(E_0) +\left(
\frac{ d \cot\delta_l}{d E}\right)_{E=E_0}(E-E_0) + \cdots$">|; 

$key = q/(3slash2),(a_0slashZ);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1696.png"
 ALT="$(3/2) (a_0/Z)$">|; 

$key = q/{displaymath}int_{-infty}^{infty}psi^{ast},(O,psi),dx=int_{-infty}^infty(O^nomath_inline{dagnomath_inline{,psi)^ast,psi,dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img545.png"
 ALT="\begin{displaymath}
\int_{-\infty}^{\infty} \psi^{\ast}  (O \psi) dx=\int_{-\infty}^\infty
(O^\dag \psi)^\ast \psi dx.
\end{displaymath}">|; 

$key = q/{displaymath}frac{dsigma}{dOmega},dOmega=frac{r^2,dOmega,|{{bf{j}'|}{|{{bf{j}|},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2867.png"
 ALT="\begin{displaymath}
\frac{d\sigma}{d\Omega} d\Omega = \frac{r^2 d\Omega \vert{\bf j}'\vert}{\vert{\bf j}\vert},
\end{displaymath}">|; 

$key = q/displaystylenablatimes{{bf{A}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2530.png"
 ALT="$\displaystyle \nabla\times {\bf A}.$">|; 

$key = q/n=1,2,3,ldots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1566.png"
 ALT="$n=1,2,3,\ldots$">|; 

$key = q/displaystylefrac{partial}{partialz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1355.png"
 ALT="$\displaystyle \frac{\partial}{\partial z}$">|; 

$key = q/0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$0$">|; 

$key = q/V=V_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2989.png"
 ALT="$V=V_0$">|; 

$key = q/{displaymath}psi(x,t)=A,cos(k,x-omega,t+varphi).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img170.png"
 ALT="\begin{displaymath}
\psi(x,t)=A \cos(k x-\omega t+\varphi).
\end{displaymath}">|; 

$key = q/rho(omega),domega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2556.png"
 ALT="$\rho(\omega) d\omega$">|; 

$key = q/3,1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="$3,1$">|; 

$key = q/V_0,a^2<pi^2,hbar^2slash8,m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1683.png"
 ALT="$V_0 a^2&lt; \pi^2 \hbar^2/8 m$">|; 

$key = q/exp(-{{rm{i},omega,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img817.png"
 ALT="$\exp(-{\rm i} \omega t)$">|; 

$key = q/H_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1095.png"
 ALT="$H_i$">|; 

$key = q/displaystylesqrt{frac{2}{a}}sin(k_y,y),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1231.png"
 ALT="$\displaystyle \sqrt{\frac{2}{a}}\sin (k_y y),$">|; 

$key = q/1.215times10^{-7};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2670.png"
 ALT="$1.215\times 10^{-7}$">|; 

$key = q/{{bf{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img307.png"
 ALT="${\bf p}$">|; 

$key = q/{displaymath}frac{d}{dt}int_{-infty}^{infty}int_{-infty}^{infty}int_{-infty}^{infty}|psi(x,y,z,t)|^2,dx,dy,dz=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="346" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1188.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \vert\psi(x,y,z,t)\vert^2 dx dy dz = 0.
\end{displaymath}">|; 

$key = q/psi(x,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img428.png"
 ALT="$\psi(x,0)$">|; 

$key = q/{displaymath}nabla^2psi=frac{2,m}{hbar^2},(V-E),psi,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1211.png"
 ALT="\begin{displaymath}
\nabla^2\psi = \frac{2 m}{\hbar^2} (V-E) \psi,
\end{displaymath}">|; 

$key = q/{displaymath}langlen,l'^{(1)},m|H_1|n,l^{(1)},mrangle=lambda_{nl},delta_{ll'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2250.png"
 ALT="\begin{displaymath}
\langle n,l'^{(1)},m\vert H_1\vert n,l^{(1)},m\rangle =\lambda_{nl}  \delta_{ll'}.
\end{displaymath}">|; 

$key = q/{displaymath}{{rm{i},hbarleft(array{{c}dot{c}_+dot{c}_-array{right)=frac{Omega,h&-1array{right)left(array{{c}c_+c_-array{right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1838.png"
 ALT="\begin{displaymath}
{\rm i} \hbar\left(\begin{array}{c}\dot{c}_+ \dot{c}_-\en...
...rray}\right)\left(\begin{array}{c}c_+ c_-\end{array}\right),
\end{displaymath}">|; 

$key = q/{displaymath}Y_{l,m}(u,phi)sim{{rm{e}^{,{{rm{i},m,phi},(1-u^2)^{-mslash2}left(frac{d}{du}right)^{l-m}(1-u^2)^l.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="404" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1459.png"
 ALT="\begin{displaymath}
Y_{l,m}(u,\phi)\sim {\rm e}^{ {\rm i} m \phi} (1-u^2)^{-m/2}\left(\frac{d}{d u}\right)^{l-m}
(1-u^2)^l.
\end{displaymath}">|; 

$key = q/{{mitSigma}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img83.png"
 ALT="${{\mit\Sigma}}$">|; 

$key = q/l=0,1,2,cdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2920.png"
 ALT="$l=0, 1, 2, \cdots$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter08slashfig02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="357" HEIGHT="326" BORDER="0"
 SRC="|."$dir".q|img1472.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter08/fig02.eps}}
\end{figure}">|; 

$key = q/displaystyleleft(array{{cc}1,<col_mark>0<row_mark>0,<col_mark>-1array{right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1814.png"
 ALT="$\displaystyle \left(\begin{array}{cc}1,&amp;0\\\\
0,&amp; -1\end{array}\right).$">|; 

$key = q/{displaymath}lambda=1.29times10^{-9},V^{-1slash2},{{rm{m},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img449.png"
 ALT="\begin{displaymath}
\lambda = 1.29\times 10^{-9} V^{-1/2} {\rm m},
\end{displaymath}">|; 

$key = q/|y|gg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1003.png"
 ALT="$\vert y\vert\gg 1$">|; 

$key = q/displaystylepsi^{(1)}_{-1,1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2001.png"
 ALT="$\displaystyle \psi^{(1)}_{-1,1/2}$">|; 

$key = q/y=(x^2+X^2-2,x,X,costheta)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2791.png"
 ALT="$y=(x^2+X^2-2 x X \cos\theta)^{1/2}$">|; 

$key = q/{displaymath}{{bf{L}cdot{{bf{S}=frac{1}{2},(J^2-L^2-S^2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img2318.png"
 ALT="\begin{displaymath}
{\bf L}\cdot {\bf S} = \frac{1}{2} (J^2-L^2-S^2).
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{S}={{bf{S}_e+{{bf{S}_p{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2376.png"
 ALT="\begin{displaymath}
{\bf S} = {\bf S}_e + {\bf S}_p
\end{displaymath}">|; 

$key = q/{displaymath}(S_+,chi_{s,m_s})^nomath_inline{dagnomath_inline{,(S_+,chi_{s,m_s})nline{dagnomath_inline{,S_-,S_+,chi_{s,m_s}geq0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="534" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img1753.png"
 ALT="\begin{displaymath}
(S_+ \chi_{s,m_s})^\dag (S_+ \chi_{s,m_s}) = \chi_{s,m_s}...
...chi_{s,m_s} = \chi_{s,m_s}^\dag S_- S_+ \chi_{s,m_s}\geq 0,
\end{displaymath}">|; 

$key = q/|psi_2|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img896.png"
 ALT="$\vert\psi_2\vert^{ 2}$">|; 

$key = q/{displaymath}H_0=frac{p^2}{2,m_e}+V_0(r).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2519.png"
 ALT="\begin{displaymath}
H_0 = \frac{p^2}{2  m_e} + V_0(r).
\end{displaymath}">|; 

$key = q/displaystylem,hbar,psi^{(1)}_{l,s;m,m_s},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1904.png"
 ALT="$\displaystyle m \hbar \psi^{(1)}_{l,s;m,m_s},$">|; 

$key = q/{displaymath}leftlangle({mitDelta}u)^2rightrangle=sum_{i=1}^MP(u_i),(u_i-langleurangle)^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img112.png"
 ALT="\begin{displaymath}
\left\langle ({\mit\Delta} u)^2\right\rangle = \sum_{i=1}^M P(u_i) (u_i - \langle u\rangle)^2,
\end{displaymath}">|; 

$key = q/displaystyleleftlanglefrac{1}{r^3}rightrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1661.png"
 ALT="$\displaystyle \left\langle\frac{1}{r^3}\right\rangle$">|; 

$key = q/{displaymath}lambda=frac{V_0}{E_0}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img979.png"
 ALT="\begin{displaymath}
\lambda = \frac{V_0}{E_0}.
\end{displaymath}">|; 

$key = q/k^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img882.png"
 ALT="$k^{-1}$">|; 

$key = q/nL_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2326.png"
 ALT="$nL_j$">|; 

$key = q/{displaymath}H_{1,2}=-frac{hbar^2}{2,m_e},nabla^{,2}_{1,2}-frac{2,e^2}{4pi,epsilon_0,r_{1,2}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2711.png"
 ALT="\begin{displaymath}
H_{1,2} = -\frac{\hbar^2}{2 m_e} \nabla^{ 2}_{1,2} -\frac{2 e^2}{4\pi \epsilon_0 r_{1,2}}.
\end{displaymath}">|; 

$key = q/displaystylelangleprangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img515.png"
 ALT="$\displaystyle \langle p\rangle,$">|; 

$key = q/{{bf{r}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2707.png"
 ALT="${\bf r}_1$">|; 

$key = q/{displaymath}int_{-infty}^inftyint_{-infty}^inftyint_{-infty}^inftypsi_i^ast,psi_j,dx,dy,dz=delta_{ij}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img2100.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty \psi_i^\ast \psi_j dx dy dz = \delta_{ij}.
\end{displaymath}">|; 

$key = q/B,psi_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img701.png"
 ALT="$B \psi_i$">|; 

$key = q/{displaymath}H_1=e,|{{bf{E}|,z.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2173.png"
 ALT="\begin{displaymath}
H_1 = e \vert{\bf E}\vert z.
\end{displaymath}">|; 

$key = q/a_pm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1067.png"
 ALT="$a_\pm$">|; 

$key = q/d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img174.png"
 ALT="$d$">|; 

$key = q/{displaymath}(nabla^2+k^2),psi=frac{2,m}{hbar^2},V,psi.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2845.png"
 ALT="\begin{displaymath}
(\nabla^2+k^2) \psi = \frac{2 m}{\hbar^2} V \psi.
\end{displaymath}">|; 

$key = q/displaystyle-leftlanglefrac{dV}{dx}rightrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img516.png"
 ALT="$\displaystyle -\left\langle \frac{dV}{dx}\right\rangle.$">|; 

$key = q/epsilon_0=8.85times10^{-12},{{rm{Fslashm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img244.png"
 ALT="$\epsilon_0= 8.85\times 10^{-12} {\rm F/m}$">|; 

$key = q/{displaymath}Y_{l,-l+1}(theta,phi)simL_+,Y_{l,-l}simfrac{{{rm{e}^{-{{rm{i},(l-1){sintheta}frac{d}{dtheta}right)(sintheta)^{2,l}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="445" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1451.png"
 ALT="\begin{displaymath}
Y_{l,-l+1}(\theta,\phi)\sim L_+ Y_{l,-l}\sim \frac{{\rm e}^...
...ac{1}{\sin\theta}\frac{d}{d\theta}\right)
(\sin\theta)^{2 l}.
\end{displaymath}">|; 

$key = q/3D_{3slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2336.png"
 ALT="$3D_{3/2}$">|; 

$key = q/k,a+delta_0=pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3016.png"
 ALT="$k a+\delta_0 = \pi/2$">|; 

$key = q/a_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img680.png"
 ALT="$a_i$">|; 

$key = q/displaystyleB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1284.png"
 ALT="$\displaystyle B$">|; 

$key = q/displaystylepsi^{(1)}_{1,-1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1995.png"
 ALT="$\displaystyle \psi^{(1)}_{1,-1/2}$">|; 

$key = q/{displaymath}H_0,psi_0=E,psi_0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2840.png"
 ALT="\begin{displaymath}
H_0 \psi_0= E \psi_0,
\end{displaymath}">|; 

$key = q/{mitDelta}k=1slash(2,{mitDelta}x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img399.png"
 ALT="${\mit\Delta}k= 1/(2 {\mit\Delta} x)$">|; 

$key = q/{displaymath}w_n=frac{omega_{n,n-1}^{,3},(d_x)_{n,n-1}^{,2}}{3pi,epsilon_0,hbar,c^3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2631.png"
 ALT="\begin{displaymath}
w_n = \frac{\omega_{n,n-1}^{ 3} (d_x)_{n,n-1}^{ 2}}{3\pi \epsilon_0 \hbar c^3}.
\end{displaymath}">|; 

$key = q/|psi_2|^{,2}<|psi_1|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img894.png"
 ALT="$\vert\psi_2\vert^{ 2} &lt; \vert\psi_1\vert^{ 2}$">|; 

$key = q/{displaymath}L^2=epsilon_{ijk},x_j,p_k,epsilon_{ilm},x_l,p_m=epsilon_{ijk},epsilon_{ilm},x_j,p_k,x_l,p_m.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1510.png"
 ALT="\begin{displaymath}
L^2 = \epsilon_{ijk} x_j p_k \epsilon_{ilm} x_l p_m =
\epsilon_{ijk} \epsilon_{ilm} x_j p_k x_l p_m.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{Chapter14slashfig01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="400" BORDER="0"
 SRC="|."$dir".q|img2777.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{Chapter14/fig01.eps}}
\end{figure}">|; 

$key = q/{displaymath}Y_{l,m}(theta,phi)=(-1)^m,left[frac{2,l+1}{4pi},frac{(l-m)!}{(l+m)!lash2}P_{l,m}(costheta),{{rm{e}^{,{{rm{i},m,phi}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="455" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1463.png"
 ALT="\begin{displaymath}
Y_{l,m}(\theta,\phi) =(-1)^m  \left[\frac{2 l+1}{4\pi} \f...
...right]^{1/2} P_{l,m}(\cos\theta) {\rm e}^{ {\rm i} m \phi}
\end{displaymath}">|; 

$key = q/1,pm1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="$1,\pm 1$">|; 

$key = q/(A,B)^ast=A^ast,B^ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img474.png"
 ALT="$(A B)^\ast = A^\ast B^\ast$">|; 

$key = q/displaystylefrac{2}{(2,a_0)^{3slash2}}left(1-frac{r}{2,a_0}right)expleft(-frac{r}{2,a_0}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1641.png"
 ALT="$\displaystyle \frac{2}{(2 a_0)^{3/2}}\left(1-\frac{r}{2 a_0}\right)
\exp\left(-\frac{r}{2 a_0}\right),$">|; 

$key = q/displaystyle-left[frac{{{rm{i},(omega-omega_{21})}{2,Omega}right]exp!left[frac{{{rm{i},(omega-omega_{21}),t}{2}right]sin(Omega,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2425.png"
 ALT="$\displaystyle - \left[\frac{{\rm i} (\omega-\omega_{21})}{2 \Omega} \right]\exp\!\left[\frac{{\rm i} (\omega-\omega_{21}) t}{2}\right]\sin(\Omega t),$">|; 

$key = q/EllE_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img924.png"
 ALT="$E\ll E_c$">|; 

$key = q/displaystyle{{rm{Re}left[psi_0,{{rm{e}^{,{{rm{i},(k,x-omega,t)}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img220.png"
 ALT="$\displaystyle {\rm Re}\left[\psi_0 {\rm e}^{ {\rm i} (k x-\omega t)}\right]$">|; 

$key = q/B,psi_iproptopsi_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img702.png"
 ALT="$B \psi_i\propto \psi_i$">|; 

$key = q/z=int_{-infty}^inftyf^ast,g,dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img615.png"
 ALT="$z=\int_{-\infty}^\infty f^\ast g dx$">|; 

$key = q/displaystyleL_z,psi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1532.png"
 ALT="$\displaystyle L_z \psi$">|; 

$key = q/N_{18};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img91.png"
 ALT="$N_{18}$">|; 

$key = q/displaystyleDeltaE_{nlm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2177.png"
 ALT="$\displaystyle \Delta E_{nlm}$">|; 

$key = q/displaystyles,(s+1),hbar^2,chi_{s_1,s_2;s,m_s}^{(2)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2055.png"
 ALT="$\displaystyle s (s+1) \hbar^2
 \chi_{s_1,s_2;s,m_s}^{(2)},$">|; 

$key = q/P(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img286.png"
 ALT="$P(y)$">|; 

$key = q/{displaymath}c_f(t)=-frac{{rm{i}{hbar}int_0^tleft[V_{fi},exp(,{{rm{i},omega,t')+,omega,t')right]exp(,{{rm{i},omega_{fi},t'),dt',{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="518" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2482.png"
 ALT="\begin{displaymath}
c_f(t) = - \frac{\rm i}{\hbar}\int_0^t\left[V_{fi} \exp( {...
... i} \omega t')\right]
\exp( {\rm i} \omega_{fi} t') dt',
\end{displaymath}">|; 

$key = q/{{bf{S}_e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2371.png"
 ALT="${\bf S}_e$">|; 

$key = q/{displaymath}H_1simeq-frac{e,{{bf{A}!cdot!{{bf{p}}{m_e},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2545.png"
 ALT="\begin{displaymath}
H_1 \simeq -\frac{e {\bf A}\!\cdot\!{\bf p}}{m_e},
\end{displaymath}">|; 

$key = q/c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img108.png"
 ALT="$c$">|; 

$key = q/displaystylee_{22};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2131.png"
 ALT="$\displaystyle e_{22}$">|; 

$key = q/{displaymath}psi_a''=frac{|c|}{sqrt{1-|c|^2}}left(psi_a-c^{-1},psi_a'right){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="61" BORDER="0"
 SRC="|."$dir".q|img675.png"
 ALT="\begin{displaymath}
\psi_a'' = \frac{\vert c\vert}{\sqrt{1-\vert c\vert^2}}\left(\psi_a - c^{-1} \psi_a'\right)
\end{displaymath}">|; 

$key = q/displaystyle-mbox{{boldmath{{mu{}cdot{{bf{B};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2311.png"
 ALT="$\displaystyle - \mbox{\boldmath$\mu$}\cdot {\bf B}$">|; 

$key = q/P_{x,in,a:b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img481.png"
 ALT="$P_{x \in a:b}$">|; 

$key = q/displaystylechi^{(2)}_{1,1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2081.png"
 ALT="$\displaystyle \chi^{(2)}_{1,1}$">|; 

$key = q/{displaymath}frac{zeta}{2}=n,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1617.png"
 ALT="\begin{displaymath}
\frac{\zeta}{2} = n,
\end{displaymath}">|; 

$key = q/{displaymath}H=frac{L_x^{,2}+L_y^{,2}}{2,I_1}+frac{L_z^{,2}}{2,I_2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1489.png"
 ALT="\begin{displaymath}
H = \frac{L_x^{ 2}+L_y^{ 2}}{2 I_1} + \frac{L_z^{ 2}}{2 I_2}.
\end{displaymath}">|; 

$key = q/chi^nomath_inline{dagnomath_inline{;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1726.png"
 ALT="$\chi^\dag $">|; 

$key = q/displaystyleL_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1308.png"
 ALT="$\displaystyle L_z$">|; 

$key = q/{displaymath}tan(k,a+delta_0)=frac{k}{kappa},tanh(kappa,a){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img3001.png"
 ALT="\begin{displaymath}
\tan(k a+ \delta_0) = \frac{k}{\kappa}  \tanh( \kappa  a)
\end{displaymath}">|; 

$key = q/dslashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img527.png"
 ALT="$d/dx$">|; 

$key = q/langlen,l,m|z|n',l',m'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2180.png"
 ALT="$\langle n,l,m\vert z\vert n',l',m'\rangle$">|; 

$key = q/displaystyle{{bf{x}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2276.png"
 ALT="$\displaystyle {\bf x}_1$">|; 

$key = q/{displaymath}langlen,l',m|H_1|n,l,mrangle=lambda_{nl},delta_{ll'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2244.png"
 ALT="\begin{displaymath}
\langle n,l',m\vert H_1\vert n,l,m\rangle = \lambda_{nl} \delta_{ll'}.
\end{displaymath}">|; 

$key = q/psi(r,theta,phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1492.png"
 ALT="$\psi(r,\theta,\phi)$">|; 

$key = q/{displaymath}tilde{calR}_0(r)=frac{sin(k,r)}{k,r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2967.png"
 ALT="\begin{displaymath}
\tilde{\cal R}_0(r) = \frac{ \sin (k r)}{k r}.
\end{displaymath}">|; 

$key = q/{displaymath}int_1^{1slashepsilon}left[frac{1}{y}-epsilonright]^{1slash2}dysimeqfrac{pi}{2,sqrt{epsilon}}-2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img925.png"
 ALT="\begin{displaymath}
\int_1^{1/\epsilon}\left[\frac{1}{y} - \epsilon\right]^{1/2} dy \simeq
\frac{\pi}{2 \sqrt{\epsilon}}-2
\end{displaymath}">|; 

$key = q/{displaymath}P=-{{rm{i},hbarleft(frac{partial}{partialx_1}+frac{partial}{partial_2}right)=-{{rm{i},hbar,frac{partial}{partialX}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1120.png"
 ALT="\begin{displaymath}
P= -{\rm i} \hbar\left(\frac{\partial}{\partial x_1} + \fra...
...l x_2}\right) = -{\rm i} \hbar \frac{\partial}{\partial X}.
\end{displaymath}">|; 

$key = q/x>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1056.png"
 ALT="$x&gt;0$">|; 

$key = q/{displaymath}psi({{bf{r})simeqsqrt{n}sum_lleft[A_l,frac{sin(k,r-l,pislash2)}{k,rac{cos(k,r-l,pislash2)}{k,r}right]P_l(costheta),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="536" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img2926.png"
 ALT="\begin{displaymath}
\psi ({\bf r} ) \simeq \sqrt{n} \sum_l\left[A_l 
\frac{\sin...
..._l \frac{\cos(k r -l \pi/2)}{k r}
\right] P_l(\cos\theta),
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{sqrt{2pi}}int_{-infty}^inftypsi(x,t),{{rm{e}^{-{{rm{i},k,x},dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img569.png"
 ALT="$\displaystyle \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^\infty \psi(x,t) {\rm e}^{-{\rm i} k x} dx,$">|; 

$key = q/{displaymath}e_{n'l'nl}=langlen',l',m|H_1|n,l,mrangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2238.png"
 ALT="\begin{displaymath}
e_{n'l'nl} = \langle n',l',m\vert H_1\vert n,l,m\rangle.
\end{displaymath}">|; 

$key = q/d=sqrt{hbarslashm,omega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1027.png"
 ALT="$d=\sqrt{\hbar/m \omega}$">|; 

$key = q/r_2=(r^2+R^2-2,r,R,costheta)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2788.png"
 ALT="$r_2=(r^2+R^2-2 r R \cos\theta)^{1/2}$">|; 

$key = q/{displaymath}frac{d^2psi(x)}{dx^2}=-k^2(x),psi(x),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img873.png"
 ALT="\begin{displaymath}
\frac{d^2\psi(x)}{dx^2} = - k^2(x) \psi(x),
\end{displaymath}">|; 

$key = q/sgeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1757.png"
 ALT="$s\geq 0$">|; 

$key = q/{displaymath}psi(y,t)proptopsi_1(t),{{rm{e}^{,{{rm{i},k,x_1}+psi_2(t),{{rm{e}^{,{{rm{i},k,x_2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img269.png"
 ALT="\begin{displaymath}
\psi(y,t) \propto \psi_1(t) {\rm e}^{ {\rm i} k x_1} + \psi_2(t) {\rm e}^{ {\rm i} k x_2},
\end{displaymath}">|; 

$key = q/{displaymath}alpha=frac{9}{2},4piepsilon_0,a_0^{,3}=4.5,,4piepsilon_0,a_0^{,3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img2234.png"
 ALT="\begin{displaymath}
\alpha = \frac{9}{2} 4\pi\epsilon_0 a_0^{ 3} = 4.5  4\pi\epsilon_0 a_0^{ 3}.
\end{displaymath}">|; 

$key = q/omega=omega_{fi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2511.png"
 ALT="$\omega=\omega_{fi}$">|; 

$key = q/{displaymath}psi_0({{bf{r})simeqsqrt{n}sum_l{{rm{i}^{,l},(2l+1)left[frac{{{rm{e}-l,pislash2)}}{2,{{rm{i},k,r}right]P_l(costheta){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="495" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img2932.png"
 ALT="\begin{displaymath}
\psi_0({\bf r}) \simeq \sqrt{n} \sum_l {\rm i}^{ l} 
(2l+1...
... (k r - l \pi/2)}}{2 {\rm i} k r} \right]P_l(\cos\theta)
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter03slashfig07.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="327" BORDER="0"
 SRC="|."$dir".q|img334.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter03/fig07.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{Chapter14slashfig02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="403" HEIGHT="385" BORDER="0"
 SRC="|."$dir".q|img2834.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{Chapter14/fig02.eps}}
\end{figure}">|; 

$key = q/mathop{mathgroupsymoperatorslog}nolimits_{10}(tau_{th})=-28.9-1.60,Z_1^{,2slash3}+1.61,Z_1slashsqrt{E};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="$\mathop {\mathgroup \symoperators log}\nolimits _{10}(\tau _{th}) = -28.9 - 1.60 Z_1^{ 2/3} + 1.61 Z_1/\sqrt {E}$">|; 

$key = q/chi^{(2)}_{s,m_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2083.png"
 ALT="$\chi^{(2)}_{s,m_s}$">|; 

$key = q/L_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1319.png"
 ALT="$L_x$">|; 

$key = q/{displaymath}P_{x,in,a:b}(t)proptoint_{a}^{b}|psi(x,t)|^{,2},dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img488.png"
 ALT="\begin{displaymath}
P_{x \in  a:b}(t) \propto \int_{a}^{b}\vert\psi(x,t)\vert^{ 2} dx.
\end{displaymath}">|; 

$key = q/{displaymath}u_l(0)=0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2956.png"
 ALT="\begin{displaymath}
u_l(0) = 0
\end{displaymath}">|; 

$key = q/chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2038.png"
 ALT="$\chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}$">|; 

$key = q/{displaymath}psi(x,t)=psi(x),{{rm{e}^{-{{rm{i},E,tslashhbar},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img786.png"
 ALT="\begin{displaymath}
\psi(x,t) = \psi(x) {\rm e}^{-{\rm i} E t/\hbar},
\end{displaymath}">|; 

$key = q/v_0-{mitDelta}vslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img407.png"
 ALT="$v_0-{\mit\Delta}v/2$">|; 

$key = q/(4slash3),mu_B,B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2357.png"
 ALT="$(4/3) \mu_B B$">|; 

$key = q/{displaymath}chi^nomath_inline{dagnomath_inline{_{s,m_s},chi_{s',m_s'}=delta_{ss'},delta_{m_sm_s'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img1750.png"
 ALT="\begin{displaymath}
\chi^\dag _{s,m_s} \chi_{s',m_s'} =\delta_{ss'} \delta_{m_s m_s'}.
\end{displaymath}">|; 

$key = q/delta_0simeqpislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3017.png"
 ALT="$\delta_0 \simeq \pi/2$">|; 

$key = q/displaystyleY(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1230.png"
 ALT="$\displaystyle Y(y)$">|; 

$key = q/{displaymath}psi_a=sum_ilanglepsi_i|psi_arangle,psi_iequivsum_ilanglei|arangle,psi_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2108.png"
 ALT="\begin{displaymath}
\psi_a = \sum_i\langle\psi_i\vert\psi_a\rangle \psi_i\equiv \sum_i \langle i\vert a\rangle \psi_i.
\end{displaymath}">|; 

$key = q/{displaymath}Z=frac{27}{16}=1.69.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img2773.png"
 ALT="\begin{displaymath}
Z = \frac{27}{16} = 1.69.
\end{displaymath}">|; 

$key = q/{displaymath}psi'_{nlm}=psi_{nlm}+sum_{n',l'=lpm1}frac{e_{n'l'nl}}{E_n-E_{n'}},psi_{n'l'm},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="313" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2237.png"
 ALT="\begin{displaymath}
\psi'_{nlm} = \psi_{nlm} + \sum_{n',l'=l\pm 1}\frac{e_{n'l'nl}}{E_n-E_{n'}} \psi_{n'l'm},
\end{displaymath}">|; 

$key = q/{displaymath}ointY_{l',m'}^{,ast},Y_{l,m},dOmega=delta_{ll'},delta_{mm'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1636.png"
 ALT="\begin{displaymath}
\oint Y_{l',m'}^{ \ast} Y_{l,m} d\Omega = \delta_{ll'} \delta_{mm'}.
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}x=x_2-x_1simeqfrac{d}{D},y,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img267.png"
 ALT="\begin{displaymath}
{\mit\Delta} x = x_2-x_1 \simeq \frac{d}{D} y,
\end{displaymath}">|; 

$key = q/psi^{(1)}_{0,1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2004.png"
 ALT="$\psi^{(1)}_{0,1/2}$">|; 

$key = q/{displaymath}E_{bind}=E_{total}-E_0=-(F_++1),E_0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2827.png"
 ALT="\begin{displaymath}
E_{bind} = E_{total} - E_0 = - (F_+ +1) E_0.
\end{displaymath}">|; 

$key = q/displaystyleV_{fi}^nomath_inline{dagnomath_inline{;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2485.png"
 ALT="$\displaystyle V_{fi}^\dag $">|; 

$key = q/h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img249.png"
 ALT="$h$">|; 

$key = q/{displaymath}{{rm{i},hbar,frac{partialpsi_X}{partialt}=-frac{hbar^2}{2,M}frac{partial^2psi_X}{partialX^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1124.png"
 ALT="\begin{displaymath}
{\rm i} \hbar \frac{\partial\psi_X}{\partial t} = -\frac{\hbar^2}{2 M}\frac{\partial^2\psi_X}{\partial X^2}.
\end{displaymath}">|; 

$key = q/g(u);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img105.png"
 ALT="$g(u)$">|; 

$key = q/y=r,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img208.png"
 ALT="$y=r \sin\theta$">|; 

$key = q/{displaymath}langleV_{ee}rangle=-frac{4,E_0}{pi^2},int{{rm{e}^{-2,r_2},I({{bf{r}_2),d^3{{bf{r}_2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2732.png"
 ALT="\begin{displaymath}
\langle V_{ee}\rangle = -\frac{4 E_0}{\pi^2} \int {\rm e}^{-2 r_2} I({\bf r}_2) d^3{\bf r}_2,
\end{displaymath}">|; 

$key = q/k=pslashhbar>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img313.png"
 ALT="$k= p/\hbar&gt;0$">|; 

$key = q/displaystyleA,{{rm{Re}left[{{rm{e}^{,{{rm{i},(k,x-omega,t+varphi)}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img223.png"
 ALT="$\displaystyle A {\rm Re}\left[{\rm e}^{ {\rm i} (k x-\omega t+\varphi)}\right].$">|; 

$key = q/{displaymath}mu=frac{m_e,m_p}{m_e+m_p}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1582.png"
 ALT="\begin{displaymath}
\mu = \frac{m_e m_p}{m_e+m_p}
\end{displaymath}">|; 

$key = q/textstyle=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img122.png"
 ALT="$\textstyle =$">|; 

$key = q/z=k,r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1545.png"
 ALT="$z=k r$">|; 

$key = q/pequiv-{{rm{i},hbar,partialslashpartialx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img623.png"
 ALT="$p\equiv-{\rm i} \hbar \partial/\partial x$">|; 

$key = q/P(x,y,z,t),dx,dy,dz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1177.png"
 ALT="$P(x,y,z,t) dx dy dz$">|; 

$key = q/displaystyle-[l,(l+1)-m,(m+1)]^{1slash2},alpha+(x+m+1),beta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="369" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1947.png"
 ALT="$\displaystyle -[l (l+1)-m (m+1)]^{1/2} \alpha +(x+m+1) \beta$">|; 

$key = q/{calR}_l(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2946.png"
 ALT="${\cal R}_l(r)$">|; 

$key = q/displaystyleS_z,chi_{s,m_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1742.png"
 ALT="$\displaystyle S_z \chi_{s,m_s}$">|; 

$key = q/{displaymath}H,psi_n=E_n,psi_n.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2690.png"
 ALT="\begin{displaymath}
H \psi_n = E_n \psi_n.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=4in{centerline{epsffile{Chapter05slashfig06.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="490" HEIGHT="435" BORDER="0"
 SRC="|."$dir".q|img948.png"
 ALT="\begin{figure}
\epsfysize =4in
\centerline{\epsffile{Chapter05/fig06.eps}}
\end{figure}">|; 

$key = q/{displaymath}int_{-infty}^inftypsi^ast(x,E_a),psi(x,E_b),dx=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img1157.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \psi^\ast(x,E_a) \psi(x,E_b) dx = 0.
\end{displaymath}">|; 

$key = q/{displaymath}langleArangle=int_{-infty}^inftypsi_a^{ast},A,psi_a,dx=a,int_{-infty}^inftypsi_a^{ast},psi_a,dx=a,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img651.png"
 ALT="\begin{displaymath}
\langle A\rangle = \int_{-\infty}^\infty \psi_a^{\ast} A \...
...a dx
= a \int_{-\infty}^\infty \psi_a^{\ast} \psi_a dx =a,
\end{displaymath}">|; 

$key = q/{displaymath}psi({{bf{r})=sqrt{n}left[exp(,{{rm{i},k,rcostheta)+frac{exp(,{{rm{i},k,r),f(theta)}{r}right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="377" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2906.png"
 ALT="\begin{displaymath}
\psi({\bf r}) = \sqrt{n}
\left[ \exp( {\rm i} k r\cos\theta) + \frac{\exp( {\rm i} k r)  f(\theta)}
{r} \right],
\end{displaymath}">|; 

$key = q/{displaymath}|R|^{,2}=frac{(k^2+q^2)^{,2},sinh^2(q,a)}{4,k^2,q^2+(k^2+q^2)^{,2},sinh^2(q,a)},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img862.png"
 ALT="\begin{displaymath}
\vert R\vert^{ 2} = \frac{(k^2+q^2)^{ 2} \sinh^2(q a)}{4 k^2 q^2 + (k^2+q^2)^{ 2} \sinh^2(q a)},
\end{displaymath}">|; 

$key = q/{displaymath}-{{rm{i},hbarfrac{dPhi_m}{dphi}=m,hbar,Phi_m.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1396.png"
 ALT="\begin{displaymath}
-{\rm i} \hbar\frac{d\Phi_m}{d\phi} = m \hbar \Phi_m.
\end{displaymath}">|; 

$key = q/{displaymath}sigmasimfrac{hbar,t}{m,{mitDelta}x}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img403.png"
 ALT="\begin{displaymath}
\sigma\sim \frac{\hbar t}{m {\mit\Delta}x}.
\end{displaymath}">|; 

$key = q/{displaymath}langle(x_1-x_2)^2rangle_{dist}=langlex^2rangle_a+langlex^2rangle_b-2,langlexrangle_a,langlexrangle_b,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1152.png"
 ALT="\begin{displaymath}
\langle (x_1-x_2)^2\rangle_{dist} = \langle x^2\rangle_a + \langle x^2\rangle_b - 2 \langle x\rangle_a \langle x\rangle_b,
\end{displaymath}">|; 

$key = q/displaystylecos(alphaslash2),exp(-{{rm{i},Omega,tslash2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1846.png"
 ALT="$\displaystyle \cos(\alpha/2) \exp(-{\rm i} \Omega t/2),$">|; 

$key = q/{displaymath}L^2=-hbar^2left[frac{1}{sintheta}frac{partial}{partialtheta}left(si{sin^2theta}frac{partial^2}{partialphi^2}right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1360.png"
 ALT="\begin{displaymath}
L^2 = -\hbar^2\left[\frac{1}{\sin\theta}\frac{\partial}{\par...
...frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right],
\end{displaymath}">|; 

$key = q/{{bf{k}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2881.png"
 ALT="${\bf k}'$">|; 

$key = q/rgeqa;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2991.png"
 ALT="$r\geq a$">|; 

$key = q/psi_p(x,p');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img724.png"
 ALT="$\psi_p(x,p')$">|; 

$key = q/{displaymath}L_+,L_-,Y_{l,m}=c^+_{l,m-1},c^-_{l,m},Y_{l,m}=[l,(l+1)-m,(m-1)],hbar^2,Y_{l,m},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="487" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img1383.png"
 ALT="\begin{displaymath}
L_+ L_- Y_{l,m} = c^+_{l,m-1} c^-_{l,m} Y_{l,m} = [l (l+1)-m (m-1)] \hbar^2 Y_{l,m},
\end{displaymath}">|; 

$key = q/{scriptstyle-sqrt{1slash3}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2021.png"
 ALT="${\scriptstyle -\sqrt{1/3}}$">|; 

$key = q/displaystyles_2,(s_2+1),hbar^2,chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2042.png"
 ALT="$\displaystyle s_2 (s_2+1) \hbar^2
 \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)},$">|; 

$key = q/{displaymath}P_n(t)=|c_n(t)|^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2469.png"
 ALT="\begin{displaymath}
P_n(t)= \vert c_n(t)\vert^2
\end{displaymath}">|; 

$key = q/E=p,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1296.png"
 ALT="$E=p c$">|; 

$key = q/{displaymath}sum_{l=1,N_n}|n,l,mranglelanglen,l,m|equiv1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2251.png"
 ALT="\begin{displaymath}
\sum_{l=1,N_n}\vert n,l,m\rangle\langle n,l,m\vert\equiv 1,
\end{displaymath}">|; 

$key = q/displaystyle[L_x,L_y];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1315.png"
 ALT="$\displaystyle [L_x, L_y]$">|; 

$key = q/{displaymath}E_n'=E_n+e_{nn}+sum_{kneqn}frac{|e_{nk}|^2}{E_n-E_k}+{calO}(epsilon^3),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img2169.png"
 ALT="\begin{displaymath}
E_n' = E_n + e_{nn} + \sum_{k\neq n}\frac{\vert e_{nk}\vert^2}{E_n-E_k}
+ {\cal O}(\epsilon^3),
\end{displaymath}">|; 

$key = q/{displaymath}C_l=(2,l+1),exp[,{{rm{i},(delta_l+l,pislash2)].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2934.png"
 ALT="\begin{displaymath}
C_l = (2 l+1) \exp[ {\rm i} (\delta_l + l \pi/2)].
\end{displaymath}">|; 

$key = q/displaystylea',psi_{a'}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img664.png"
 ALT="$\displaystyle a' \psi_{a'}.$">|; 

$key = q/displaystyleexp!left[frac{{{rm{i},(omega-omega_{21}),t}{2}right]cos(Omega,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2424.png"
 ALT="$\displaystyle \exp\!\left[\frac{{\rm i} (\omega-\omega_{21}) t}{2}\right]
\cos(\Omega t)$">|; 

$key = q/{displaymath}langlem|Eranglesimeq-frac{e_{mn}}{E_m-E_n}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img2165.png"
 ALT="\begin{displaymath}
\langle m\vert E\rangle \simeq - \frac{e_{mn}}{E_m-E_n}.
\end{displaymath}">|; 

$key = q/E_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1588.png"
 ALT="$E_0$">|; 

$key = q/displaystyleexp(,{{rm{i},delta_0),left[cosdelta_0,j_0(k,r)-sindelta_0,y_0(k,r)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="314" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2993.png"
 ALT="$\displaystyle \exp( {\rm i}  \delta_0) \left[
\cos\delta_0 j_0(k r) - \sin\delta_0 y_0(k r) \right]$">|; 

$key = q/varphi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1140.png"
 ALT="$\varphi=0$">|; 

$key = q/displaystyle1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2159.png"
 ALT="$\displaystyle 1,$">|; 

$key = q/|psi_{000}(0)|^2=1slash(pi,a_0^{,3});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2374.png"
 ALT="$\vert\psi_{000}(0)\vert^2=
1/(\pi a_0^{ 3})$">|; 

$key = q/psi^ast,psiequiv|psi|^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1406.png"
 ALT="$\psi^\ast \psi\equiv \vert\psi\vert^2$">|; 

$key = q/lambda=633;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img444.png"
 ALT="$\lambda = 633$">|; 

$key = q/hbar,omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1023.png"
 ALT="$\hbar \omega$">|; 

$key = q/P_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2396.png"
 ALT="$P_n$">|; 

$key = q/displaystyleS_-,chi_{s,m_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1935.png"
 ALT="$\displaystyle S_- \chi_{s,m_s}$">|; 

$key = q/{displaymath}sigma_A^{,2},sigma_B^{,2}geqleft(frac{1}{2,{{rm{i}}langle[A,B]rangleright)^{,2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img620.png"
 ALT="\begin{displaymath}
\sigma_A^{ 2} \sigma_B^{ 2} \geq \left(\frac{1}{2 {\rm i}}\langle[A,B]\rangle\right)^{ 2},
\end{displaymath}">|; 

$key = q/{displaymath}F_pm(X)=-1+frac{2}{X}left[frac{(1+X),{{rm{e}^{-2,X}pm(1-2,X^2slash3e}^{-X}}{1pm(1+X+X^2slash3),{{rm{e}^{-X}}right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="449" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2822.png"
 ALT="\begin{displaymath}
F_\pm(X) = -1 + \frac{2}{X}\left[\frac{(1+X) {\rm e}^{-2 X...
...X^2/3)
 {\rm e}^{-X}}{1\pm (1+X+X^2/3) {\rm e}^{-X}}\right].
\end{displaymath}">|; 

$key = q/L_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$L_z$">|; 

$key = q/E_i=E_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1165.png"
 ALT="$E_i=E_j$">|; 

$key = q/{displaymath}j_{x}=frac{{{rm{i},hbar}{2,m}left(psi,frac{partialpsi^ast}{partialx}-psi^ast,frac{partialpsi}{partialx}right){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1185.png"
 ALT="\begin{displaymath}
j_{x} =\frac{{\rm i} \hbar}{2 m}\left(\psi \frac{\partial...
...partial x} - \psi^\ast \frac{\partial\psi}{\partial x}\right)
\end{displaymath}">|; 

$key = q/displaystyleJ^2,psi_{j,m_j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1885.png"
 ALT="$\displaystyle J^2 \psi_{j,m_j}$">|; 

$key = q/{displaymath}z=int_{-infty}^{infty}[(A-langleArangle),psi]^{,ast},[(B-langleBran^ast,(A-langleArangle),(B-langleBrangle),psi,dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="583" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img617.png"
 ALT="\begin{displaymath}
z = \int_{-\infty}^{\infty} [(A-\langle A\rangle) \psi]^{ ...
...si^\ast (A-\langle A\rangle) (B-\langle B\rangle) \psi dx,
\end{displaymath}">|; 

$key = q/{displaymath}left(frac{1}{z^2},frac{d}{dz},z^2,frac{d}{dz}-frac{l,(l+1)}{z^2}+frac{2,n}{z}-1right)R_{n,l}=0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1629.png"
 ALT="\begin{displaymath}
\left(\frac{1}{z^2} \frac{d}{dz} z^2 \frac{d}{dz}- \frac{l (l+1)}{z^2}
+ \frac{2 n}{z}-1\right) R_{n,l} = 0
\end{displaymath}">|; 

$key = q/psi_E(x,t,E_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img738.png"
 ALT="$\psi_E(x,t,E_i)$">|; 

$key = q/omega=Eslashhbar>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img314.png"
 ALT="$\omega = E/\hbar&gt;0$">|; 

$key = q/{displaymath}[{{bf{r},H_0]=frac{{{rm{i},hbar,{{bf{p}}{m_e},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2578.png"
 ALT="\begin{displaymath}[{\bf r}, H_0]= \frac{{\rm i} \hbar {\bf p}}{m_e},
\end{displaymath}">|; 

$key = q/langleprangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img747.png"
 ALT="$\langle p\rangle$">|; 

$key = q/displaystylelangle1,0,0|z|2,1,0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2665.png"
 ALT="$\displaystyle \langle 1,0,0\vert z\vert 2,1,0\rangle$">|; 

$key = q/lambdaslashalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img417.png"
 ALT="$\lambda/\alpha$">|; 

$key = q/Lslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img756.png"
 ALT="$L/2$">|; 

$key = q/{displaymath}k^2(x)=frac{2,m,[E-V(x)]}{hbar^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img874.png"
 ALT="\begin{displaymath}
k^2(x) = \frac{2 m [E-V(x)]}{\hbar^2}.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{n}cdot{{bf{r}=d,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img177.png"
 ALT="\begin{displaymath}
{\bf n}\cdot{\bf r} = d,
\end{displaymath}">|; 

$key = q/{scriptstylesqrt{1slash3}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2018.png"
 ALT="${\scriptstyle\sqrt{1/3}}$">|; 

$key = q/{displaymath}langleA^2rangle=int_{-infty}^inftypsi_a^{ast},A^2,psi_a,dx=a,int_{-=a^2,int_{-infty}^inftypsi_a^{ast},psi_a,dx=a^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="550" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img652.png"
 ALT="\begin{displaymath}
\langle A^2\rangle = \int_{-\infty}^\infty \psi_a^{\ast} A^...
...dx =a^2 \int_{-\infty}^\infty \psi_a^{\ast} \psi_a dx =a^2,
\end{displaymath}">|; 

$key = q/a_0=(hbar^2slashm_e),(4pi,epsilon_0slashe^2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1691.png"
 ALT="$a_0=(\hbar^2/m_e) (4\pi \epsilon_0/e^2)$">|; 

$key = q/{displaymath}E=E_n+{calO}(epsilon),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2157.png"
 ALT="\begin{displaymath}
E = E_n + {\cal O}(\epsilon),
\end{displaymath}">|; 

$key = q/{displaymath}D=frac{1}{X}left(1-[1+X],{{rm{e}^{-2,X}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img2813.png"
 ALT="\begin{displaymath}
D = \frac{1}{X} \left( 1-[1+X] {\rm e}^{-2 X}\right).
\end{displaymath}">|; 

$key = q/{displaymath}|T|^{,2}=expleft(-frac{2,sqrt{2,m}}{hbar}int_{x_1}^{x_2}sqrt{V(x)-E},dxright),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img905.png"
 ALT="\begin{displaymath}
\vert T\vert^{ 2} = \exp\left(-\frac{2 \sqrt{2 m}}{\hbar}\int_{x_1}^{x_2}
\sqrt{V(x)- E} dx\right),
\end{displaymath}">|; 

$key = q/r>a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2942.png"
 ALT="$r&gt;a$">|; 

$key = q/alpha=e^2slash(4pi,epsilon_0,hbar,c)simeq1slash137;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1625.png"
 ALT="$\alpha = e^2/ (4\pi \epsilon_0 \hbar c)\simeq 1/137$">|; 

$key = q/{displaymath}DeltaE=-frac{alpha^2}{16},E_0=4.53times10^{-5},{{rm{eV}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img2677.png"
 ALT="\begin{displaymath}
\Delta E = - \frac{\alpha^2}{16} E_0 = 4.53\times 10^{-5} {\rm eV}.
\end{displaymath}">|; 

$key = q/A_n=sqrt{2slasha};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img808.png"
 ALT="$A_n = \sqrt{2/a}$">|; 

$key = q/{displaymath}langleArangle=left(frac{a_1+a_2}{2}right)+left(frac{a_1-a_2}{2}right)cosleft(frac{[E_1-E_2],t}{hbar}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="400" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img785.png"
 ALT="\begin{displaymath}
\langle A\rangle = \left(\frac{a_1+a_2}{2}\right) + \left(\f...
...a_1-a_2}{2}\right)\cos\left(\frac{[E_1-E_2] t}{\hbar}\right).
\end{displaymath}">|; 

$key = q/displaystyleoint(L_+,Y_{l,m})^ast,(L_+,Y_{l,m}),dOmega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1407.png"
 ALT="$\displaystyle \oint (L_+ Y_{l,m})^\ast (L_+ Y_{l,m}) d\Omega$">|; 

$key = q/{displaymath}int|psi_pm|^2,d^3{{bf{r}=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img2782.png"
 ALT="\begin{displaymath}
\int \vert\psi_\pm\vert^2 d^3{\bf r} = 1.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{p}=hbar,{{bf{k},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img306.png"
 ALT="\begin{displaymath}
{\bf p} = \hbar {\bf k},
\end{displaymath}">|; 

$key = q/{displaymath}psi^{(2)}_{j,m_j}=left(frac{j+m_j}{2,l+1}right)^{1slash2}psi^{(1)}_ight)^{1slash2},psi^{(1)}_{m_j+1slash2,-1slash2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="486" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2344.png"
 ALT="\begin{displaymath}
\psi^{(2)}_{j,m_j} = \left(\frac{j+m_j}{2 l+1}\right)^{1/2}...
...t(\frac{j-m_j}{2 l+1}\right)^{1/2} \psi^{(1)}_{m_j+1/2,-1/2}
\end{displaymath}">|; 

$key = q/{displaymath}N(t)=N(0),exp(-nu,|T|^2,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img936.png"
 ALT="\begin{displaymath}
N(t) = N(0) \exp(-\nu \vert T\vert^2 t).
\end{displaymath}">|; 

$key = q/H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img556.png"
 ALT="$H$">|; 

$key = q/{displaymath}U_{jk}=langlen,j,m|H_1|n,k,mrangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2256.png"
 ALT="\begin{displaymath}
U_{jk} = \langle n,j,m\vert H_1\vert n,k,m\rangle.
\end{displaymath}">|; 

$key = q/P_y(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img284.png"
 ALT="$P_y(y)$">|; 

$key = q/A(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img606.png"
 ALT="$A(x)$">|; 

$key = q/{displaymath}E_y(x,t)=|bar{psi}|,cosleft(k,[x-(omegaslashk),t]+varphiright)=|bar{psi}|,cosleft(k,[x-c,t]+varphiright).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="525" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img237.png"
 ALT="\begin{displaymath}
E_y(x,t) =\vert\bar{\psi}\vert \cos\left(k [x-(\omega/k) ...
...)= \vert\bar{\psi}\vert \cos\left(k [x-c t]+\varphi\right).
\end{displaymath}">|; 

$key = q/displaystylej_0(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1551.png"
 ALT="$\displaystyle j_0(z)$">|; 

$key = q/{{rm{sinh}(0)=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2506.png"
 ALT="${\rm sinh} (0)=1$">|; 

$key = q/{displaymath}|T|^{,2}=expleft(-2,sqrt{2},frac{m^{1slash2},W^{,3slash2}}{hbar,e,{calE}}int_{0}^{1}sqrt{1-y},dyright),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="376" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img907.png"
 ALT="\begin{displaymath}
\vert T\vert^{ 2} = \exp\left(-2 \sqrt{2} \frac{m^{1/2} W^{ 3/2}}{\hbar e {\cal E}}\int_{0}^{1}
\sqrt{1-y} dy\right),
\end{displaymath}">|; 

$key = q/displaystyle{{rm{i},hbar,delta_{ij}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1086.png"
 ALT="$\displaystyle {\rm i} \hbar \delta_{ij}.$">|; 

$key = q/m_{s,min}<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1765.png"
 ALT="$m_{s min}&lt;0$">|; 

$key = q/{displaymath}E-V_0=frac{hbar^2,k'^{,2}}{2,m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2996.png"
 ALT="\begin{displaymath}
E - V_0 = \frac{\hbar^2  k'^{ 2}}{2 m}.
\end{displaymath}">|; 

$key = q/v_g=domegaslashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img392.png"
 ALT="$v_g=d\omega/dt$">|; 

$key = q/{displaymath}v_p=frac{p}{2,m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img311.png"
 ALT="\begin{displaymath}
v_p = \frac{p}{2 m}.
\end{displaymath}">|; 

$key = q/{displaymath}Y_{l,-m}simY^{,ast}_{l,m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img1460.png"
 ALT="\begin{displaymath}
Y_{l,-m} \sim Y^{ \ast}_{l,m}.
\end{displaymath}">|; 

$key = q/{displaymath}psi(x,y,z)=X(x),Y(y),Z(z).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1213.png"
 ALT="\begin{displaymath}
\psi(x,y,z) = X(x) Y(y) Z(z).
\end{displaymath}">|; 

$key = q/{displaymath}frac{d}{dt}int_{-infty}^{infty}|psi|^{,2},dx=frac{{{rm{i},hbar}{2,mtialpsi^ast}{partialx}right]_{-infty}^{infty}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="384" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img478.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\int_{-\infty}^{\infty}\vert\psi\vert^{ 2} dx=...
...{\partial\psi^\ast}{\partial x}\right]_{-\infty}^{\infty} = 0.
\end{displaymath}">|; 

$key = q/T=2pislashomega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img155.png"
 ALT="$T=2\pi/\omega$">|; 

$key = q/displaystyleL_z,(L_+,Y_{l,m});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1372.png"
 ALT="$\displaystyle L_z (L_+ Y_{l,m})$">|; 

$key = q/lambda<(pislash2)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img991.png"
 ALT="$\lambda &lt; (\pi/2)^2$">|; 

$key = q/displaystyleint_{-infty}^{infty}psi^ast(x,t)left(-{{rm{i},hbar,frac{partial}{partialx}right)psi(x,t),dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img589.png"
 ALT="$\displaystyle \int_{-\infty}^{\infty} \psi^\ast(x,t)\left(-{\rm i} \hbar 
\frac{\partial}{\partial x}\right)\psi(x,t) dx$">|; 

$key = q/{displaymath}psi(x)=frac{{{rm{e}^{,{{rm{i},varphi}}{(2pi,sigma^2)^{1slash4}},{{rm{e}^{-(x-x_0)^{,2}slash(4,sigma^2)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img494.png"
 ALT="\begin{displaymath}
\psi(x) = \frac{{\rm e}^{ {\rm i} \varphi}}{(2\pi \sigma^2)^{1/4}} {\rm e}^{-(x-x_0)^{ 2}/(4 \sigma^2)}.
\end{displaymath}">|; 

$key = q/{displaymath}m_{s,max}=+s.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1767.png"
 ALT="\begin{displaymath}
m_{s max} = +s.
\end{displaymath}">|; 

$key = q/displaystyle28.5,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img942.png"
 ALT="$\displaystyle 28.5,$">|; 

$key = q/m_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1887.png"
 ALT="$m_j$">|; 

$key = q/{displaymath}S^2,chi_{s,m_{s,min}}=(S_+,S_-+S_z^{,2}-hbar,S_z),chi_{s,m_{s,min}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1763.png"
 ALT="\begin{displaymath}
S^2 \chi_{s,m_{s min}} = (S_+ S_- +S_z^{ 2}-\hbar S_z) \chi_{s,m_{s min}},
\end{displaymath}">|; 

$key = q/mbox{{boldmath{{sigma{}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2089.png"
 ALT="$\mbox{\boldmath$\sigma$}_2$">|; 

$key = q/{displaymath}|psi_0|^{,2}int_{-infty}^{infty}{{rm{e}^{-(x-x_0)^{,2}slash(2,sigma^2)},dx=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img463.png"
 ALT="\begin{displaymath}
\vert\psi_0\vert^{ 2}\int_{-\infty}^{\infty}{\rm e}^{-(x-x_0)^{ 2}/(2 \sigma^2)} dx = 1.
\end{displaymath}">|; 

$key = q/m=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1573.png"
 ALT="$m=1$">|; 

$key = q/displaystyleDeltaE_{l,1slash2;j,m_j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2319.png"
 ALT="$\displaystyle \Delta E_{l,1/2;j,m_j}$">|; 

$key = q/{displaymath}sigma_A^{,2},sigma_B^{,2}geqleft[frac{1}{2,{{rm{i}},(z-z^ast)right]^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img616.png"
 ALT="\begin{displaymath}
\sigma_A^{ 2} \sigma_B^{ 2} \geq \left[\frac{1}{2 {\rm i}} (z-z^\ast)\right]^{ 2}.
\end{displaymath}">|; 

$key = q/k_x^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1225.png"
 ALT="$k_x^{ 2}$">|; 

$key = q/{displaymath}H_0=frac{p^2}{2,m_e}+V_0(r),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2544.png"
 ALT="\begin{displaymath}
H_0 = \frac{p^2}{2 m_e} + V_0(r),
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{sqrt{2pi}}int_{-infty}^{infty}bar{psi}(k,t),{{rm{e}^{+{{rm{i},k,x},dk,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img567.png"
 ALT="$\displaystyle \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{\infty} \bar{\psi}(k,t) {\rm e}^{+{\rm i} k x} dk,$">|; 

$key = q/L_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1314.png"
 ALT="$L_i$">|; 

$key = q/1slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img76.png"
 ALT="$1/3$">|; 

$key = q/displaystyle-{{rm{i},hbar,frac{partial}{partialz}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1198.png"
 ALT="$\displaystyle -{\rm i} \hbar \frac{\partial}{\partial z}.$">|; 

$key = q/{displaymath}P_{irightarrowf}^{abs}(t)=frac{t^2,e^2}{2,epsilon_0,hbar^2,m_e^{,2}{,2},u,{{rm{sinc}^2[(omega-omega_{fi}),tslash2].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="553" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2554.png"
 ALT="\begin{displaymath}
P_{i\rightarrow f}^{abs}(t) = \frac{t^2 e^2}{2 \epsilon_0\...
...\right\vert^{ 2} u {\rm sinc}^2[(\omega-\omega_{fi}) t/2].
\end{displaymath}">|; 

$key = q/j_0(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="$j_0(z)$">|; 

$key = q/displaystylehbar,(m-m'),langlen,l,m|z|n',l',m'rangle=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2186.png"
 ALT="$\displaystyle \hbar (m-m') \langle n,l,m\vert z\vert n',l',m'\rangle = 0,$">|; 

$key = q/rho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1300.png"
 ALT="$\rho$">|; 

$key = q/{displaymath}{{bf{k}cdot{{bf{r}-omega,t+varphi=j,2pi,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img184.png"
 ALT="\begin{displaymath}
{\bf k}\cdot{\bf r} -\omega t +\varphi= j 2\pi,
\end{displaymath}">|; 

$key = q/1slash36;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img89.png"
 ALT="$1/36$">|; 

$key = q/A^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2546.png"
 ALT="$A^2$">|; 

$key = q/displaystylee_{12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2133.png"
 ALT="$\displaystyle e_{12}$">|; 

$key = q/chi_{s,-s+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1770.png"
 ALT="$\chi_{s,-s+1}$">|; 

$key = q/aneqa';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img671.png"
 ALT="$a\neq a'$">|; 

$key = q/tau=hbarslashGamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3032.png"
 ALT="$\tau = \hbar/
\Gamma$">|; 

$key = q/k^2=(2mslashhbar^2),E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1058.png"
 ALT="$k^2= (2m/\hbar^2) E$">|; 

$key = q/c_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1725.png"
 ALT="$c_2$">|; 

$key = q/{displaymath}DeltaE=frac{mu_0,g_p,e^2}{3pi,m_p,m_e,a_0^{,3}},langle{{bf{S}_pcdot{{bf{S}_erangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2375.png"
 ALT="\begin{displaymath}
\Delta E = \frac{\mu_0 g_p e^2}{3\pi m_p m_e a_0^{ 3}} \langle{\bf S}_p\cdot{\bf S}_e\rangle.
\end{displaymath}">|; 

$key = q/{displaymath}frac{e_{mm}}{E_m}sim{calO}(epsilon){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img2156.png"
 ALT="\begin{displaymath}
\frac{e_{mm}}{E_m}\sim {\cal O}(\epsilon)
\end{displaymath}">|; 

$key = q/e,a_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2615.png"
 ALT="$e a_0$">|; 

$key = q/{displaymath}J=intpsi_0({{bf{r}_1),psi_0({{bf{r}_2),d^3{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img2786.png"
 ALT="\begin{displaymath}
J = \int \psi_0({\bf r}_1) \psi_0({\bf r}_2) d^3{\bf r}.
\end{displaymath}">|; 

$key = q/displaystyle[sigma_z,sigma_x];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1806.png"
 ALT="$\displaystyle [\sigma_z,\sigma_x]$">|; 

$key = q/displaystyleH-q,phi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2524.png"
 ALT="$\displaystyle H - q \phi,$">|; 

$key = q/displaystylesqrt{frac{2}{3}},psi^{(2)}_{3slash2,1slash2}+sqrt{frac{1}{3}},psi^{(2)}_{1slash2,1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1998.png"
 ALT="$\displaystyle \sqrt{\frac{2}{3}} \psi^{(2)}_{3/2,1/2} +
\sqrt{ \frac{1}{3}} \psi^{(2)}_{1/2,1/2},$">|; 

$key = q/n>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2623.png"
 ALT="$n&gt;0$">|; 

$key = q/psi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img790.png"
 ALT="$\psi=0$">|; 

$key = q/k_B,TggE_F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1259.png"
 ALT="$k_B T\gg E_F$">|; 

$key = q/l=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1434.png"
 ALT="$l=2$">|; 

$key = q/{displaymath}leftlangle({mitDelta}u)^2rightrangle=leftlangleu^2rightrangle-langleurangle^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img114.png"
 ALT="\begin{displaymath}
\left\langle({\mit\Delta} u)^2\right\rangle= \left\langle u^2\right\rangle-\langle u\rangle^2.
\end{displaymath}">|; 

$key = q/A(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1006.png"
 ALT="$A(y)$">|; 

$key = q/dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img138.png"
 ALT="$dt$">|; 

$key = q/v_p=omegaslashk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img315.png"
 ALT="$v_p=\omega/k$">|; 

$key = q/{displaymath}Y_{l,l}(theta,phi)sim(sintheta)^l,{{rm{e}^{,{{rm{i},l,phi}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1442.png"
 ALT="\begin{displaymath}
Y_{l,l}(\theta,\phi) \sim (\sin\theta)^l {\rm e}^{ {\rm i} l \phi}.
\end{displaymath}">|; 

$key = q/{displaymath}int_{-infty}^{infty}f(x),delta(x),dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img578.png"
 ALT="\begin{displaymath}
\int_{-\infty}^{\infty}f(x) \delta(x) dx.
\end{displaymath}">|; 

$key = q/langlepsi|tilde{psi}_0rangle=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2704.png"
 ALT="$\langle \psi\vert\tilde{\psi}_0\rangle=0$">|; 

$key = q/{displaymath}psi(x,t)propto{{rm{e}^{,{{rm{i},(k_0,x-omega_0,t)}int_{-infty}^{inf,{{rm{i},beta_1,y-(1+{{rm{i},beta_2),y^{,2}},dy,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="347" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img374.png"
 ALT="\begin{displaymath}
\psi(x,t)\propto {\rm e}^{ {\rm i} (k_0 x-\omega_0 t)}
\...
...e}^{  {\rm i} \beta_1 y-(1+{\rm i} \beta_2) y^{ 2}} dy,
\end{displaymath}">|; 

$key = q/{displaymath}[a_+,a_-]=-1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1034.png"
 ALT="\begin{displaymath}[a_+,a_-]= -1.
\end{displaymath}">|; 

$key = q/delta(x-x');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img708.png"
 ALT="$\delta(x-x')$">|; 

$key = q/displaystyleS^2,chi_pm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1780.png"
 ALT="$\displaystyle S^2 \chi_\pm$">|; 

$key = q/{{bf{r}_{1,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2781.png"
 ALT="${\bf r}_{1,2}$">|; 

$key = q/R(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1531.png"
 ALT="$R(r)$">|; 

$key = q/{displaymath}chi_+^nomath_inline{dagnomath_inline{,chi_+=chi_-^nomath_inline{dagnomath_inline{,chi_-=1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1786.png"
 ALT="\begin{displaymath}
\chi_+^\dag \chi_+ = \chi_-^\dag \chi_- = 1,
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{B}=frac{mu_0}{4pi,r^3},left[3,(mbox{{boldmath{{mu{}_pcdot{{bf{u_0}{3},mbox{{boldmath{{mu{}_p,delta^3({{bf{r}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="379" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img2366.png"
 ALT="\begin{displaymath}
{\bf B} = \frac{\mu_0}{4\pi r^3} \left[3 (\mbox{\boldmath...
...frac{2 \mu_0}{3} \mbox{\boldmath$\mu$}_p \delta^3({\bf r}),
\end{displaymath}">|; 

$key = q/{displaymath}L_pm=L_xpm{{rm{i},L_y.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1332.png"
 ALT="\begin{displaymath}
L_\pm = L_x\pm {\rm i} L_y.
\end{displaymath}">|; 

$key = q/n,l=3,0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="$n,l=3,0$">|; 

$key = q/{displaymath}psi({{bf{r}_1,{{bf{r}_2)=psi_0({{bf{r}_1),psi_0({{bf{r_2})=frac{8}{a_0^{,3}},expleft(-frac{2,[r_1+r_2]}{a_0}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="429" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2726.png"
 ALT="\begin{displaymath}
\psi({\bf r}_1, {\bf r}_2) = \psi_0({\bf r}_1) \psi_0({\bf ...
...{\pi a_0^{ 3}} \exp\left(- \frac{2 [r_1+r_2]}{a_0}\right).
\end{displaymath}">|; 

$key = q/displaystyle{{bf{B};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2529.png"
 ALT="$\displaystyle {\bf B}$">|; 

$key = q/N_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2603.png"
 ALT="$N_i$">|; 

$key = q/{displaymath}A_{ij}=psi_i^nomath_inline{dagnomath_inline{,A,psi_j.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2114.png"
 ALT="\begin{displaymath}
A_{ij}=\psi_i^\dag A \psi_j.
\end{displaymath}">|; 

$key = q/displaystyleC_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img943.png"
 ALT="$\displaystyle C_2$">|; 

$key = q/{displaymath}int_{-infty}^{infty}psi_1^ast,psi_2,dx=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img658.png"
 ALT="\begin{displaymath}
\int_{-\infty}^{\infty}\psi_1^\ast \psi_2 dx = 0.
\end{displaymath}">|; 

$key = q/psi_{200};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2268.png"
 ALT="$\psi_{200}$">|; 

$key = q/{displaymath}exp(,{{rm{i},{{bf{k}!cdot!{{bf{r})=1+{{rm{i},{{bf{k}!cdot!{{bf{r}+cdots,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2576.png"
 ALT="\begin{displaymath}
\exp( {\rm i} {\bf k}\!\cdot\!{\bf r}) = 1 + {\rm i}  {\bf k}\!\cdot\!{\bf r}+\cdots,
\end{displaymath}">|; 

$key = q/displaystyle-frac{{{rm{i},t}{hbar}left(V_{fi},expleft[,{{rm{i},(omega+omega_{fi}h2right]{{rm{sinc}left[(omega+omega_{fi}),tslash2right]right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="390" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2489.png"
 ALT="$\displaystyle - \frac{{\rm i} t}{\hbar}\left(V_{fi} \exp\left[ {\rm i} (\om...
...\omega_{fi}) t/2\right]{\rm sinc}\left[(\omega+\omega_{fi}) t/2\right]\right.$">|; 

$key = q/displaystyleleft{l^2,(l+1)^2-2,l,(l+1),l',(l'+1)+l'^2,(l'+1)^2right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="375" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2206.png"
 ALT="$\displaystyle \left\{l^2 (l+1)^2-2 l (l+1) l' (l'+1) + l'^2 (l'+1)^2\right.$">|; 

$key = q/2,gamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2441.png"
 ALT="$2 \gamma$">|; 

$key = q/displaystyle(m+1),hbar,(L_+,Y_{l,m}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1374.png"
 ALT="$\displaystyle (m+1) \hbar (L_+ Y_{l,m}),$">|; 

$key = q/{displaymath}oint|Y_{l,m}(theta,phi)|^2,dOmega=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1462.png"
 ALT="\begin{displaymath}
\oint \vert Y_{l,m}(\theta,\phi)\vert^2 d\Omega = 1.
\end{displaymath}">|; 

$key = q/|f({{bf{k},{{bf{k}')|^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2869.png"
 ALT="$\vert f({\bf k},{\bf k}')\vert^2$">|; 

$key = q/97slash169;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img776.png"
 ALT="$97/169$">|; 

$key = q/m_s=-1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2008.png"
 ALT="$m_s=-1/2$">|; 

$key = q/partial|psi|^{,2}slashpartialt=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img826.png"
 ALT="$\partial \vert\psi\vert^{ 2}/\partial t = 0$">|; 

$key = q/{displaymath}psi(x,t)=sum_{n=0,infty}c_n,psi_n(x),{{rm{e}^{-{{rm{i},E_n,tslashhbar},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img810.png"
 ALT="\begin{displaymath}
\psi(x,t) = \sum_{n=0,\infty} c_n \psi_n(x) {\rm e}^{-{\rm i} E_n t/\hbar},
\end{displaymath}">|; 

$key = q/displaystyleA,{{rm{e}^{,{{rm{i},q,a}+B,{{rm{e}^{-{{rm{i},q,a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img842.png"
 ALT="$\displaystyle A  {\rm e}^{ {\rm i} q a}+ B  {\rm e}^{-{\rm i} q a}$">|; 

$key = q/displaystylelanglerrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1655.png"
 ALT="$\displaystyle \langle r\rangle$">|; 

$key = q/displaystylepsi_2';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2145.png"
 ALT="$\displaystyle \psi_2'$">|; 

$key = q/displaystylem_s,hbar,psi^{(1)}_{l,s;m,m_s}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1906.png"
 ALT="$\displaystyle m_s \hbar \psi^{(1)}_{l,s;m,m_s}.$">|; 

$key = q/S_z=hbarslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2090.png"
 ALT="$S_z=\hbar/2$">|; 

$key = q/{displaymath}int_{-infty}^inftypsi_a^ast,psi_{a'},dx=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img672.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty\psi_a^\ast \psi_{a'} dx = 0.
\end{displaymath}">|; 

$key = q/Y_{l,l-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1445.png"
 ALT="$Y_{l,l-1}$">|; 

$key = q/displaystyle1.61.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img951.png"
 ALT="$\displaystyle 1.61.$">|; 

$key = q/L_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1333.png"
 ALT="$L_+$">|; 

$key = q/displaystylefrac{partial}{partialx_2}=frac{m_2}{m_1+m_2}frac{partial}{partialX}-frac{partial}{partialx'}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1116.png"
 ALT="$\displaystyle \frac{\partial}{\partial x_2} = \frac{m_2}{m_1+m_2}\frac{\partial}{\partial X} - \frac{\partial}{\partial x'}.$">|; 

$key = q/displaystylephi_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img779.png"
 ALT="$\displaystyle \phi_2$">|; 

$key = q/displaystylex_i,x_i,p_j,p_j-{{rm{i},hbar,delta_{ij},x_i,p_j-x_i,p_j,p_i,x_j-{{rm{i},hbar,delta_{ij},x_i,p_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="385" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1519.png"
 ALT="$\displaystyle x_i x_i p_j p_j - {\rm i} \hbar \delta_{ij} x_i p_j
-x_i p_j p_i x_j - {\rm i} \hbar \delta_{ij} x_i p_j$">|; 

$key = q/x+dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1068.png"
 ALT="$x+dx$">|; 

$key = q/2,1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="$2,1$">|; 

$key = q/Z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="$Z$">|; 

$key = q/u_l=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2959.png"
 ALT="$u_l =0$">|; 

$key = q/r_2=Z_1,Z_2,e^2slash(4pi,epsilon_0,E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img919.png"
 ALT="$r_2 = Z_1 Z_2 e^2/(4\pi \epsilon_0 E)$">|; 

$key = q/qrightarrow-{{rm{i},q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img864.png"
 ALT="$q\rightarrow -{\rm i} q$">|; 

$key = q/{displaymath}l,(l+1)=m_-,(m_--1).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1425.png"
 ALT="\begin{displaymath}
l (l+1) = m_- (m_-1).
\end{displaymath}">|; 

$key = q/a<b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img457.png"
 ALT="$a&lt;b$">|; 

$key = q/mneqn;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2162.png"
 ALT="$m\neq n$">|; 

$key = q/w_{irightarrowf}^{abs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2565.png"
 ALT="$w_{i\rightarrow f}^{abs}$">|; 

$key = q/displaystyleR_{3,1}(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1646.png"
 ALT="$\displaystyle R_{3,1}(r)$">|; 

$key = q/log_{10}(tau_{th})=-28.9-1.60,Z_1^{,2slash3}+1.61,Z_1slashsqrt{E};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img947.png"
 ALT="$\log_{10}(\tau_{th}) = -28.9 - 1.60 Z_1^{ 2/3} + 1.61 Z_1/\sqrt{E}$">|; 

$key = q/{displaymath}x,frac{df}{dx}neqfrac{d}{dx}left(x,fright).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img529.png"
 ALT="\begin{displaymath}
x \frac{df}{dx} \neq \frac{d}{dx}\left(x f\right).
\end{displaymath}">|; 

$key = q/alpha>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1061.png"
 ALT="$\alpha&gt;0$">|; 

$key = q/lambda=1.5,pi^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="$\lambda = 1.5 \pi ^2$">|; 

$key = q/zrightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1613.png"
 ALT="$z\rightarrow\infty$">|; 

$key = q/{displaymath}k=q,tan(q,aslash2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img974.png"
 ALT="\begin{displaymath}
k = q \tan(q a/2).
\end{displaymath}">|; 

$key = q/displaystyle{{rm{i},hbar,S_y.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1705.png"
 ALT="$\displaystyle {\rm i} \hbar S_y.$">|; 

$key = q/{displaymath}H=frac{left(p^2-e,{{bf{A}!cdot!{{bf{p}-e,{{bf{p}!cdot!{{bf{A}+e^2A^2right)}{2,m_e}+e,phi+V_0(r).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="394" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2533.png"
 ALT="\begin{displaymath}
H = \frac{ \left(p^2 -e  {\bf A}\!\cdot \! {\bf p}
-e  {\b...
...\!\cdot\!{\bf A} + e^2 A^2\right)}{2 m_e}+ e  \phi + V_0(r).
\end{displaymath}">|; 

$key = q/{displaymath}bar{psi}=|bar{psi}|,{{rm{e}^{,{{rm{i},varphi},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img230.png"
 ALT="\begin{displaymath}
\bar{\psi} = \vert\bar{\psi}\vert {\rm e}^{ {\rm i} \varphi},
\end{displaymath}">|; 

$key = q/displaystyle[s,(s+1)-m_s,(m_s-1)]^{1slash2},hbar,chi_{s,m_s-1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1936.png"
 ALT="$\displaystyle [s (s+1)-m_s (m_s-1)]^{1/2} \hbar \chi_{s,m_s-1}.$">|; 

$key = q/V_0<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2992.png"
 ALT="$V_0&lt;0$">|; 

$key = q/{displaymath}[L^2,L_x]=[L^2,L_y]=[L^2,L_z]=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1331.png"
 ALT="\begin{displaymath}[L^2, L_x]= [L^2, L_y] = [L^2,L_z] = 0.
\end{displaymath}">|; 

$key = q/displaystyle2,{{rm{i},sigma_x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1805.png"
 ALT="$\displaystyle 2 {\rm i} \sigma_x,$">|; 

$key = q/{displaymath}k,a=z_{n,l},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1565.png"
 ALT="\begin{displaymath}
k a = z_{n,l},
\end{displaymath}">|; 

$key = q/rggr';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2852.png"
 ALT="$r\gg r'$">|; 

$key = q/langle1rangle=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img520.png"
 ALT="$\langle 1\rangle =1$">|; 

$key = q/{displaymath}|m|leql<n,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1631.png"
 ALT="\begin{displaymath}
\vert m\vert \leq l &lt; n,
\end{displaymath}">|; 

$key = q/displaystylem_{s_2},hbar,chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2046.png"
 ALT="$\displaystyle m_{s_2} \hbar
 \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)},$">|; 

$key = q/displaystyle{{bf{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2521.png"
 ALT="$\displaystyle {\bf p}$">|; 

$key = q/E_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img734.png"
 ALT="$E_i$">|; 

$key = q/S^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="$S^2$">|; 

$key = q/{displaymath}I(y)proptocos^2left(frac{k,{mitDelta}x}{2}right)simeqcos^2left(frac{k,d}{2,D},yright).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img276.png"
 ALT="\begin{displaymath}
I(y) \propto \cos^2\left(\frac{k {\mit\Delta} x}{2}\right) \simeq \cos^2\left(
\frac{k d}{2 D} y\right).
\end{displaymath}">|; 

$key = q/P_{x,in,-infty:infty}=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img460.png"
 ALT="$P_{x \in  -\infty:\infty}=1$">|; 

$key = q/displaystyleL_+,Y_{l,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1378.png"
 ALT="$\displaystyle L_+ Y_{l,m}$">|; 

$key = q/R_{n,l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2660.png"
 ALT="$R_{n,l}$">|; 

$key = q/P(x,t),dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1173.png"
 ALT="$P(x,t) dx$">|; 

$key = q/Lslashsqrt{12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img759.png"
 ALT="$L/\sqrt{12}$">|; 

$key = q/W_{irightarrowf}=W_{frightarrowi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2606.png"
 ALT="$W_{i\rightarrow f}=W_{f\rightarrow i}$">|; 

$key = q/J^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1882.png"
 ALT="$J^2$">|; 

$key = q/{displaymath}langlexrangle=x_0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img497.png"
 ALT="\begin{displaymath}
\langle x\rangle =x_0.
\end{displaymath}">|; 

$key = q/{displaymath}E_f-E_i=-hbar,omega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2510.png"
 ALT="\begin{displaymath}
E_f - E_i = -\hbar \omega.
\end{displaymath}">|; 

$key = q/h=6.6261times10^{-34},{{rm{J,s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img248.png"
 ALT="$h = 6.6261\times 10^{-34} {\rm J s}$">|; 

$key = q/lgeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1417.png"
 ALT="$l\geq 0$">|; 

$key = q/displaystyle[l,(l+1)+3slash4+m],hbar^2,Y_{l,m},chi_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1941.png"
 ALT="$\displaystyle [l (l+1)+3/4+m] \hbar^2 Y_{l,m} \chi_+$">|; 

$key = q/{displaymath}V^nomath_inline{dagnomath_inline{=-frac{e,A_0,mbox{{boldmath{{epsil{p}}{2,m_e},exp(,{{rm{i},{{bf{k}!cdot!{{bf{r},).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2549.png"
 ALT="\begin{displaymath}
V^\dag = - \frac{e  A_0  \mbox{\boldmath$\epsilon$}\!\cdot\!{\bf p} }{2 m_e}  \exp( {\rm i} {\bf k}\!\cdot\!{\bf r} ).
\end{displaymath}">|; 

$key = q/m_s=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2066.png"
 ALT="$m_s=0$">|; 

$key = q/{displaymath}H_1=-frac{p^4}{8,m_e^{,3},c^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2292.png"
 ALT="\begin{displaymath}
H_1 = - \frac{p^4}{8 m_e^{ 3} c^2}.
\end{displaymath}">|; 

$key = q/displaystyleqleft(A,{{rm{e}^{,q,a}-B,{{rm{e}^{-q,a}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img860.png"
 ALT="$\displaystyle q\left(A  {\rm e}^{ q a}-B  {\rm e}^{-q a}\right)$">|; 

$key = q/chi_+leftrightarrowchi_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2460.png"
 ALT="$\chi_+\leftrightarrow\chi_-$">|; 

$key = q/displaystylembox{{boldmath{{epsilon{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2590.png"
 ALT="$\displaystyle \mbox{\boldmath$\epsilon$}$">|; 

$key = q/T=4,m,a^2slashpi,hbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1048.png"
 ALT="$T=4 m a^2/\pi \hbar$">|; 

$key = q/P(u_2),cdots,P(u_M);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img99.png"
 ALT="$P(u_2),\cdots, P(u_M)$">|; 

$key = q/{displaymath}int_{-infty}^inftypsi_i^ast,psi_j,dx=delta_{ij}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img681.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \psi_i^\ast \psi_j  dx = \delta_{ij}.
\end{displaymath}">|; 

$key = q/displaystyle[S_-,S_z];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1717.png"
 ALT="$\displaystyle [S_-,S_z]$">|; 

$key = q/displaystyleJ^2,Y_{l,m+1},chi_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1943.png"
 ALT="$\displaystyle J^2 Y_{l,m+1} \chi_-$">|; 

$key = q/{displaymath}H=-frac{hbar^2}{2,m}left(frac{partial^2}{partialr^2}+frac{2}{r}fracial}{partialr}-frac{L^2}{hbar^2,r^2}right)+V(r).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1529.png"
 ALT="\begin{displaymath}
H = -\frac{\hbar^2}{2 m}\left(\frac{\partial^2}{\partial r^...
...{\partial}{\partial r}- \frac{L^2}{\hbar^2 r^2}\right)
+V(r).
\end{displaymath}">|; 

$key = q/{displaymath}c(x')=int_{-infty}^inftypsi_x^ast(x,x'),psi(x),dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img717.png"
 ALT="\begin{displaymath}
c(x') = \int_{-\infty}^\infty \psi_x^\ast(x,x') \psi(x) dx.
\end{displaymath}">|; 

$key = q/{displaymath}langlexrangle=int_{-infty}^{infty}x,|psi|^{,2},dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img489.png"
 ALT="\begin{displaymath}
\langle x\rangle = \int_{-\infty}^{\infty} x \vert\psi\vert^{ 2} dx.
\end{displaymath}">|; 

$key = q/s,m_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2075.png"
 ALT="$s, m_s$">|; 

$key = q/{displaymath}mbox{{boldmath{{mu{}=-frac{e}{2,m_e},{{bf{L},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img1824.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\mu$}= -\frac{e}{2 m_e} {\bf L},
\end{displaymath}">|; 

$key = q/nu_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img453.png"
 ALT="$\nu_0$">|; 

$key = q/z^ast=x-{{rm{i},y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img200.png"
 ALT="$z^\ast = x-{\rm i} y$">|; 

$key = q/{displaymath}f(x)=frac{1}{sqrt{2pi}}int_{-infty}^{infty}bar{f}(k),{{rm{e}^{,{{rm{i},k,x},dk,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img323.png"
 ALT="\begin{displaymath}
f(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \bar{f}(k) {\rm e}^{ {\rm i} k x} dk,
\end{displaymath}">|; 

$key = q/{displaymath}c(p')=int_{-infty}^inftypsi_p^ast(x,p'),psi(x),dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img730.png"
 ALT="\begin{displaymath}
c(p') = \int_{-\infty}^\infty \psi_p^\ast(x,p') \psi(x) dx.
\end{displaymath}">|; 

$key = q/displaystyleL_-,Y_{l,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1380.png"
 ALT="$\displaystyle L_- Y_{l,m}$">|; 

$key = q/z=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1560.png"
 ALT="$z=0$">|; 

$key = q/{displaymath}j_r=v,|T|^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img825.png"
 ALT="\begin{displaymath}
j_r = v  \vert T\vert^{ 2}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{d}{dt}int_{-infty}^{infty}|psi(x,t)|^2,dx=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1182.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\int_{-\infty}^{\infty} \vert\psi(x,t)\vert^2 dx = 0.
\end{displaymath}">|; 

$key = q/{{bf{r}={{bf{0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2896.png"
 ALT="${\bf r} = {\bf0}$">|; 

$key = q/psi_i(x_i,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1100.png"
 ALT="$\psi_i(x_i,t)$">|; 

$key = q/{displaymath}frac{dlangleprangle}{dt}=-int_{-infty}^{infty}frac{dV}{dx},|psi|^{,2},dx=-leftlanglefrac{dV}{dx}rightrangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="309" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img513.png"
 ALT="\begin{displaymath}
\frac{d\langle p\rangle}{dt} =-\int_{-\infty}^{\infty} \frac...
...psi\vert^{ 2} dx =- \left\langle \frac{dV}{dx}\right\rangle.
\end{displaymath}">|; 

$key = q/{mitSigma};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="${\mit\Sigma}$">|; 

$key = q/partialjslashpartialx=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img827.png"
 ALT="$\partial j/\partial x = 0$">|; 

$key = q/3slash2,-1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2017.png"
 ALT="$3/2, -1/2$">|; 

$key = q/y_0(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="$y_0(z)$">|; 

$key = q/{{bf{k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img186.png"
 ALT="${\bf k}$">|; 

$key = q/Z_1=Z-2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$Z_1=Z-2$">|; 

$key = q/E_aneqE_b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1156.png"
 ALT="$E_a\neq E_b$">|; 

$key = q/{displaymath}frac{R_ast}{R_odot}=0.010left(frac{M_odot}{M}right)^{1slash3},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1289.png"
 ALT="\begin{displaymath}
\frac{R_\ast}{R_\odot}= 0.010\left(\frac{M_\odot}{M}\right)^{1/3},
\end{displaymath}">|; 

$key = q/displaystyle(L_x+{{rm{i},L_y),(L_x-{{rm{i},L_y)=L_x^{,2}+L_y^{,2}-{{rm{i},[L_x,L_y]=L_x^{,2}+L_y^{,2}+hbar,L_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="519" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1337.png"
 ALT="$\displaystyle (L_x+{\rm i} L_y) (L_x-{\rm i} L_y)
= L_x^{ 2} + L_y^{ 2} - {\rm i} [L_x, L_y]
= L_x^{ 2} + L_y^{ 2} + \hbar L_z$">|; 

$key = q/displaystylefrac{e^2,hbar^2}{16pi,epsilon_0,m_e^{,2},c^2},left[j,(j+1)-l,(l+1)-3slash4right],leftlanglefrac{1}{r^3}rightrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="385" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2322.png"
 ALT="$\displaystyle \frac{e^2 \hbar^2}{16\pi \epsilon_0 m_e^{ 2} c^2} \left[j (j+1)-l (l+1)-3/4\right] \left\langle\frac{1}{r^3}\right\rangle.$">|; 

$key = q/v=omegaslashk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img167.png"
 ALT="$v=\omega/k$">|; 

$key = q/{displaymath}{{rm{i},hbar,frac{partialpsi_{x'}}{partialt}=-frac{hbar^2}{2,mu}frartial^2psi_{x'}}{partialx'^{,2}}+V(x'),psi_{x'},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1123.png"
 ALT="\begin{displaymath}
{\rm i} \hbar \frac{\partial\psi_{x'}}{\partial t} = -\fr...
...c{\partial^2\psi_{x'}}{\partial x'^{ 2}} + V(x') \psi_{x'},
\end{displaymath}">|; 

$key = q/{displaymath}J={{rm{e}^{-X}left(1+X+frac{X^3}{3}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2799.png"
 ALT="\begin{displaymath}
J = {\rm e}^{-X}\left(1+X+\frac{X^3}{3}\right).
\end{displaymath}">|; 

$key = q/{displaymath}z=x+{{rm{i},y,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img196.png"
 ALT="\begin{displaymath}
z = x + {\rm i} y,
\end{displaymath}">|; 

$key = q/{displaymath}frac{dP_{irightarrowf}^{abs}}{dt}+w_{irightarrowf}^{abs},P_{irightarrowf}^{abs}=w_{irightarrowf}^{abs},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img2569.png"
 ALT="\begin{displaymath}
\frac{dP_{i\rightarrow f}^{abs}}{dt} + w_{i\rightarrow f}^{abs} P_{i\rightarrow f}^{abs} = w_{i\rightarrow f}^{abs},
\end{displaymath}">|; 

$key = q/{displaymath}langleHrangle=left[1+2,frac{(DpmE)}{(1pmJ)}right]E_0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2818.png"
 ALT="\begin{displaymath}
\langle H\rangle = \left[1+ 2 \frac{(D\pm E)}{(1\pm J)}\right] E_0,
\end{displaymath}">|; 

$key = q/{displaymath}frac{dTheta_{l,l}}{dtheta}-l,cottheta,Theta_{l,l}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1440.png"
 ALT="\begin{displaymath}
\frac{d\Theta_{l,l}}{d\theta} - l \cot\theta \Theta_{l,l} = 0.
\end{displaymath}">|; 

$key = q/displaystylem_{s_1},hbar,chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2044.png"
 ALT="$\displaystyle m_{s_1} \hbar
 \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)},$">|; 

$key = q/{displaymath}int_{-infty}^{infty}y^2,{{rm{e}^{-y^2},dy=frac{sqrt{pi}}{2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img501.png"
 ALT="\begin{displaymath}
\int_{-\infty}^{\infty} y^2 {\rm e}^{-y^2} dy = \frac{\sqrt{\pi}}{2},
\end{displaymath}">|; 

$key = q/3,{mitDelta}x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img331.png"
 ALT="$3 {\mit\Delta} x$">|; 

$key = q/displaystyle[L^2,z];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2188.png"
 ALT="$\displaystyle [L^2,z]$">|; 

$key = q/{displaymath}frac{dsigma}{dOmega}simeqleft(frac{2,m,V_0}{hbar^2,mu}right)^2frac{1}{[2,k^2,(1-costheta)+mu^2]^{,2}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img2886.png"
 ALT="\begin{displaymath}
\frac{d\sigma}{d \Omega} \simeq \left(\frac{2 m  V_0}{ \hb...
...u}\right)^2
\frac{1}{[2 k^2  (1-\cos\theta) + \mu^2]^{ 2}},
\end{displaymath}">|; 

$key = q/{displaymath}psi^{(1)}_{l,1slash2;m,pm1slash2}=Y_{l,m},chi_pm.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="37" BORDER="0"
 SRC="|."$dir".q|img1922.png"
 ALT="\begin{displaymath}
\psi^{(1)}_{l,1/2;m,\pm 1/2} = Y_{l,m} \chi_\pm.
\end{displaymath}">|; 

$key = q/x=d,y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1032.png"
 ALT="$x = d y$">|; 

$key = q/{displaymath}alpha=frac{e^2}{4piepsilon_0,hbar,c}simeqfrac{1}{137}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2303.png"
 ALT="\begin{displaymath}
\alpha = \frac{e^2}{4\pi\epsilon_0 \hbar c}\simeq \frac{1}{137}
\end{displaymath}">|; 

$key = q/{displaymath}psi(x,t)propto(1+{{rm{i},beta_2)^{-1slash2},{{rm{e}^{,{{rm{i},(k_0,^{,2}slash4}int_{-infty}^infty{{rm{e}^{-z^2},dz.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img384.png"
 ALT="\begin{displaymath}
\psi(x,t)\propto (1+{\rm i} \beta_2)^{-1/2} {\rm e}^{ {\r...
...ta_2) \beta^{ 2}/4}\int_{-\infty}^\infty
{\rm e}^{-z^2} dz.
\end{displaymath}">|; 

$key = q/Eproptol^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1255.png"
 ALT="$E\propto l^2$">|; 

$key = q/nneqn';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1578.png"
 ALT="$n\neq n'$">|; 

$key = q/0,0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2072.png"
 ALT="$0, 0$">|; 

$key = q/{displaymath}S_z,psi^{(1)}_{m,pm1slash2}=pmfrac{hbar}{2},psi^{(1)}_{m,pm1slash2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2346.png"
 ALT="\begin{displaymath}
S_z \psi^{(1)}_{m,\pm 1/2} = \pm \frac{\hbar}{2} \psi^{(1)}_{m,\pm 1/2}.
\end{displaymath}">|; 

$key = q/{displaymath}H_0=frac{p^2}{2,m_e}-frac{e^2}{4piepsilon_0,r},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2172.png"
 ALT="\begin{displaymath}
H_0 =\frac{p^2}{2 m_e} -\frac{e^2}{4\pi\epsilon_0 r},
\end{displaymath}">|; 

$key = q/{displaymath}psi(x_1,x_2,t)={{rm{e}^{,{{rm{i},varphi},psi(x_2,x_1,t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1137.png"
 ALT="\begin{displaymath}
\psi(x_1,x_2,t) = {\rm e}^{ {\rm i} \varphi} \psi(x_2,x_1,t),
\end{displaymath}">|; 

$key = q/w_{irightarrowf}^{abs},tll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2572.png"
 ALT="$w_{i\rightarrow f}^{abs} t \ll 1$">|; 

$key = q/{displaymath}psi(x)=frac{{{rm{e}^{,{{rm{i},varphi}}{(2pi,sigma^2)^{1slash4}},{{rm{e}^{-(x-x_0)^{,2}slash(4,sigma^2)},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img468.png"
 ALT="\begin{displaymath}
\psi(x) = \frac{{\rm e}^{ {\rm i} \varphi}}{(2\pi \sigma^2)^{1/4}} {\rm e}^{-(x-x_0)^{ 2}/(4 \sigma^2)},
\end{displaymath}">|; 

$key = q/m_{s_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2035.png"
 ALT="$m_{s_1}$">|; 

$key = q/displaystyleA,{{rm{e}^{,q,a}+B,{{rm{e}^{-q,a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img859.png"
 ALT="$\displaystyle A  {\rm e}^{ q a}+ B  {\rm e}^{-q a}$">|; 

$key = q/{displaymath}k,x-omega,t+varphi=j,2pi,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img171.png"
 ALT="\begin{displaymath}
k x-\omega t+\varphi = j 2\pi,
\end{displaymath}">|; 

$key = q/displaystylepsi_1';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2143.png"
 ALT="$\displaystyle \psi_1'$">|; 

$key = q/{displaymath}H,psi_E=E,psi_E.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1090.png"
 ALT="\begin{displaymath}
H \psi_E = E \psi_E.
\end{displaymath}">|; 

$key = q/displaystyleY_{0,0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1473.png"
 ALT="$\displaystyle Y_{0,0}$">|; 

$key = q/displaystyleAleft[-frac{hbar^2}{2,m_e},nabla^2-frac{e^2}{4pi,epsilon_0}left(fracr_2}right)right]left[psi_0({{bf{r}_1)pmpsi_0({{bf{r}_2)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="415" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2804.png"
 ALT="$\displaystyle A\left[-\frac{\hbar^2}{2 m_e} \nabla^2 - \frac{e^2}{4\pi \epsi...
...frac{1}{r_2}\right)\right]
\left[\psi_0({\bf r}_1) \pm \psi_0({\bf r}_2)\right]$">|; 

$key = q/displaystylepsi^{(2)}_{1slash2,1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1987.png"
 ALT="$\displaystyle \psi^{(2)}_{1/2,1/2}$">|; 

$key = q/{displaymath}{{bf{J}={{bf{L}+{{bf{S}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img2316.png"
 ALT="\begin{displaymath}
{\bf J} = {\bf L} + {\bf S}
\end{displaymath}">|; 

$key = q/chi^nomath_inline{dagnomath_inline{,chigeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1752.png"
 ALT="$\chi^\dag \chi\geq 0$">|; 

$key = q/{displaymath}c_i=int_{-infty}^inftypsi_i^ast,psi_a,dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img2104.png"
 ALT="\begin{displaymath}
c_i = \int_{-\infty}^\infty\psi_i^\ast \psi_a dx,
\end{displaymath}">|; 

$key = q/{displaymath}H=H_1+H_2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2710.png"
 ALT="\begin{displaymath}
H = H_1 + H_2,
\end{displaymath}">|; 

$key = q/{displaymath}chi_-equivleft(array{{c}01array{right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1797.png"
 ALT="\begin{displaymath}
\chi_- \equiv \left(\begin{array}{c}0 1\end{array}\right),
\end{displaymath}">|; 

$key = q/{displaymath}K=h,nu-W.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img252.png"
 ALT="\begin{displaymath}
K = h \nu - W.
\end{displaymath}">|; 

$key = q/{displaymath}frac{dP_{x,in,a:b}}{dt}+j(b,t)-j(a,t)=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img480.png"
 ALT="\begin{displaymath}
\frac{d P_{x \in a:b}}{dt} + j(b,t) - j(a,t) = 0,
\end{displaymath}">|; 

$key = q/{displaymath}a_+,a_-,psi_n=n,psi_n,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1035.png"
 ALT="\begin{displaymath}
a_+ a_- \psi_n = n \psi_n,
\end{displaymath}">|; 

$key = q/U;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1273.png"
 ALT="$U$">|; 

$key = q/i=1,N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1071.png"
 ALT="$i=1, N$">|; 

$key = q/psi^{(2)}_{3slash2,pm1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1981.png"
 ALT="$\psi^{(2)}_{3/2,\pm 1/2}$">|; 

$key = q/{displaymath}psi(x)=frac{{{rm{e}^{,{{rm{i},varphi}}{(2pi,sigma_x^{,2})^{1slash4}^{-(x-langlexrangle)^{,2}slash(4,sigma_x^{,2})}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img503.png"
 ALT="\begin{displaymath}
\psi(x) = \frac{{\rm e}^{ {\rm i} \varphi}}{(2\pi \sigma_...
...}} {\rm e}^{-(x-\langle x\rangle)^{ 2}/(4 \sigma_x^{ 2})}.
\end{displaymath}">|; 

$key = q/{displaymath}int_{-infty}^infty|f|^{,2},dx=int_{-infty}^inftypsi^ast,(A-langleArangle)^{,2},psi,dx=sigma_A^{,2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="357" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img608.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \vert f\vert^{ 2} dx=\int_{-\infty}^...
...^\ast (A-\langle A \rangle)^{ 2} \psi dx = \sigma_A^{ 2},
\end{displaymath}">|; 

$key = q/r_1=r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2787.png"
 ALT="$r_1=r$">|; 

$key = q/{displaymath}V(r)=V_1(r)+V_2(r)left(3,frac{(mbox{{boldmath{sigma}_1cdot{{bf{r}),x{{boldmath{sigma}_1cdotmbox{{boldmath{sigma}_2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="543" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2087.png"
 ALT="\begin{displaymath}
V(r) = V_1(r) + V_2(r)\left(3 \frac{(\mbox{\boldmath $\sigm...
... \mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2,
\end{displaymath}">|; 

$key = q/{displaymath}psi(t)=sum_mc_m,expleft(-{{rm{i},E_m,tslashhbarright)psi_m.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2391.png"
 ALT="\begin{displaymath}
\psi(t) = \sum_m c_m \exp\left(-{\rm i} E_m t/\hbar\right)\psi_m.
\end{displaymath}">|; 

$key = q/{displaymath}f({{bf{k},{{bf{k}')=-frac{m}{2pi,sqrt{n},hbar^2}int{{rm{e}^{-{{rm{i{{bf{r}'},V({{bf{r}'),psi({{bf{r}'),d^3{{bf{r}'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="379" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2859.png"
 ALT="\begin{displaymath}
f({\bf k},{\bf k}') = -\frac{m}{2\pi \sqrt{n} \hbar^2}\int...
...f k}'\cdot{\bf r}'} V({\bf r}') \psi({\bf r}') d^3{\bf r}'.
\end{displaymath}">|; 

$key = q/{displaymath}|R|^{,2}+|T|^{,2}=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img829.png"
 ALT="\begin{displaymath}
\vert R\vert^{ 2} + \vert T\vert^{ 2} = 1.
\end{displaymath}">|; 

$key = q/{displaymath}tandelta_l=frac{j_l(k,a)}{y_l(k,a)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2961.png"
 ALT="\begin{displaymath}
\tan \delta_l = \frac{j_l(k a)}{y_l(k a)}.
\end{displaymath}">|; 

$key = q/{displaymath}sigma_A^{,2}=langleA^2rangle-langleArangle=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img687.png"
 ALT="\begin{displaymath}
\sigma_A^{ 2} = \langle A^2\rangle - \langle A\rangle = 0.
\end{displaymath}">|; 

$key = q/{displaymath}psi_{n,l,m}(r,theta,phi)=R_{n,l}(r),Y_{l,m}(theta,phi),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2659.png"
 ALT="\begin{displaymath}
\psi_{n,l,m}(r,\theta,\phi) = R_{n,l}(r) Y_{l,m}(\theta,\phi),
\end{displaymath}">|; 

$key = q/{displaymath}psi(x,t)=bar{psi},{{rm{e}^{,{{rm{i},(k,x-omega,t)},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img312.png"
 ALT="\begin{displaymath}
\psi(x,t) = \bar{\psi} {\rm e}^{ {\rm i} (k x-\omega t)},
\end{displaymath}">|; 

$key = q/displaystylephi_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img777.png"
 ALT="$\displaystyle \phi_1$">|; 

$key = q/|x|rightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img805.png"
 ALT="$\vert x\vert\rightarrow\infty$">|; 

$key = q/{scriptstyle-1slashsqrt{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2073.png"
 ALT="${\scriptstyle -1/\sqrt{2}}$">|; 

$key = q/k',a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3010.png"
 ALT="$k' a$">|; 

$key = q/{displaymath}frac{d^2psi}{dx^2}=frac{2,m}{hbar^2}left(frac{1}{2},K,x^2-Eright)psi.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img997.png"
 ALT="\begin{displaymath}
\frac{d^2\psi}{dx^2} = \frac{2 m}{\hbar^2}\left(\frac{1}{2} K x^2-E\right)\psi.
\end{displaymath}">|; 

$key = q/displaystylefrac{a_0^{,2},n^2}{2},[5,n^2+1-3,l,(l+1)],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1654.png"
 ALT="$\displaystyle \frac{a_0^{ 2} n^2}{2} [5 n^2+1-3 l (l+1)],$">|; 

$key = q/{displaymath}DeltaE_{2P}simfrac{hbar}{tau_{2P}}sim4times10^{-7},{{rm{eV}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2675.png"
 ALT="\begin{displaymath}
\Delta E_{2P} \sim \frac{\hbar}{\tau_{2P}}\sim 4\times 10^{-7} {\rm eV}.
\end{displaymath}">|; 

$key = q/displaystyle-2,hbar^2,(L^2,z-z,L^2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2198.png"
 ALT="$\displaystyle -2 \hbar^2 (L^2 z-z L^2),$">|; 

$key = q/l_{{rm{max}=k,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2983.png"
 ALT="$l_{\rm max} = k a$">|; 

$key = q/bar{psi}(k);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img319.png"
 ALT="$\bar{\psi}(k)$">|; 

$key = q/displaystyle{{rm{i},hbar,L_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1320.png"
 ALT="$\displaystyle {\rm i} \hbar L_z,$">|; 

$key = q/{displaymath}frac{d^2u_l}{dr^2}+left[k^2-frac{l,(l+1)}{r^2}-frac{2,m}{hbar^2},Vright]u_l=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2955.png"
 ALT="\begin{displaymath}
\frac{d^2 u_l}{d r^2} +\left[k^2 -\frac{l (l+1)}{r^2} -\frac{2 m}{\hbar^2}  V\right] u_l = 0.
\end{displaymath}">|; 

$key = q/displaystyle1+R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img838.png"
 ALT="$\displaystyle 1 + R$">|; 

$key = q/{figure}{epsfysize{=2.8in{centerline{epsffile{Chapter05slashfig01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="304" BORDER="0"
 SRC="|."$dir".q|img849.png"
 ALT="\begin{figure}
\epsfysize =2.8in
\centerline{\epsffile{Chapter05/fig01.eps}}
\end{figure}">|; 

$key = q/l=(0,1,2,3,cdots);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2328.png"
 ALT="$l=(0,1,2,3,\cdots)$">|; 

$key = q/nu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$\nu $">|; 

$key = q/psi_i(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img737.png"
 ALT="$\psi_i(x)$">|; 

$key = q/{displaymath}x,psi_x(x,x')=x',psi_x(x,x'){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img707.png"
 ALT="\begin{displaymath}
x \psi_x(x,x') = x' \psi_x(x,x')
\end{displaymath}">|; 

$key = q/z=R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2778.png"
 ALT="$z=R$">|; 

$key = q/{displaymath}H_0,psi_m=E_m,psi_m.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2388.png"
 ALT="\begin{displaymath}
H_0 \psi_m = E_m \psi_m.
\end{displaymath}">|; 

$key = q/psi_{nlm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2174.png"
 ALT="$\psi_{nlm}$">|; 

$key = q/y+dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img293.png"
 ALT="$y+dy$">|; 

$key = q/{displaymath}j_l(y)=frac{(-{{rm{i})^l}{2}int_{-1}^1exp(,{{rm{i},y,mu),P_l(mu),dmu,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img2919.png"
 ALT="\begin{displaymath}
j_l(y) = \frac{(-{\rm i})^l}{2} \int_{-1}^1 \exp( {\rm i}  y \mu)
 P_l(\mu) d\mu,
\end{displaymath}">|; 

$key = q/{displaymath}p=frac{E}{c}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img258.png"
 ALT="\begin{displaymath}
p = \frac{E}{c}.
\end{displaymath}">|; 

$key = q/I(y)propto|psi(y,t)|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img291.png"
 ALT="$I(y) \propto \vert\psi(y,t)\vert^{ 2}$">|; 

$key = q/{displaymath}w_{2Prightarrow1S}=left(frac{2}{3}right)^8,alpha^5,frac{m_e,c^2}{hbar}=6.27times10^8,{{rm{s}^{-1}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="340" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2672.png"
 ALT="\begin{displaymath}
w_{2P\rightarrow 1S} = \left(\frac{2}{3}\right)^8 \alpha^5 \frac{m_e c^2}{\hbar} = 6.27\times 10^8 {\rm s}^{-1}
\end{displaymath}">|; 

$key = q/x_3=z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1201.png"
 ALT="$x_3=z$">|; 

$key = q/v_g=domegaslashdk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img398.png"
 ALT="$v_g=d\omega/dk$">|; 

$key = q/{displaymath}intpsi^ast_{n',l',m'},psi_{n,l,m},dV=delta_{nn'},delta_{ll'},delta_{mm'},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1632.png"
 ALT="\begin{displaymath}
\int \psi^\ast_{n',l',m'} \psi_{n,l,m} dV = \delta_{nn'} \delta_{ll'} \delta_{mm'},
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{S}times{{bf{S}={{rm{i},hbar,{{bf{S}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1868.png"
 ALT="\begin{displaymath}
{\bf S}\times{\bf S} = {\rm i} \hbar {\bf S}.
\end{displaymath}">|; 

$key = q/varphi=pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1141.png"
 ALT="$\varphi=\pi$">|; 

$key = q/dOmega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2861.png"
 ALT="$d\Omega$">|; 

$key = q/{displaymath}psi_i^nomath_inline{dagnomath_inline{,psi_j=delta_{ij}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2101.png"
 ALT="\begin{displaymath}
\psi_i^\dag \psi_j = \delta_{ij}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{d^2psi}{dx^2}=-k^2,psi{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img795.png"
 ALT="\begin{displaymath}
\frac{d^2 \psi}{d x^2} = - k^2 \psi
\end{displaymath}">|; 

$key = q/{displaymath}a_0=frac{4piepsilon_0,hbar^2}{m_e,e^2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2224.png"
 ALT="\begin{displaymath}
a_0 = \frac{4\pi\epsilon_0 \hbar^2}{m_e e^2}
\end{displaymath}">|; 

$key = q/displaystylez^lleft(-frac{1}{z}frac{d}{dz}right)^lleft(frac{sinz}{z}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1548.png"
 ALT="$\displaystyle z^l\left(-\frac{1}{z}\frac{d}{dz}\right)^l\left(\frac{\sin z}{z}\right),$">|; 

$key = q/p^4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2304.png"
 ALT="$p^4$">|; 

$key = q/{displaymath}w_{irightarrowf}^{spn}=left(frac{N_f}{N_i}-1right)frac{pi}{3,epsilon_0,hbar^2},d^{,2}_{if},rho(omega_{if}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2608.png"
 ALT="\begin{displaymath}
w_{i\rightarrow f}^{spn} = \left(\frac{N_f}{N_i}-1\right)\frac{\pi}{3 \epsilon_0 \hbar^2} d^{ 2}_{if} \rho(\omega_{if}).
\end{displaymath}">|; 

$key = q/{displaymath}w_n=frac{n,e^2,omega_0^{,2}}{6pi,epsilon_0,m_e,c^3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2632.png"
 ALT="\begin{displaymath}
w_n = \frac{n e^2 \omega_0^{ 2}}{6\pi \epsilon_0 m_e c^3}.
\end{displaymath}">|; 

$key = q/-hbarslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1820.png"
 ALT="$-\hbar/2$">|; 

$key = q/{displaymath}H_{1,2},psi_{1,2}=E_{1,2},psi_{1,2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2714.png"
 ALT="\begin{displaymath}
H_{1,2} \psi_{1,2} = E_{1,2} \psi_{1,2},
\end{displaymath}">|; 

$key = q/{displaymath}psi_E=langle1|Erangle,psi_1+langle2|Erangle,psi_2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2124.png"
 ALT="\begin{displaymath}
\psi_E = \langle 1\vert E\rangle \psi_1 + \langle 2\vert E\rangle \psi_2.
\end{displaymath}">|; 

$key = q/z=infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1594.png"
 ALT="$z= \infty$">|; 

$key = q/O^{16};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1278.png"
 ALT="$O^{16}$">|; 

$key = q/{mitDelta}t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img637.png"
 ALT="${\mit\Delta} t$">|; 

$key = q/{{bf{L};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1500.png"
 ALT="${\bf L}$">|; 

$key = q/3,2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="$3,2$">|; 

$key = q/|psi_1|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img891.png"
 ALT="$\vert\psi_1\vert^{ 2}$">|; 

$key = q/{displaymath}int_{-infty}^inftyf(x),delta(x-x'),dx=f(x'),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img714.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty f(x) \delta(x-x') dx = f(x'),
\end{displaymath}">|; 

$key = q/displaystylec_+(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1841.png"
 ALT="$\displaystyle c_+(0)$">|; 

$key = q/hbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img434.png"
 ALT="$\hbar$">|; 

$key = q/{displaymath}langlepsi_i|psi_jrangleequivlanglei|jrangle=delta_{ij}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2102.png"
 ALT="\begin{displaymath}
\langle \psi_i\vert\psi_j\rangle \equiv \langle i\vert j\rangle = \delta_{ij}.
\end{displaymath}">|; 

$key = q/{displaymath}I=2,(1pmJ),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2785.png"
 ALT="\begin{displaymath}
I = 2 (1\pm J),
\end{displaymath}">|; 

$key = q/{displaymath}B,psi_i=b_i,psi_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img703.png"
 ALT="\begin{displaymath}
B \psi_i = b_i \psi_i,
\end{displaymath}">|; 

$key = q/S_pm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1709.png"
 ALT="$S_\pm$">|; 

$key = q/displaystyleS_1^{,2},chi_{s_1,s_2;s,m_s}^{(2)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2050.png"
 ALT="$\displaystyle S_1^{ 2}  \chi_{s_1,s_2;s,m_s}^{(2)}$">|; 

$key = q/displaystyleY_{2,0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1479.png"
 ALT="$\displaystyle Y_{2,0}$">|; 

$key = q/|psi_i(x_i,t)|^2,dx_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1098.png"
 ALT="$\vert\psi_i(x_i,t)\vert^2 dx_i$">|; 

$key = q/{displaymath}langleArangleequivlanglea|A|arangle=sum_{i,j}langlea|iranglelanglei|A|jranglelanglej|arangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2116.png"
 ALT="\begin{displaymath}
\langle A\rangle \equiv\langle a\vert A\vert a\rangle= \sum_...
...rangle
\langle i\vert A\vert j\rangle \langle j\vert a\rangle.
\end{displaymath}">|; 

$key = q/displaystylea',int_{-infty}^{infty}psi_a^ast,psi_{a'},dx.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img669.png"
 ALT="$\displaystyle a' \int_{-\infty}^{\infty}\psi_a^\ast \psi_{a'} dx.$">|; 

$key = q/psi(y)=h(y),exp(-y^2slash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1017.png"
 ALT="$\psi(y) = h(y) \exp(-y^2/2)$">|; 

$key = q/m_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1428.png"
 ALT="$m_+$">|; 

$key = q/displaystylem,,mpm1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2652.png"
 ALT="$\displaystyle m, m\pm 1.$">|; 

$key = q/{displaymath}langleS_xrangle=frac{hbar}{2}left(c_+^ast,c_-^astright)left(array{{,&0array{right)left(array{{c}c_+c_-array{right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1853.png"
 ALT="\begin{displaymath}
\langle S_x\rangle= \frac{\hbar}{2}\left(c_+^\ast, c_-^\ast\...
...rray}\right)\left(\begin{array}{c}c_+ c_-\end{array}\right),
\end{displaymath}">|; 

$key = q/mbox{{boldmath{{sigma{}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2088.png"
 ALT="$\mbox{\boldmath$\sigma$}_1$">|; 

$key = q/Z<2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2755.png"
 ALT="$Z&lt;2$">|; 

$key = q/displaystyleint_{-infty}^infty(A,psi_a)^ast,psi_{a'},dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img666.png"
 ALT="$\displaystyle \int_{-\infty}^\infty (A \psi_a)^\ast \psi_{a'} dx$">|; 

$key = q/{displaymath}w_{irightarrowf}^{spn}simalpha^3,omega_{if},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img2618.png"
 ALT="\begin{displaymath}
w_{i\rightarrow f}^{spn} \sim \alpha^3 \omega_{if},
\end{displaymath}">|; 

$key = q/|c_a|^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1741.png"
 ALT="$\vert c_a\vert^2$">|; 

$key = q/{displaymath}psi_n'=psi_n+sum_{kneqn}frac{e_{kn}}{E_n-E_k},psi_k+{calO}(epsilon^2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2170.png"
 ALT="\begin{displaymath}
\psi_n' = \psi_n + \sum_{k\neq n} \frac{e_{kn}}{E_n-E_k} \psi_k + {\cal O}(\epsilon^2).
\end{displaymath}">|; 

$key = q/{displaymath}delta_0=-k,a.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img2963.png"
 ALT="\begin{displaymath}
\delta_0 = -k a.
\end{displaymath}">|; 

$key = q/2,0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="$2,0$">|; 

$key = q/displaystyle1.83,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img944.png"
 ALT="$\displaystyle 1.83,$">|; 

$key = q/psi_0({{bf{r}_{1,2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2802.png"
 ALT="$\psi_0({\bf r}_{1,2})$">|; 

$key = q/displaystyleE_2-frac{|e_{12}|^2}{E_1-E_2}+cdots.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2142.png"
 ALT="$\displaystyle E_2 - \frac{\vert e_{12}\vert^2}{E_1-E_2}+\cdots.$">|; 

$key = q/displaystylefrac{{{rm{e}^{+{{rm{i},p_0,xslashhbar}}{(2pi,sigma_x^{,2})^{1slash4}},{{rm{e}^{-(x-x_0)^{,2}slash4,sigma_x^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img628.png"
 ALT="$\displaystyle \frac{{\rm e}^{+{\rm i} p_0 x/\hbar}}{(2\pi \sigma_x^{ 2})^{1/4}} {\rm e}^{-(x-x_0)^{ 2}/4 \sigma_x^{ 2}},$">|; 

$key = q/{displaymath}L^2=x_i,x_i,p_j,p_j-x_i,p_ileft(x_j,p_j-[x_j,p_j]right)-2,{{rm{i},hbar,x_i,p_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="403" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1522.png"
 ALT="\begin{displaymath}
L^2 = x_i x_i p_j p_j - x_i p_i\left(x_j p_j - [x_j,p_j]\right) - 2 {\rm i} \hbar x_i p_i.
\end{displaymath}">|; 

$key = q/hat{{bf{r}slashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2855.png"
 ALT="$\hat{\bf r}/r$">|; 

$key = q/w_{irightarrowf}^{spn}llomega_{if};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2619.png"
 ALT="$w_{i\rightarrow f}^{spn} \ll \omega_{if}$">|; 

$key = q/|R|^{,2}=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img868.png"
 ALT="$\vert R\vert^{ 2}=1$">|; 

$key = q/{displaymath}-l,,-l+1,,cdots,0,,cdots,,l-1,,l.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1430.png"
 ALT="\begin{displaymath}
-l, -l+1, \cdots  0,  \cdots,  l-1,  l.
\end{displaymath}">|; 

$key = q/{{bf{S};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1866.png"
 ALT="${\bf S}$">|; 

$key = q/displaystylec_2(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2421.png"
 ALT="$\displaystyle c_2(t)$">|; 

$key = q/displaystylembox{hspace{1cm}}timessindelta_l,sindelta_{l'},P_l(mu),P_{l'}(mu),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2939.png"
 ALT="$\displaystyle \mbox{\hspace{1cm}}\times \sin\delta_l  \sin\delta_{l'} 
P_l(\mu)  P_{l'}(\mu),$">|; 

$key = q/2000,AA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img446.png"
 ALT="$2000 \AA$">|; 

$key = q/{displaymath}int_{-infty}^inftypsi_1^ast,(A,psi_2),dx=int_{-infty}^infty(A,psi_1)^ast,psi_2,dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img654.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \psi_1^\ast (A \psi_2) dx = \int_{-\infty}^\infty
(A \psi_1)^\ast \psi_2 dx.
\end{displaymath}">|; 

$key = q/{displaymath}psi(r,theta)=sum_lR_l(r),P_l(costheta),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2908.png"
 ALT="\begin{displaymath}
\psi(r,\theta) = \sum_l R_l(r)  P_l(\cos\theta),
\end{displaymath}">|; 

$key = q/displaystyle{{rm{i},hbarleft(-L_y,L_z-L_z,L_y+L_z,L_y+L_y,L_zright)=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1330.png"
 ALT="$\displaystyle {\rm i} \hbar\left(-L_y L_z - L_z L_y + L_z L_y + L_y L_z\right) = 0,$">|; 

$key = q/{displaymath}U(Z)=frac{e^2}{4pi,epsilon_0}left(frac{[Z-2]}{r_1}+frac{[Z-2]}{r_2}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2760.png"
 ALT="\begin{displaymath}
U(Z) = \frac{e^2}{4\pi \epsilon_0}\left(\frac{[Z-2]}{r_1} + \frac{[Z-2]}{r_2}\right).
\end{displaymath}">|; 

$key = q/displaystylefrac{d^2omega(k_0)}{dk^2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img372.png"
 ALT="$\displaystyle \frac{d^2\omega(k_0)}{dk^2}.$">|; 

$key = q/{displaymath}w_{irightarrowf}simalpha^3,omega_{if}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2685.png"
 ALT="\begin{displaymath}
w_{i\rightarrow f} \sim \alpha^3 \omega_{if}.
\end{displaymath}">|; 

$key = q/x_1=x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1199.png"
 ALT="$x_1 =x$">|; 

$key = q/{{bf{d}_{if};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2585.png"
 ALT="${\bf d}_{if}$">|; 

$key = q/displaystylesqrt{frac{1}{3}},psi^{(2)}_{3slash2,-1slash2}+sqrt{frac{2}{3}},psi^{(2)}_{1slash2,-1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2002.png"
 ALT="$\displaystyle \sqrt{\frac{1}{3}} \psi^{(2)}_{3/2,-1/2} +
\sqrt{ \frac{2}{3}} \psi^{(2)}_{1/2,-1/2},$">|; 

$key = q/l_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1241.png"
 ALT="$l_y$">|; 

$key = q/{displaymath}|{{bf{r}-{{bf{r}'|simeqr-hat{{bf{r}cdot{{bf{r}'{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2853.png"
 ALT="\begin{displaymath}
\vert{\bf r}-{\bf r}'\vert \simeq r - \hat{\bf r}\cdot {\bf r}'
\end{displaymath}">|; 

$key = q/l'=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2242.png"
 ALT="$l'=1$">|; 

$key = q/{displaymath}psi_{n,l,m}(r,theta,phi)=R_{n,l}(r),Y_{l,m}(theta,phi).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1627.png"
 ALT="\begin{displaymath}
\psi_{n,l,m}(r,\theta,\phi) = R_{n,l}(r) Y_{l,m}(\theta,\phi).
\end{displaymath}">|; 

$key = q/displaystyleleftlanglefrac{1}{r^2}rightrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1659.png"
 ALT="$\displaystyle \left\langle\frac{1}{r^2}\right\rangle$">|; 

$key = q/{displaymath}V(x)=left{array{{lcl}0&mbox{hspace{1cm}}&mbox{for{0leqxleqa{}[0.5ex]infty&&mbox{otherwise}array{right..{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img788.png"
 ALT="\begin{displaymath}
V(x) = \left\{\begin{array}{lcl}
0&amp;\mbox{\hspace{1cm}}&amp;\mbox...
...leq a$} [0.5ex]
\infty&amp;&amp;\mbox{otherwise}
\end{array}\right..
\end{displaymath}">|; 

$key = q/q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2520.png"
 ALT="$q$">|; 

$key = q/{displaymath}int_{-infty}^infty|g|^{,2},dx=sigma_B^{,2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img611.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \vert g\vert^{ 2} dx = \sigma_B^{ 2},
\end{displaymath}">|; 

$key = q/phi=2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1402.png"
 ALT="$\phi=2\pi$">|; 

$key = q/{displaymath}L_+,Y_{l,l}(theta,phi)=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1436.png"
 ALT="\begin{displaymath}
L_+ Y_{l,l}(\theta,\phi) = 0,
\end{displaymath}">|; 

$key = q/s+ds;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img136.png"
 ALT="$s+ds$">|; 

$key = q/{displaymath}psi^{(2)}_{j,m_j}=left(frac{j+1-m_j}{2,l+1}right)^{1slash2}psi^{(1)ight)^{1slash2},psi^{(1)}_{m_j+1slash2,-1slash2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="552" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2345.png"
 ALT="\begin{displaymath}
\psi^{(2)}_{j,m_j} = \left(\frac{j+1-m_j}{2 l+1}\right)^{1/...
...\frac{j+1+m_j}{2 l+1}\right)^{1/2} \psi^{(1)}_{m_j+1/2,-1/2}
\end{displaymath}">|; 

$key = q/{displaymath}psi(t)=sum_nc_n(t),exp(-{{rm{i},omega_n,t),psi_n,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2464.png"
 ALT="\begin{displaymath}
\psi(t)= \sum_n c_n(t) \exp(-{\rm i} \omega_n t) \psi_n,
\end{displaymath}">|; 

$key = q/{displaymath}G=frac{U}{c}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img246.png"
 ALT="\begin{displaymath}
G = \frac{U}{c}.
\end{displaymath}">|; 

$key = q/displaystyle{{rm{i},hbar,L_x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1322.png"
 ALT="$\displaystyle {\rm i} \hbar L_x,$">|; 

$key = q/{displaymath}H_1=-mbox{{boldmath{{mu{}cdot{{bf{B},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2339.png"
 ALT="\begin{displaymath}
H_1 = -\mbox{\boldmath$\mu$}\cdot{\bf B},
\end{displaymath}">|; 

$key = q/{displaymath}B=-V,frac{partialP}{partialV}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1266.png"
 ALT="\begin{displaymath}
B = - V \frac{\partial P}{\partial V}
\end{displaymath}">|; 

$key = q/{displaymath}psi(x,t)=psi_0,{{rm{e}^{,{{rm{i},(k,x-omega,t)}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img487.png"
 ALT="\begin{displaymath}
\psi(x,t) = \psi_0 {\rm e}^{ {\rm i} (k x-\omega t)}
\end{displaymath}">|; 

$key = q/l_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1240.png"
 ALT="$l_x$">|; 

$key = q/{displaymath}chi_pmequivchi_{1slash2,pm1slash2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1777.png"
 ALT="\begin{displaymath}
\chi_\pm \equiv \chi_{1/2,\pm 1/2}.
\end{displaymath}">|; 

$key = q/H_0,psi_n=E_n,psi_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2461.png"
 ALT="$H_0 \psi_n=E_n \psi_n$">|; 

$key = q/{{bf{k}'=k,hat{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2856.png"
 ALT="${\bf k}'=k \hat{\bf r}$">|; 

$key = q/x_3equivz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1497.png"
 ALT="$x_3\equiv z$">|; 

$key = q/{displaymath}bar{psi}(k)propto{{rm{e}^{-{{rm{i},k,x_0}int_{-infty}^{infty}expleft[-{{rm{i},beta,y-y^2right]dy,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img339.png"
 ALT="\begin{displaymath}
\bar{\psi}(k) \propto {\rm e}^{-{\rm i} k x_0}
\int_{-\infty}^{\infty}\exp\left[-{\rm i} \beta y - y^2\right] dy,
\end{displaymath}">|; 

$key = q/{displaymath}{calE}=K+U,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1272.png"
 ALT="\begin{displaymath}
{\cal E} = K + U,
\end{displaymath}">|; 

$key = q/Y_{l,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1369.png"
 ALT="$Y_{l,m}$">|; 

$key = q/2slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2006.png"
 ALT="$2/3$">|; 

$key = q/k,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2968.png"
 ALT="$k a$">|; 

$key = q/displaystyle[L_z,L_x];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1323.png"
 ALT="$\displaystyle [L_z,L_x]$">|; 

$key = q/displaystylegamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2458.png"
 ALT="$\displaystyle \gamma$">|; 

$key = q/{displaymath}psi_x(x,x')=delta(x-x').{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img711.png"
 ALT="\begin{displaymath}
\psi_x(x,x') = \delta(x-x').
\end{displaymath}">|; 

$key = q/{displaymath}nabla^2equivfrac{partial^2}{partialx^2}+frac{partial^2}{partialy^2}+frac{partial^2}{partialz^2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1207.png"
 ALT="\begin{displaymath}
\nabla^2 \equiv \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial
y^2} + \frac{\partial^2}{\partial z^2}
\end{displaymath}">|; 

$key = q/|psi|^2=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1608.png"
 ALT="$\vert\psi\vert^2=0$">|; 

$key = q/{displaymath}psi(x_2,x_1,t)=psi(x_1,x_2,t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1142.png"
 ALT="\begin{displaymath}
\psi(x_2,x_1,t) = \psi(x_1,x_2,t),
\end{displaymath}">|; 

$key = q/t=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img142.png"
 ALT="$t=0$">|; 

$key = q/{displaymath}z=r,{{rm{e}^{,{{rm{i},theta},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img216.png"
 ALT="\begin{displaymath}
z = r {\rm e}^{ {\rm i} \theta},
\end{displaymath}">|; 

$key = q/displaystylesqrt{frac{1}{3}},psi^{(1)}_{-1,1slash2}+sqrt{frac{2}{3}},psi^{(1)}_{0,-1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1992.png"
 ALT="$\displaystyle \sqrt{\frac{1}{3}} \psi^{(1)}_{-1,1/2} +
\sqrt{ \frac{2}{3}} \psi^{(1)}_{0,-1/2},$">|; 

$key = q/{displaymath}L^2=x_i,x_i,p_j,p_j-x_i,p_i,x_j,p_j+{{rm{i},hbar,x_i,p_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1524.png"
 ALT="\begin{displaymath}
L^2 = x_i x_i p_j p_j - x_i p_i x_j p_j + {\rm i} \hbar x_i p_i.
\end{displaymath}">|; 

$key = q/displaystyleS_z,chi_pm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1778.png"
 ALT="$\displaystyle S_z \chi_\pm$">|; 

$key = q/{displaymath}psi(x)=A,{{rm{e}^{-k,x}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img968.png"
 ALT="\begin{displaymath}
\psi(x) = A {\rm e}^{-k x}.
\end{displaymath}">|; 

$key = q/{displaymath}{{rm{i},hbar,frac{partialpsi}{partialt}=-frac{hbar^2}{2,mu}frac{partial^2psi}{partialx^{,2}}+V(x),psi.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1131.png"
 ALT="\begin{displaymath}
{\rm i} \hbar \frac{\partial\psi}{\partial t} = -\frac{\h...
... \mu}
\frac{\partial^2\psi}{\partial x^{ 2}} + V(x) \psi.
\end{displaymath}">|; 

$key = q/displaystylepsi^{(1)}_{m,1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1959.png"
 ALT="$\displaystyle \psi^{(1)}_{m,1/2}$">|; 

$key = q/{displaymath}omega(k)=frac{hbar,k^2}{2,m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img429.png"
 ALT="\begin{displaymath}
\omega(k) = \frac{\hbar k^2}{2 m}.
\end{displaymath}">|; 

$key = q/P=n,k_B,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1263.png"
 ALT="$P= n k_B T$">|; 

$key = q/l=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2076.png"
 ALT="$l=1/2$">|; 

$key = q/{displaymath}frac{dsigma}{dOmega}=|f({{bf{k},{{bf{k}')|^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2868.png"
 ALT="\begin{displaymath}
\frac{d\sigma}{d\Omega} = \vert f({\bf k},{\bf k}')\vert^2.
\end{displaymath}">|; 

$key = q/n',l',m';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2181.png"
 ALT="$n',l',m'$">|; 

$key = q/displaystylez;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1349.png"
 ALT="$\displaystyle z$">|; 

$key = q/{figure}{epsfysize{=3.25in{centerline{epsffile{Chapter03slashfig03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="371" HEIGHT="369" BORDER="0"
 SRC="|."$dir".q|img210.png"
 ALT="\begin{figure}
\epsfysize =3.25in
\centerline{\epsffile{Chapter03/fig03.eps}}
\end{figure}">|; 

$key = q/{displaymath}U=frac{E_y^{,2}}{epsilon_0},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img243.png"
 ALT="\begin{displaymath}
U = \frac{E_y^{ 2}}{\epsilon_0},
\end{displaymath}">|; 

$key = q/{displaymath}phi(k)=k,x-omega(k),t.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img355.png"
 ALT="\begin{displaymath}
\phi(k) = k x - \omega(k) t.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter05slashfig08.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="337" HEIGHT="327" BORDER="0"
 SRC="|."$dir".q|img989.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter05/fig08.eps}}
\end{figure}">|; 

$key = q/{displaymath}L^2=L_+,L_-+L_z^{,2}-hbar,L_z{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1422.png"
 ALT="\begin{displaymath}
L^2 = L_+ L_-+L_z^{ 2} - \hbar L_z
\end{displaymath}">|; 

$key = q/{{bf{n}=(1,,0,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img178.png"
 ALT="${\bf n} = (1, 0, 0)$">|; 

$key = q/displaystylephi(p);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img629.png"
 ALT="$\displaystyle \phi(p)$">|; 

$key = q/psi(x)=delta(x-x_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img582.png"
 ALT="$\psi(x) = \delta(x-x_0)$">|; 

$key = q/chi^nomath_inline{dagnomath_inline{,chi'=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1732.png"
 ALT="$\chi^\dag \chi' = 0$">|; 

$key = q/{displaymath}H(x_1,x_2)=-frac{hbar^2}{2,m_1}frac{partial^2}{partialx_1^{,2}}-fram_2}frac{partial^2}{partialx_2^{,2}}+V(x_1-x_2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="406" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1112.png"
 ALT="\begin{displaymath}
H(x_1,x_2) = -\frac{\hbar^2}{2 m_1}\frac{\partial^2}{\part...
...^2}{2 m_2}\frac{\partial^2}{\partial x_2^{ 2}}+ V(x_1-x_2).
\end{displaymath}">|; 

$key = q/psi^{(2)}_{l,1slash2;j,m_j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1924.png"
 ALT="$\psi^{(2)}_{l,1/2;j,m_j}$">|; 

$key = q/psi_0({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2873.png"
 ALT="$\psi_0({\bf r})$">|; 

$key = q/V=-e^2slash(4piepsilon_0,r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2296.png"
 ALT="$V=-e^2/(4\pi\epsilon_0 r)$">|; 

$key = q/a>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1682.png"
 ALT="$a&gt;0$">|; 

$key = q/K;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="$K$">|; 

$key = q/{displaymath}(S_pm)^nomath_inline{dagnomath_inline{=S_mp.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1710.png"
 ALT="\begin{displaymath}
(S_\pm)^\dag = S_\mp.
\end{displaymath}">|; 

$key = q/omega=k,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2540.png"
 ALT="$\omega=k c$">|; 

$key = q/{displaymath}psi({{bf{r})_pm=Aleft[psi_0({{bf{r}_1)pmpsi_0({{bf{r}_2)right]{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2779.png"
 ALT="\begin{displaymath}
\psi({\bf r})_\pm = A\left[\psi_0({\bf r}_1) \pm \psi_0({\bf r}_2)\right]
\end{displaymath}">|; 

$key = q/|x|stackrel{_{{normalsize{<}}{_{{normalsize{sim}}pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2495.png"
 ALT="$\vert x\vert\stackrel {_{\normalsize &lt;}}{_{\normalsize\sim}}\pi$">|; 

$key = q/int|psi|^2,dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1616.png"
 ALT="$\int\vert\psi\vert^2 dV$">|; 

$key = q/E_0=(m_eslash2,hbar^2),(e^2slash4pi,epsilon_0)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1693.png"
 ALT="$E_0 = (m_e/2 \hbar^2) (e^2/4\pi \epsilon_0)^2$">|; 

$key = q/{displaymath}psi(x,t)=A,cos(k,x-omega,t+varphi),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img144.png"
 ALT="\begin{displaymath}
\psi(x,t) = A \cos(k x-\omega t+\varphi),
\end{displaymath}">|; 

$key = q/displaystylefrac{2,sqrt{2}}{27,sqrt{5},(3,a_0)^{3slash2}}left(frac{r}{a_0}right)^2expleft(-frac{r}{3,a_0}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1649.png"
 ALT="$\displaystyle \frac{2 \sqrt{2}}{27 \sqrt{5} (3 a_0)^{3/2}}
\left(\frac{r}{a_0}\right)^2 \exp\left(-\frac{r}{3 a_0}\right).$">|; 

$key = q/{displaymath}langlef|mbox{{boldmath{{epsilon{}!cdot!{{bf{p},exp(,{{rm{i},{{bf{k}{}!cdot!{{bf{p}),({{bf{k}!cdot!{{bf{r})|irangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2684.png"
 ALT="\begin{displaymath}
\langle f\vert\mbox{\boldmath$\epsilon$}\!\cdot\!{\bf p} \e...
...n$}\!\cdot\!{\bf p}) ({\bf k}\!\cdot\!{\bf r})\vert i\rangle.
\end{displaymath}">|; 

$key = q/{displaymath}c_i=int_{-infty}^{infty}psi_i^ast(x),psi(x,0),dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img743.png"
 ALT="\begin{displaymath}
c_i = \int_{-\infty}^{\infty} \psi_i^\ast(x) \psi(x,0) dx.
\end{displaymath}">|; 

$key = q/{displaymath}S_-,chi_{s,-s}=S_+,chi_{s,s}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1768.png"
 ALT="\begin{displaymath}
S_- \chi_{s,-s} = S_+ \chi_{s,s} = 0.
\end{displaymath}">|; 

$key = q/Xsimeq2.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2828.png"
 ALT="$X\simeq 2.5$">|; 

$key = q/gamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2412.png"
 ALT="$\gamma$">|; 

$key = q/displaystylefrac{X''}{X};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1219.png"
 ALT="$\displaystyle \frac{X''}{X}$">|; 

$key = q/{displaymath}psi({{bf{r})=sqrt{n}sum_lleft[A_l,j_l(k,r)+B_l,y_l(k,r)right]P_l(costheta),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="373" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2923.png"
 ALT="\begin{displaymath}
\psi({\bf r}) = \sqrt{n}\sum_l\left[
A_l j_l(k r) + B_l y_l(k r)\right] P_l(\cos\theta),
\end{displaymath}">|; 

$key = q/2times2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1793.png"
 ALT="$2\times 2$">|; 

$key = q/{displaymath}a_l={{rm{i}^{,l},(2,l+1),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2921.png"
 ALT="\begin{displaymath}
a_l = {\rm i}^{ l}  (2 l+1),
\end{displaymath}">|; 

$key = q/0,1,2,cdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2622.png"
 ALT="$0,1,2,\cdots$">|; 

$key = q/displaystyleleft[frac{sqrt{r_1^{,2}+r_2^{,2}-2,r_1,r_2,mu}}{r_1,r_2}right]_{+1}^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="88" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2742.png"
 ALT="$\displaystyle \left[\frac{\sqrt{r_1^{ 2}+r_2^{ 2}-2 r_1 r_2 \mu}}{r_1 r_2}\right]_{+1}^{-1}$">|; 

$key = q/displaystyleleft(frac{-{{rm{i},gamma}{Omega}right)exp!left[frac{-{{rm{i},(omega-omega_{21}),t}{2}right]sin(Omega,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2422.png"
 ALT="$\displaystyle \left(\frac{-{\rm i} \gamma}{\Omega}\right)\exp\!\left[\frac{-{\rm i} (\omega-\omega_{21}) t}{2}\right]\sin(\Omega t),$">|; 

$key = q/l^2=l_x^{,2}+l_y^{,2}+l_z^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1245.png"
 ALT="$l^2 = l_x^{ 2} + l_y^{ 2}+ l_z^{ 2}$">|; 

$key = q/(m+1),hbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1375.png"
 ALT="$(m+1) \hbar$">|; 

$key = q/{displaymath}{{rm{i},frac{partialpsi}{partialt}+frac{hbar}{2,m},frac{partial^2psr{psi}(k),{{rm{e}^{,{{rm{i},(k,x-omega,t)},dk=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="466" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img433.png"
 ALT="\begin{displaymath}
{\rm i} \frac{\partial\psi}{\partial t} +\frac{\hbar}{2 m}...
...t)\bar{\psi}(k) {\rm e}^{ {\rm i} (k x-\omega t)} dk= 0,
\end{displaymath}">|; 

$key = q/|k'|llk^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img880.png"
 ALT="$\vert k'\vert\ll k^2$">|; 

$key = q/d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1635.png"
 ALT="$d{\mit\Omega}$">|; 

$key = q/v_0+{mitDelta}vslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img406.png"
 ALT="$v_0+ {\mit\Delta}v/2$">|; 

$key = q/S_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1698.png"
 ALT="$S_x$">|; 

$key = q/displaystylel,(l+1),hbar^2,psi^{(1)}_{l,s;m,m_s},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1900.png"
 ALT="$\displaystyle l (l+1) \hbar^2 \psi^{(1)}_{l,s;m,m_s},$">|; 

$key = q/a_0rightarrow(2slashZ),a_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2764.png"
 ALT="$a_0\rightarrow (2/Z) a_0$">|; 

$key = q/{displaymath}v_g=frac{domega}{dk},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img391.png"
 ALT="\begin{displaymath}
v_g = \frac{d\omega}{dk},
\end{displaymath}">|; 

$key = q/|V_0|ggE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3002.png"
 ALT="$\vert V_0\vert\gg E$">|; 

$key = q/theta=j,2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img165.png"
 ALT="$\theta=j 2\pi$">|; 

$key = q/displaystylepsi_{E}(x_1,x_2,ldotsx_N);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1162.png"
 ALT="$\displaystyle \psi_{E}(x_1,x_2,\ldots x_N)$">|; 

$key = q/i^ast=-i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img476.png"
 ALT="$i^\ast= -i$">|; 

$key = q/|c(x')|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img718.png"
 ALT="$\vert c(x')\vert^{ 2}$">|; 

$key = q/d=(j-varphislash2pi),lambda+v,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img173.png"
 ALT="$d=(j-\varphi/2\pi) \lambda + v t$">|; 

$key = q/{displaymath}langleHrangle(1.69)=frac{1}{2}left(frac{3}{2}right)^6E_0=-77.5,{{rm{eV}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2774.png"
 ALT="\begin{displaymath}
\langle H\rangle(1.69) = \frac{1}{2}\left(\frac{3}{2}\right)^6 E_0 = -77.5 {\rm eV}.
\end{displaymath}">|; 

$key = q/{displaymath}beta_{l-}=beta_{l+}=infty,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2960.png"
 ALT="\begin{displaymath}
\beta_{l-} = \beta_{l+} = \infty,
\end{displaymath}">|; 

$key = q/O(p);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img587.png"
 ALT="$O(p)$">|; 

$key = q/displaystyleint_{-infty}^inftypsi_m,x,psi_n,dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1045.png"
 ALT="$\displaystyle \int_{-\infty}^\infty \psi_m x \psi_n dx$">|; 

$key = q/{displaymath}E=frac{E_0}{n^2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1620.png"
 ALT="\begin{displaymath}
E = \frac{E_0}{n^2}
\end{displaymath}">|; 

$key = q/{displaymath}H_1=frac{mu_0,g_p,e^2}{8pi,m_p,m_e},frac{3,({{bf{S}_pcdot{{bf{e}_r)_p,m_e},{{bf{S}_pcdot{{bf{S}_e,delta^3({{bf{r}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="546" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2372.png"
 ALT="\begin{displaymath}
H_1=\frac{\mu_0 g_p e^2}{8\pi m_p m_e} 
\frac{3 ({\bf ...
...e^2}{3 m_p m_e} {\bf S}_p\cdot{\bf S}_e \delta^3({\bf r}).
\end{displaymath}">|; 

$key = q/chi_{s_1,s_2;s,m_s}^{(2)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2049.png"
 ALT="$\chi_{s_1,s_2;s,m_s}^{(2)}$">|; 

$key = q/l'=m'=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2655.png"
 ALT="$l'=m'=0$">|; 

$key = q/{displaymath}[J^2,S^2]=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1893.png"
 ALT="\begin{displaymath}[J^2,S^2]=0.
\end{displaymath}">|; 

$key = q/N_f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2605.png"
 ALT="$N_f$">|; 

$key = q/displaystylecos^2(gamma,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2434.png"
 ALT="$\displaystyle \cos^2(\gamma t),$">|; 

$key = q/E_{ion}=13.60;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img445.png"
 ALT="$E_{ion} = 13.60$">|; 

$key = q/S_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1749.png"
 ALT="$S_-$">|; 

$key = q/H_0+H_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2098.png"
 ALT="$H_0+H_1$">|; 

$key = q/R_odot=7times10^5,{{rm{km};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1290.png"
 ALT="$R_\odot= 7\times 10^5 {\rm km}$">|; 

$key = q/{displaymath}psi(y)simeqA(y),{{rm{e}^{pmy^2slash2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1005.png"
 ALT="\begin{displaymath}
\psi(y) \simeq A(y) {\rm e}^{\pm y^2/2},
\end{displaymath}">|; 

$key = q/S_{1z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2032.png"
 ALT="$S_{1z}$">|; 

$key = q/displaystyleP_1(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2429.png"
 ALT="$\displaystyle P_1(t)$">|; 

$key = q/psi({{bf{r})simeqpsi_0({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2895.png"
 ALT="$\psi({\bf r})
\simeq \psi_0({\bf r})$">|; 

$key = q/x>x_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img895.png"
 ALT="$x&gt;x_2$">|; 

$key = q/displaystyleleft.+V_{fi}^nomath_inline{dagnomath_inline{,expleft[-{{rm{i},(omega2right]{{rm{sinc}left[(omega-omega_{fi}),tslash2right]right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="390" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2490.png"
 ALT="$\displaystyle \left.+V_{fi}^\dag \exp\left[-{\rm i} (\omega-\omega_{fi}) t/2\right]{\rm sinc}\left[(\omega-\omega_{fi}) t/2\right]\right),$">|; 

$key = q/z^l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1630.png"
 ALT="$z^l$">|; 

$key = q/{displaymath}{{rm{i},hbar,frac{partialpsi_i}{partialt}=H_i,psi_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1102.png"
 ALT="\begin{displaymath}
{\rm i} \hbar \frac{\partial \psi_i}{\partial t} = H_i \psi_i.
\end{displaymath}">|; 

$key = q/u_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img781.png"
 ALT="$u_2$">|; 

$key = q/{displaymath}P_n=hbar,omega_0,w_n=frac{e^2,omega_0^{,2}}{6pi,epsilon_0,m_e,c^3},[E_n-(1slash2),hbar,omega_0].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2633.png"
 ALT="\begin{displaymath}
P_n = \hbar \omega_0 w_n = \frac{e^2 \omega_0^{ 2}}{6\pi \epsilon_0 m_e c^3} [E_n -(1/2) \hbar \omega_0].
\end{displaymath}">|; 

$key = q/y=(x-x_0)slash(sqrt{2},sigma);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img464.png"
 ALT="$y=(x-x_0)/(\sqrt{2} \sigma)$">|; 

$key = q/psi^{(2)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1928.png"
 ALT="$\psi^{(2)}$">|; 

$key = q/displaystylesqrt{frac{2}{3}},psi^{(2)}_{3slash2,-1slash2}-sqrt{frac{1}{3}},psi^{(2)}_{1slash2,-1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2000.png"
 ALT="$\displaystyle \sqrt{\frac{2}{3}} \psi^{(2)}_{3/2,-1/2}
-\sqrt{ \frac{1}{3}} \psi^{(2)}_{1/2,-1/2},$">|; 

$key = q/displaystyles_1,(s_1+1),hbar^2,chi_{s_1,s_2;s,m_s}^{(2)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2051.png"
 ALT="$\displaystyle s_1 (s_1+1) \hbar^2
 \chi_{s_1,s_2;s,m_s}^{(2)},$">|; 

$key = q/displaystyleleftlangle({mitDelta}u)^2rightrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img126.png"
 ALT="$\displaystyle \left\langle({\mit\Delta} u)^2\right\rangle$">|; 

$key = q/{displaymath}int_{-infty}^inftypsi_p^ast(x,p'),psi_p(x,p''),dx=delta(p'-p'').{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="309" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img725.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \psi_p^\ast(x,p') \psi_p(x,p'') dx = \delta(p'-p'').
\end{displaymath}">|; 

$key = q/2,pm1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="$2,\pm 1$">|; 

$key = q/displaystyleleft(array{{cc}0,<col_mark>1<row_mark>1,<col_mark>0array{right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1810.png"
 ALT="$\displaystyle \left(\begin{array}{cc}0,&amp;1\\\\
1,&amp; 0\end{array}\right),$">|; 

$key = q/displaystyleq,(A-B).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img841.png"
 ALT="$\displaystyle q (A-B).$">|; 

$key = q/c_n,z^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1618.png"
 ALT="$c_n z^n$">|; 

$key = q/t';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2487.png"
 ALT="$t'$">|; 

$key = q/displaystylepsi^{(2)}_{3slash2,1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1985.png"
 ALT="$\displaystyle \psi^{(2)}_{3/2,1/2}$">|; 

$key = q/E_Fpropton^{2slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1265.png"
 ALT="$E_F\propto n^{2/3}$">|; 

$key = q/m=-1,0,1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2265.png"
 ALT="$m=-1,0,1$">|; 

$key = q/psi(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img327.png"
 ALT="$\psi(x)$">|; 

$key = q/n_f=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1680.png"
 ALT="$n_f=3$">|; 

$key = q/{{bf{B};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1831.png"
 ALT="${\bf B}$">|; 

$key = q/r>R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img915.png"
 ALT="$r&gt;R$">|; 

$key = q/displaystylefrac{partial}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1353.png"
 ALT="$\displaystyle \frac{\partial}{\partial y}$">|; 

$key = q/displaystyleR_{3,2}(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1648.png"
 ALT="$\displaystyle R_{3,2}(r)$">|; 

$key = q/langlen|mrangle=delta_{nm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2462.png"
 ALT="$\langle n\vert m\rangle=\delta_{nm}$">|; 

$key = q/|T|^{,2}=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img852.png"
 ALT="$\vert T\vert^{ 2}=1$">|; 

$key = q/{displaymath}frac{partial^2psi}{partialx^2}=int_{-infty}^{infty}(-k^2),bar{psi}(k),{{rm{e}^{,{{rm{i},(k,x-omega,t)},dk,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img431.png"
 ALT="\begin{displaymath}
\frac{\partial^2\psi}{\partial x^2} = \int_{-\infty}^{\infty...
...^2) \bar{\psi}(k) {\rm e}^{ {\rm i} (k x-\omega t)} dk,
\end{displaymath}">|; 

$key = q/E_b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1147.png"
 ALT="$E_b$">|; 

$key = q/Y(0)=Y(a)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1215.png"
 ALT="$Y(0)=Y(a)=0$">|; 

$key = q/{displaymath}langlef(u)rangleequivsum_{i=1}^{M}P(u_i),f(u_i).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img104.png"
 ALT="\begin{displaymath}
\langle f(u)\rangle \equiv \sum_{i=1}^{M} P(u_i)  f(u_i).
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}psimeq2,hbar,k,sinalphasimeqfrac{4pi,hbar,alpha}{lambda}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img425.png"
 ALT="\begin{displaymath}
{\mit\Delta} p \simeq 2 \hbar k \sin\alpha\simeq \frac{4\pi \hbar \alpha}{\lambda}.
\end{displaymath}">|; 

$key = q/displaystyleS_{2z},chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2045.png"
 ALT="$\displaystyle S_{2z}  \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}$">|; 

$key = q/|psi|^2rightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img957.png"
 ALT="$\vert\psi\vert^2\rightarrow 0$">|; 

$key = q/{displaymath}E=hbar,omega,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img255.png"
 ALT="\begin{displaymath}
E = \hbar \omega,
\end{displaymath}">|; 

$key = q/{displaymath}psi^{ast},frac{partialpsi}{partialt}=frac{{{rm{i},hbar}{2,m},psi^asi}{partialx^2}-frac{{{rm{i}}{hbar},V,|psi|^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img472.png"
 ALT="\begin{displaymath}
\psi^{\ast} \frac{\partial\psi}{\partial t}= \frac{{\rm i}\...
...partial x^2} - \frac{{\rm i}}{\hbar} V \vert\psi\vert^{ 2}.
\end{displaymath}">|; 

$key = q/{displaymath}(E_m-E_n),langlem|Erangle+e_{mn}simeq0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2164.png"
 ALT="\begin{displaymath}
(E_m-E_n) \langle m\vert E\rangle + e_{mn} \simeq 0,
\end{displaymath}">|; 

$key = q/displaystyle[x_i,p_j];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1085.png"
 ALT="$\displaystyle [x_i,p_j]$">|; 

$key = q/epsilon_{ilm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1511.png"
 ALT="$\epsilon_{ilm}$">|; 

$key = q/{displaymath}E=E_0,y^2-V_0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img976.png"
 ALT="\begin{displaymath}
E = E_0 y^2 - V_0,
\end{displaymath}">|; 

$key = q/psi^{(2)}_{1slash2,pm1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1982.png"
 ALT="$\psi^{(2)}_{1/2,\pm 1/2}$">|; 

$key = q/P_y(y)=P(y),dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img288.png"
 ALT="$P_y(y) = P(y) dy$">|; 

$key = q/{displaymath}beta_{l+}=k,aleft[frac{cosdelta_l,j_l'(k,a)-sindelta_l,y_l'(k,a)}{cosdelta_l,j_l(k,a)-sindelta_l,y_l(k,a)}right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2947.png"
 ALT="\begin{displaymath}
\beta_{l+} = k a \left[\frac{ \cos\delta_l j_l'(k a) -
\s...
...}{\cos\delta_l  
j_l(k a) - \sin\delta_l y_l(k a)}\right],
\end{displaymath}">|; 

$key = q/{displaymath}psi_n(x)=sqrt{frac{2}{a}},sinleft(n,pi,frac{x}{a}right){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img809.png"
 ALT="\begin{displaymath}
\psi_n(x) = \sqrt{\frac{2}{a}} \sin\left(n \pi \frac{x}{a}\right)
\end{displaymath}">|; 

$key = q/1slashmu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2883.png"
 ALT="$1/\mu$">|; 

$key = q/l,(l+1),hbar^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1368.png"
 ALT="$l (l+1) \hbar^2$">|; 

$key = q/V_0slashmurightarrowZ,Z',e^2slash4pi,epsilon_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2889.png"
 ALT="$V_0/\mu \rightarrow
Z Z'  e^2 / 4\pi \epsilon_0$">|; 

$key = q/n_f=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1678.png"
 ALT="$n_f=1$">|; 

$key = q/{displaymath}E_{total}=langleHrangle+frac{e^2}{4pi,epsilon_0,R}=langleHrangle-frac{2}{X},E_0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2820.png"
 ALT="\begin{displaymath}
E_{total} = \langle H\rangle + \frac{e^2}{4\pi \epsilon_0 R} = \langle H\rangle - \frac{2}{X} E_0,
\end{displaymath}">|; 

$key = q/{displaymath}sigma^2_x=int_{-infty}^{infty}left(x-langlexrangleright)^{,2},|psi|^{,2},dxequivlanglex^2rangle-langlexrangle^{2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="346" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img492.png"
 ALT="\begin{displaymath}
\sigma^2_x = \int_{-\infty}^{\infty} \left(x-\langle x\rangl...
...ert^{ 2} dx \equiv \langle x^2\rangle -\langle x\rangle^{2},
\end{displaymath}">|; 

$key = q/{{bf{e}_r={{bf{r}slashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2367.png"
 ALT="${\bf e}_r = {\bf r}/r$">|; 

$key = q/langlen,l,m|x|n',l',m'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2638.png"
 ALT="$\langle n,l,m\vert x\vert n',l',m'\rangle$">|; 

$key = q/{mitDelta}p=hbar,{mitDelta}k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img401.png"
 ALT="${\mit\Delta}p=\hbar {\mit\Delta}k$">|; 

$key = q/{displaymath}(d_x)_{n,n'}=langlen|e,x|n'rangle=eint_{-infty}^{infty}psi_n(x),x,psi_{n'}(x),dx{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="367" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img2625.png"
 ALT="\begin{displaymath}
(d_x)_{n,n'} = \langle n\vert e x\vert n'\rangle = e\int_{-\infty}^{\infty} \psi_n(x) x \psi_{n'}(x) dx
\end{displaymath}">|; 

$key = q/c_{k_{min}},z^{k_{min}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1603.png"
 ALT="$c_{k_{min}} z^{k_{min}}$">|; 

$key = q/du;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img117.png"
 ALT="$du$">|; 

$key = q/{displaymath}left(frac{d^2}{dz^2}+frac{2}{z}frac{d}{dz}-frac{l,(l+1)}{z^2}+frac{zeta}{z}-1right)R_{n,l}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1590.png"
 ALT="\begin{displaymath}
\left(\frac{d^2}{dz^2} + \frac{2}{z}\frac{d}{dz}-\frac{l (l+1)}{z^2}+ \frac{\zeta}{z}-1\right)
R_{n,l} = 0,
\end{displaymath}">|; 

$key = q/alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$\alpha $">|; 

$key = q/P_{irightarrowf}^{abs}(0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2570.png"
 ALT="$P_{i\rightarrow f}^{abs}(0)=0$">|; 

$key = q/l'=lpm1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2211.png"
 ALT="$l'=l\pm 1$">|; 

$key = q/n_f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1670.png"
 ALT="$n_f$">|; 

$key = q/0<y<sqrt{lambda};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img980.png"
 ALT="$0&lt; y&lt; \sqrt{\lambda}$">|; 

$key = q/Y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="$Y$">|; 

$key = q/P(y)proptoI(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img290.png"
 ALT="$P(y)\propto I(y)$">|; 

$key = q/{displaymath}int_0^aj_l(z_{n,l},rslasha),j_{l}(z_{n',l},rslasha),r^2,dr=0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img1577.png"
 ALT="\begin{displaymath}
\int_0^a j_l(z_{n,l} r/a) j_{l}(z_{n',l} r/a)  r^2 dr = 0
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{J}times{{bf{J}={{rm{i},hbar,{{bf{J}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1877.png"
 ALT="\begin{displaymath}
{\bf J}\times{\bf J} = {\rm i} \hbar {\bf J}.
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}xsimeqf,frac{lambda}{D}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img413.png"
 ALT="\begin{displaymath}
{\mit\Delta}x \simeq f \frac{\lambda}{D}.
\end{displaymath}">|; 

$key = q/{displaymath}p^2=-hbar^2left[frac{1}{r}frac{partial}{partialr}left(r,frac{partia{partial}{partialr}-frac{L^2}{hbar^2,r^2}right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1528.png"
 ALT="\begin{displaymath}
p^2 = -\hbar^2\left[\frac{1}{r}\frac{\partial}{\partial r}\l...
...}\frac{\partial}{\partial r}- \frac{L^2}{\hbar^2 r^2}\right].
\end{displaymath}">|; 

$key = q/{displaymath}psi_0({{bf{r})=sqrt{n},{{rm{e}^{,{{rm{i},{{bf{k}cdot{{bf{r}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2838.png"
 ALT="\begin{displaymath}
\psi_0({\bf r}) = \sqrt{n} {\rm e}^{ {\rm i} {\bf k}\cdot {\bf r}}
\end{displaymath}">|; 

$key = q/hbarslash{mitDelta}x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img404.png"
 ALT="$\hbar/{\mit\Delta}x$">|; 

$key = q/pislash(2,gamma);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2436.png"
 ALT="$\pi/(2 \gamma)$">|; 

$key = q/{displaymath}[x_j,p_j]equiv[x_1,p_1]+[x_2,p_2]+[x_3,p_3]=3,{{rm{i},hbar.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1523.png"
 ALT="\begin{displaymath}[x_j,p_j]\equiv [x_1,p_1]+[x_2,p_2]+[x_3,p_3] = 3 {\rm i} \hbar.
\end{displaymath}">|; 

$key = q/H_{nm}(t)=langlen|H_1(t)|mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2467.png"
 ALT="$H_{nm}(t)=\langle n\vert H_1(t)\vert m\rangle$">|; 

$key = q/e_{nl'nl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2239.png"
 ALT="$e_{nl'nl}$">|; 

$key = q/{displaymath}lambda=frac{h}{p}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img300.png"
 ALT="\begin{displaymath}
\lambda = \frac{h}{p}.
\end{displaymath}">|; 

$key = q/{scriptstyle-sqrt{(l+m+1)slash(2,l+1)}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1970.png"
 ALT="${\scriptstyle-\sqrt{(l+m+1)/(2 l+1)}}$">|; 

$key = q/(3slash4),hbar^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2380.png"
 ALT="$(3/4) \hbar^2$">|; 

$key = q/displaystylepsi^{(1)}_{0,1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1997.png"
 ALT="$\displaystyle \psi^{(1)}_{0,1/2}$">|; 

$key = q/{displaymath}psi({{bf{r})=psi_0({{bf{r})-frac{2,m}{hbar^2}intfrac{{{rm{e}^{,{{rm{bf{r}'|},V({{bf{r}'),psi({{bf{r}'),d^3{{bf{r}'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="394" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2850.png"
 ALT="\begin{displaymath}
\psi({\bf r}) = \psi_0({\bf r})- \frac{2 m}{\hbar^2}
\int\f...
...f r}-{\bf r}'\vert} V({\bf r}') \psi({\bf r}') d^3{\bf r}'.
\end{displaymath}">|; 

$key = q/x_1-x_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1151.png"
 ALT="$x_1-x_2$">|; 

$key = q/{displaymath}A,psi_a(x)=a,psi_a(x),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img649.png"
 ALT="\begin{displaymath}
A \psi_a(x) = a \psi_a(x),
\end{displaymath}">|; 

$key = q/B_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2925.png"
 ALT="$B_l$">|; 

$key = q/displaystylee,|{{bf{E}|,langlen,l,m|z|n,l,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2178.png"
 ALT="$\displaystyle e \vert{\bf E}\vert \langle n,l,m\vert z\vert n,l,m\rangle$">|; 

$key = q/delta_0rightarrowpi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3012.png"
 ALT="$\delta_0\rightarrow \pi$">|; 

$key = q/2,j+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2352.png"
 ALT="$2 j+1$">|; 

$key = q/{displaymath}p_requivfrac{{{bf{p}cdot{{bf{r}}{r}=-{{rm{i},hbar,frac{partial}{partialr}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1499.png"
 ALT="\begin{displaymath}
p_r \equiv \frac{{\bf p}\cdot{\bf r}}{r} = -{\rm i} \hbar \frac{\partial}{\partial r}.
\end{displaymath}">|; 

$key = q/displaystyleint_{-infty}^inftyP(u),(u-langleurangle)^2,du=leftlangleu^2rightrangle-langleurangle^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="314" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img127.png"
 ALT="$\displaystyle \int_{-\infty}^\infty P(u)  (u-\langle u\rangle)^2 du = \left\langle u^2\right\rangle-\langle u\rangle^2,$">|; 

$key = q/{displaymath}A^nomath_inline{dagnomath_inline{equivleft(array{{cc}A_{11}^ast,&A_{21}^astA_{12}^ast,&A_{22}^astarray{right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1795.png"
 ALT="\begin{displaymath}
A^\dag\equiv \left(\begin{array}{cc}A_{11}^\ast,&amp; A_{21}^\ast\\\\
A_{12}^\ast,&amp; A_{22}^\ast\end{array}\right).
\end{displaymath}">|; 

$key = q/x,y,z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1799.png"
 ALT="$x,y,z$">|; 

$key = q/displaystyles,(s+1),hbar^2,psi^{(2)}_{l,s;j,m_j},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1915.png"
 ALT="$\displaystyle s (s+1) \hbar^2 \psi^{(2)}_{l,s;j,m_j},$">|; 

$key = q/hbar,k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2891.png"
 ALT="$\hbar  k$">|; 

$key = q/beta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img767.png"
 ALT="$\beta$">|; 

$key = q/{displaymath}f({{bf{k}',{{bf{k})simeq-frac{2,m}{hbar^2,q}int_0^inftyr',V(r')sin(q,r'),dr'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="327" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img2878.png"
 ALT="\begin{displaymath}
f({\bf k}', {\bf k}) \simeq - \frac{2 m}{\hbar^2 q}
\int_0^\infty r'  V(r') \sin(q  r')  dr'.
\end{displaymath}">|; 

$key = q/displaystyleleft.-2,l,(l+1)-2,l',(l'+1)right}langlen,l,m|z|n',l',mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2207.png"
 ALT="$\displaystyle \left.-2 l (l+1) - 2 l' (l'+1)\right\}\langle n,l,m\vert z\vert n',l',m\rangle$">|; 

$key = q/m_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1073.png"
 ALT="$m_i$">|; 

$key = q/{displaymath}l^2=l_x^{,2}+l_y^{,2}+l_z^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img1244.png"
 ALT="\begin{displaymath}
l^2 = l_x^{ 2} + l_y^{ 2}+ l_z^{ 2}.
\end{displaymath}">|; 

$key = q/{displaymath}S_z,(S_+,chi_{s,m_s})=(m_s+1),hbar,(S_+,chi_{s,m_s}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="301" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1746.png"
 ALT="\begin{displaymath}
S_z (S_+ \chi_{s,m_s}) = (m_s+1) \hbar (S_+ \chi_{s,m_s}),
\end{displaymath}">|; 

$key = q/S_y=hbarslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2091.png"
 ALT="$S_y=\hbar/2$">|; 

$key = q/l';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2213.png"
 ALT="$l'$">|; 

$key = q/langleV_{ee}rangle(Z)=-(5,Zslash4),E_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2763.png"
 ALT="$\langle V_{ee}\rangle(Z) = -(5 Z/4) E_0$">|; 

$key = q/{displaymath}psi(x_1,x_2,ldots,x_N,t)=psi_1(x_1,t),psi_2(x_2,t)ldotspsi_N(x_N,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="427" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1097.png"
 ALT="\begin{displaymath}
\psi(x_1,x_2,\ldots, x_N, t) = \psi_1(x_1,t) \psi_2(x_2,t)\ldots\psi_N(x_N,t).
\end{displaymath}">|; 

$key = q/{displaymath}chi^nomath_inline{dagnomath_inline{equiv(c_+^ast,c_-^ast).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1790.png"
 ALT="\begin{displaymath}
\chi^\dag\equiv (c_+^\ast, c_-^\ast).
\end{displaymath}">|; 

$key = q/{displaymath}e_{11}=e_{22}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img2135.png"
 ALT="\begin{displaymath}
e_{11}= e_{22} = 0.
\end{displaymath}">|; 

$key = q/displaystylesigma_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1811.png"
 ALT="$\displaystyle \sigma_y$">|; 

$key = q/l=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1570.png"
 ALT="$l=3$">|; 

$key = q/displaystyleY_{2,pm2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1483.png"
 ALT="$\displaystyle Y_{2,\pm 2}$">|; 

$key = q/displaystylehbar^2,left[l,(l+1)-m,(m+1)right]geq0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1413.png"
 ALT="$\displaystyle \hbar^2 \left[l (l+1) -m (m+1)\right]\geq 0.$">|; 

$key = q/{displaymath}int_0^apsi_n(x),psi_m(x),dx=delta_{nm}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img807.png"
 ALT="\begin{displaymath}
\int_0^a \psi_n(x) \psi_m(x) dx = \delta_{nm}.
\end{displaymath}">|; 

$key = q/-78.98,{{rm{eV};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2725.png"
 ALT="$-78.98 {\rm eV}$">|; 

$key = q/{displaymath}|z|^{,2}=[{{rm{Re}(z)]^{,2}+[{{rm{Im}(z)]^{,2}geq[{{rm{Im}(z)]^{,2}=left[frac{1}{2,{{rm{i}},(z-z^ast)right]^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="449" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img614.png"
 ALT="\begin{displaymath}
\vert z\vert^{ 2} = [{\rm Re}(z)]^{ 2} + [{\rm Im}(z)]^{ ...
...]^{ 2} = \left[\frac{1}{2 {\rm i}} (z-z^\ast)\right]^{ 2}.
\end{displaymath}">|; 

$key = q/pllm,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2289.png"
 ALT="$p\ll m c$">|; 

$key = q/0,1,2,3,cdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1429.png"
 ALT="$0, 1, 2, 3, \cdots$">|; 

$key = q/displaystyle[p_i,p_j];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1084.png"
 ALT="$\displaystyle [p_i,p_j]$">|; 

$key = q/{displaymath}S_+,chi_+=S_-,chi_-=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1938.png"
 ALT="\begin{displaymath}
S_+ \chi_+=S_- \chi_- = 0,
\end{displaymath}">|; 

$key = q/(k,sintheta,kcostheta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img420.png"
 ALT="$(k \sin\theta, k\cos\theta)$">|; 

$key = q/{displaymath}int_{-infty}^{infty}psi^ast,(O,psi),dx=int_{-infty}^{infty}(O,psi)^ast,psi,dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img542.png"
 ALT="\begin{displaymath}
\int_{-\infty}^{\infty} \psi^\ast (O \psi) dx = \int_{-\infty}^{\infty}(O \psi)^\ast \psi dx,
\end{displaymath}">|; 

$key = q/c_pm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1849.png"
 ALT="$c_\pm$">|; 

$key = q/q=sqrt{2,m,(E-V_0)slashhbar^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img835.png"
 ALT="$q=\sqrt{2 m (E-V_0)/\hbar^2}$">|; 

$key = q/k_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img350.png"
 ALT="$k_0$">|; 

$key = q/{displaymath}psi(x)=A,{{rm{e}^{,k,x},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img965.png"
 ALT="\begin{displaymath}
\psi(x) = A {\rm e}^{ k x},
\end{displaymath}">|; 

$key = q/displaystylelanglexrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img533.png"
 ALT="$\displaystyle \langle x\rangle$">|; 

$key = q/-l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1576.png"
 ALT="$-l$">|; 

$key = q/displaystylelpm1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2650.png"
 ALT="$\displaystyle l\pm 1,$">|; 

$key = q/{displaymath}P_{irightarrowf}^{abs}(t)=frac{t^2}{hbar^2},frac{e^2,|A_0|^2}{4,m_e|^{,2},{{rm{sinc}^2[(omega-omega_{fi}),tslash2].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="523" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2550.png"
 ALT="\begin{displaymath}
P_{i\rightarrow f}^{abs}(t) = \frac{t^2}{\hbar^2} \frac{e^2...
...gle\right\vert^{ 2} {\rm sinc}^2[(\omega-\omega_{fi}) t/2].
\end{displaymath}">|; 

$key = q/P_{irightarrowf}=|c_f|^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2500.png"
 ALT="$P_{i\rightarrow f}=\vert c_f\vert^2$">|; 

$key = q/{displaymath}sum_kc_kleft{k,(k-1),z^{k-2}-2,k,z^{k-1}-l,(l+1),z^{k-2}+zeta,z^{k-1}right}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="493" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1596.png"
 ALT="\begin{displaymath}
\sum_k c_k\left\{k (k-1) z^{k-2} - 2 k z^{k-1} - l (l+1) z^{k-2}
+ \zeta z^{k-1}\right\} = 0.
\end{displaymath}">|; 

$key = q/N_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2245.png"
 ALT="$N_n$">|; 

$key = q/n(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img281.png"
 ALT="$n(y)$">|; 

$key = q/{mitDelta}k_x=k,(sintheta-1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img421.png"
 ALT="${\mit\Delta}k_x= k (\sin\theta-1)$">|; 

$key = q/langlexrangle_{ab};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1159.png"
 ALT="$\langle x\rangle_{ab}$">|; 

$key = q/displaystylem_s,hbar,chi_{s,m_s},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1743.png"
 ALT="$\displaystyle m_s \hbar \chi_{s,m_s},$">|; 

$key = q/{displaymath}langlepsi|H|psiranglegeqE_1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2705.png"
 ALT="\begin{displaymath}
\langle \psi \vert H\vert\psi\rangle \geq E_1.
\end{displaymath}">|; 

$key = q/|{{bf{r}|rightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1187.png"
 ALT="$\vert{\bf r}\vert\rightarrow\infty$">|; 

$key = q/cos(theta+2pi)=costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img150.png"
 ALT="$\cos(\theta+2\pi)=\cos\theta$">|; 

$key = q/{displaymath}Y_{l,-m}=(-1)^m,Y^{,ast}_{l,m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img1470.png"
 ALT="\begin{displaymath}
Y_{l,-m} = (-1)^m Y^{ \ast}_{l,m}.
\end{displaymath}">|; 

$key = q/displaystyle-4,hbar^2,L_y,(y,L_z-L_y,z)+4,hbar^2,L_x,(L_x,z-x,L_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="369" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2197.png"
 ALT="$\displaystyle -4 \hbar^2 L_y (y L_z-L_y z) + 4 \hbar^2 L_x (L_x z-x L_z)$">|; 

$key = q/{displaymath}-frac{hbar^2}{2,m_e}left(frac{d^2}{dr^2}+frac{2}{r}frac{d}{dr}-fraceft(frac{e^2}{4pi,epsilon_0,r}+Eright)R_{n,l}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="509" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1585.png"
 ALT="\begin{displaymath}
-\frac{\hbar^2}{2 m_e}\left(\frac{d^2}{dr^2} + \frac{2}{r}\...
...} -\left(\frac{e^2}{4\pi \epsilon_0 r}+E
\right) R_{n,l}= 0.
\end{displaymath}">|; 

$key = q/(2pi,hbar)^{-1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img726.png"
 ALT="$(2\pi \hbar)^{-1/2}$">|; 

$key = q/{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1302.png"
 ALT="${\bf r}$">|; 

$key = q/a_0slashZ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1697.png"
 ALT="$a_0/Z$">|; 

$key = q/{displaymath}H_1=-mbox{{boldmath{{mu{}_ecdot{{bf{B},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2368.png"
 ALT="\begin{displaymath}
H_1 = - \mbox{\boldmath$\mu$}_e\cdot {\bf B},
\end{displaymath}">|; 

$key = q/Y_{l,l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1444.png"
 ALT="$Y_{l,l}$">|; 

$key = q/k(x)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img898.png"
 ALT="$k(x)=0$">|; 

$key = q/+z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1783.png"
 ALT="$+z$">|; 

$key = q/{displaymath}left(frac{ddelta_l(E)}{dE}right)_{E=E_0}=frac{2}{Gamma},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img3026.png"
 ALT="\begin{displaymath}
\left(\frac{d \delta_l(E)}{d E} \right)_{E=E_0} = \frac{2}{\Gamma},
\end{displaymath}">|; 

$key = q/displaystyle-frac{1}{2,m_e,c^2}left[E_n^{,2}+2,E_nleft(frac{e^2}{4piepsilon_0}ri4piepsilon_0}right)^2frac{1}{(l+1slash2),n^3,a_0^{,2}}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="527" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2301.png"
 ALT="$\displaystyle -\frac{1}{2 m_e c^2}\left[
E_n^{ 2} + 2 E_n\left(\frac{e^2}{4...
...ft(\frac{e^2}{4\pi\epsilon_0}\right)^2\frac{1}{(l+1/2) n^3 a_0^{ 2}}\right].$">|; 

$key = q/P_{l,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1465.png"
 ALT="$P_{l,m}$">|; 

$key = q/{displaymath}f(theta)=sum_{l=0}^infty(2,l+1),frac{exp(,{{rm{i},delta_l)}{k},sindelta_l,P_l(costheta).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="345" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img2935.png"
 ALT="\begin{displaymath}
f(\theta) = \sum_{l=0}^\infty (2 l+1) \frac{\exp( {\rm i} \delta_l)}
{k}  \sin\delta_l P_l(\cos\theta).
\end{displaymath}">|; 

$key = q/displaystyleJ_z,psi^{(1)}_{l,s;m,m_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1907.png"
 ALT="$\displaystyle J_z \psi^{(1)}_{l,s;m,m_s}$">|; 

$key = q/{displaymath}v_p=frac{omega}{k},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img242.png"
 ALT="\begin{displaymath}
v_p = \frac{\omega}{k},
\end{displaymath}">|; 

$key = q/{displaymath}H(x',X)=-frac{hbar^2}{2,M}frac{partial^2}{partialX^2}-frac{hbar^2}{2,mu}frac{partial^2}{partialx'^{,2}}+V(x'),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1117.png"
 ALT="\begin{displaymath}
H(x',X) = -\frac{\hbar^2}{2 M} \frac{\partial^2}{\partial ...
...\hbar^2}{2 \mu}\frac{\partial^2}{\partial x'^{ 2}} + V(x'),
\end{displaymath}">|; 

$key = q/displaystyle-left(frac{1}{sin^2delta_l}frac{ddelta_l}{dE}right)_{E=E_0}(E-E_0)+cdots.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3025.png"
 ALT="$\displaystyle - \left(\frac{1}{\sin^2\delta_l}\frac{d\delta_l}{d E}\right)_{E=E_0}
(E-E_0)+\cdots.$">|; 

$key = q/{displaymath}m_{s,min}=-s.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1766.png"
 ALT="\begin{displaymath}
m_{s min} = -s.
\end{displaymath}">|; 

$key = q/displaystyleH_0,psi_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2121.png"
 ALT="$\displaystyle H_0 \psi_2$">|; 

$key = q/displaystyleL_x,[L_x,z]+[L_x,z],L_x+L_y,[L_y,z]+[L_y,z],L_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="367" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2190.png"
 ALT="$\displaystyle L_x [L_x,z] + [L_x,z] L_x + L_y [L_y,z]+[L_y,z] L_y$">|; 

$key = q/{displaymath}sigma_l=frac{4pi}{k^2},(2,l+1),sin^2delta_l=frac{4pi}{k^2},(2,l+1),frac{1}{1+cot^2delta_l}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img3028.png"
 ALT="\begin{displaymath}
\sigma_l = \frac{4\pi}{k^2}  (2 l+1) \sin^2\delta_l
= \frac{4\pi}{k^2}  (2 l+1) \frac{1}{1+\cot^2\delta_l}.
\end{displaymath}">|; 

$key = q/{displaymath}E_{n00}-E_{100}geqE_{200}-E_{100}=frac{3}{4},frac{e^2}{8pi,epsilon_0,a_0},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2225.png"
 ALT="\begin{displaymath}
E_{n00}-E_{100} \geq E_{200}-E_{100} = \frac{3}{4} \frac{e^2}{8\pi \epsilon_0 a_0},
\end{displaymath}">|; 

$key = q/u+du;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img116.png"
 ALT="$u+du$">|; 

$key = q/{displaymath}int_{-infty}^inftyf(x),delta(x-x_0),dx=f(x_0),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img581.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty f(x) \delta(x-x_0) dx = f(x_0),
\end{displaymath}">|; 

$key = q/n=Nslasha^3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1253.png"
 ALT="$n=N/a^3$">|; 

$key = q/s=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1775.png"
 ALT="$s=1/2$">|; 

$key = q/-1slash2,1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2068.png"
 ALT="$-1/2, 1/2$">|; 

$key = q/langlep^2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img748.png"
 ALT="$\langle p^2\rangle$">|; 

$key = q/c_n(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2400.png"
 ALT="$c_n(t)$">|; 

$key = q/{displaymath}bar{psi}(k)propto{{rm{e}^{-{{rm{i},k,x_0-beta^2slash4}int_{-infty}^{infty}{{rm{e}^{-(y-y_0)^{,2}},dy,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img341.png"
 ALT="\begin{displaymath}
\bar{\psi}(k) \propto {\rm e}^{-{\rm i} k x_0 - \beta^2/4}\int_{-\infty}^{\infty} {\rm e}^{-(y-y_0)^{ 2}} dy,
\end{displaymath}">|; 

$key = q/displaystylelanglen,l,m|[L_z,z]|n',l',m'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2184.png"
 ALT="$\displaystyle \langle n,l,m\vert[L_z,z]\vert n',l',m'\rangle$">|; 

$key = q/{scriptstyle1slashsqrt{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2071.png"
 ALT="${\scriptstyle 1/\sqrt{2}}$">|; 

$key = q/{displaymath}|T|^{,2}=expleft(-frac{2sqrt{2,m}}{hbar}int_{r_1}^{r_2}sqrt{V(r)-E},drright),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img917.png"
 ALT="\begin{displaymath}
\vert T\vert^{ 2} = \exp\left(-\frac{2\sqrt{2 m}}{\hbar}\int_{r_1}^{r_2}
\sqrt{V(r)-E} dr\right),
\end{displaymath}">|; 

$key = q/{displaymath}psi_1=psi_0,expleft(int_0^{x_1}{{rm{i},k(x'),dx'right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img889.png"
 ALT="\begin{displaymath}
\psi_1=\psi_0 \exp\left(\int_0^{x_1}{\rm i} k(x') dx'\right).
\end{displaymath}">|; 

$key = q/Lgga;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1064.png"
 ALT="$L\gg a$">|; 

$key = q/{displaymath}H_1=-frac{e,A_0,mbox{{boldmath{{epsilon{}!cdot!{{bf{p}}{2,m_e}left[m{i},{{bf{k}!cdot!{{bf{r}+{{rm{i},omegat)right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="452" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2548.png"
 ALT="\begin{displaymath}
H_1 = - \frac{e  A_0  \mbox{\boldmath$\epsilon$}\!\cdot\!{...
...{\rm i} {\bf k}\!\cdot\!{\bf r} + {\rm i} 
\omega t)\right].
\end{displaymath}">|; 

$key = q/y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img195.png"
 ALT="$y$">|; 

$key = q/omegaslashk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img168.png"
 ALT="$\omega/k$">|; 

$key = q/displaystyle(2,phi_1+3,phi_2)leftslashsqrt{13},right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img773.png"
 ALT="$\displaystyle (2 \phi_1+3 \phi_2) \left/ \sqrt{13},\right.$">|; 

$key = q/displaystyleE_0,psi-A,left(frac{e^2}{4pi,epsilon_0}right)left[frac{psi_0({{bf{r}_1)}{r_2}pmfrac{psi_0({{bf{r}_2)}{r_1}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2805.png"
 ALT="$\displaystyle E_0 \psi - A \left(\frac{e^2}{4\pi \epsilon_0}\right)\left[
\frac{\psi_0({\bf r}_1)}{r_2}\pm \frac{\psi_0({\bf r}_2)}{r_1}\right].$">|; 

$key = q/displaystylex,p_y-y,p_x.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1309.png"
 ALT="$\displaystyle x p_y - y p_x.$">|; 

$key = q/lambdagg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img984.png"
 ALT="$\lambda\gg 1$">|; 

$key = q/{displaymath}p^2,psi_{n,l,m}=2,m_e,(E_n-V),psi_{n,l,m},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2295.png"
 ALT="\begin{displaymath}
p^2 \psi_{n,l,m} = 2 m_e (E_n-V) \psi_{n,l,m},
\end{displaymath}">|; 

$key = q/{displaymath}psi(x,0)proptoexpleft[{{rm{i},k_0,x-frac{(x-x_0)^{,2}}{4,({mitDelta}x)^{,2}}right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img328.png"
 ALT="\begin{displaymath}
\psi(x,0) \propto \exp\left[{\rm i} k_0 x - \frac{(x-x_0)^{ 2}}{4 ({\mit\Delta}x)^{ 2}}\right].
\end{displaymath}">|; 

$key = q/{displaymath}psirightarrow0mbox{hspace{2cm}as{|x|rightarrowinfty{}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img961.png"
 ALT="\begin{displaymath}
\psi\rightarrow 0\mbox{\hspace{2cm}as $\vert x\vert\rightarrow\infty$}.
\end{displaymath}">|; 

$key = q/displaystylec^+_{l,m},c^-_{l,m+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1384.png"
 ALT="$\displaystyle c^+_{l,m} c^-_{l,m+1}$">|; 

$key = q/P_{irightarrowf}(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2504.png"
 ALT="$P_{i\rightarrow f}(t)$">|; 

$key = q/{displaymath}a=a^ast,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img657.png"
 ALT="\begin{displaymath}
a=a^\ast,
\end{displaymath}">|; 

$key = q/{displaymath}frac{partial|psi|^2}{partialt}+frac{partialj_x}{partialx}+frac{partialj_y}{partialy}+frac{partialj_z}{partialz}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1184.png"
 ALT="\begin{displaymath}
\frac{\partial\vert\psi\vert^2}{\partial t} + \frac{\partial...
...partial j_y}{\partial y} + \frac{\partial j_z}{\partial z}= 0.
\end{displaymath}">|; 

$key = q/zrightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1600.png"
 ALT="$z\rightarrow 0$">|; 

$key = q/{displaymath}DeltaE_{l,1slash2;j,m_j}=E_n,frac{alpha^2}{n^2}left[frac{n,left{3sl+1)-j,(j+1)right}}{2,l,(l+1slash2),(l+1)}right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="437" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2324.png"
 ALT="\begin{displaymath}
\Delta E_{l,1/2;j,m_j}= E_n \frac{\alpha^2}{n^2}\left[
\fra...
...\{3/4+l (l+1)-j (j+1)\right\}}{2 l (l+1/2) (l+1)}\right],
\end{displaymath}">|; 

$key = q/{displaymath}c^pm_{l,m}=[l,(l+l)-m,(mpm1)]^{1slash2},hbar.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img1388.png"
 ALT="\begin{displaymath}
c^\pm_{l,m} = [l (l+l) - m (m\pm 1)]^{1/2} \hbar.
\end{displaymath}">|; 

$key = q/displaystyle[l,(l+1)-m,(m-1)],hbar^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1387.png"
 ALT="$\displaystyle [l (l+1)-m (m-1)] \hbar^2.$">|; 

$key = q/langlen,l,m|z|n',l',mrangleneq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2218.png"
 ALT="$\langle n,l,m\vert z\vert n',l',m\rangle\neq 0$">|; 

$key = q/d_{if}equiv(d_x)_{n,n'}^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2626.png"
 ALT="$d_{if}\equiv (d_x)_{n,n'}^{ 2}$">|; 

$key = q/{displaymath}cotdelta_l(E)=-frac{2}{Gamma},(E-E_0)+cdots.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img3027.png"
 ALT="\begin{displaymath}
\cot\delta_l(E) = - \frac{2}{\Gamma}  (E-E_0) + \cdots.
\end{displaymath}">|; 

$key = q/1.5,{{rm{kV};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img451.png"
 ALT="$1.5 {\rm kV}$">|; 

$key = q/displaystyleointY_{l,m}^{,ast},L_-,L_+,Y_{l,m},dOmegageq0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1409.png"
 ALT="$\displaystyle \oint Y_{l,m}^{ \ast} L_- L_+ Y_{l,m} d\Omega\geq 0.$">|; 

$key = q/1slash8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1248.png"
 ALT="$1/8$">|; 

$key = q/{displaymath}K=N,bar{E}=frac{3}{5},N,E_F=frac{3}{5},N,frac{pi^2,hbar^2}{2,m_e}left(frac{3,N}{pi,V}right)^{2slash3},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="357" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1276.png"
 ALT="\begin{displaymath}
K = N \bar{E} = \frac{3}{5} N E_F = \frac{3}{5} N \frac{\pi^2 \hbar^2}{2 m_e}\left(\frac{3 N}{\pi V}\right)^{2/3},
\end{displaymath}">|; 

$key = q/displaystyleJ^2,psi^{(2)}_{l,s;j,m_j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1916.png"
 ALT="$\displaystyle J^2 \psi^{(2)}_{l,s;j,m_j}$">|; 

$key = q/dN=rho(E),dE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1299.png"
 ALT="$dN = \rho(E) dE$">|; 

$key = q/{displaymath}frac{alpha}{beta}=frac{[(l-m),(l+m+1)]^{1slash2}}{x-m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1950.png"
 ALT="\begin{displaymath}
\frac{\alpha}{\beta} = \frac{[(l-m) (l+m+1)]^{1/2}}{x-m}.
\end{displaymath}">|; 

$key = q/2Prightarrow1S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="$2P\rightarrow 1S$">|; 

$key = q/X;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="$X$">|; 

$key = q/displaystyle-z^lleft(-frac{1}{z}frac{d}{dz}right)^lleft(frac{cosz}{z}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1550.png"
 ALT="$\displaystyle -z^l\left(-\frac{1}{z}\frac{d}{dz}\right)^l\left(\frac{\cos z}{z}\right).$">|; 

$key = q/R_{n,l}(z=rslasha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1628.png"
 ALT="$R_{n,l}(z=r/a)$">|; 

$key = q/nu=-{mitDelta}Eslashh;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1672.png"
 ALT="$\nu=- {\mit\Delta}E/h$">|; 

$key = q/{displaymath}P_{x,in,a:b}(t)=int_{a}^{b}|psi(x,t)|^{,2},dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img458.png"
 ALT="\begin{displaymath}
P_{x \in  a:b}(t) = \int_{a}^{b}\vert\psi(x,t)\vert^{ 2} dx.
\end{displaymath}">|; 

$key = q/-{{rm{i},partialslashpartialx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img437.png"
 ALT="$-{\rm i} \partial/\partial x$">|; 

$key = q/displaystyleint_{-infty}^{infty}psi^ast(x,t),x,psi(x,t),dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img595.png"
 ALT="$\displaystyle \int_{-\infty}^{\infty} \psi^\ast(x,t) x \psi(x,t) dx$">|; 

$key = q/{displaymath}H_i=frac{p_i^{,2}}{2,m_i}+V(x_i,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1094.png"
 ALT="\begin{displaymath}
H_i = \frac{p_i^{ 2}}{2 m_i} + V(x_i,t).
\end{displaymath}">|; 

$key = q/{displaymath}[A^2,B]=A,[A,B]+[A,B],A.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1326.png"
 ALT="\begin{displaymath}[A^2,B]= A [A, B] +[A,B] A.
\end{displaymath}">|; 

$key = q/psi({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2851.png"
 ALT="$\psi({\bf r})$">|; 

$key = q/{displaymath}psi_n(x)=A_n,sin(k_n,x),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img799.png"
 ALT="\begin{displaymath}
\psi_n(x) = A_n \sin(k_n x),
\end{displaymath}">|; 

$key = q/{displaymath}psi_E(x_1,x_2,ldots,x_N)=psi_{E_1}(x_1),psi_{E_2}(x_2)ldotspsi_{E_N}(x_N),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="405" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1107.png"
 ALT="\begin{displaymath}
\psi_E(x_1,x_2,\ldots, x_N) = \psi_{E_1}(x_1) \psi_{E_2}(x_2)\ldots\psi_{E_N}(x_N),
\end{displaymath}">|; 

$key = q/{displaymath}E_{n,l}=z_{n,l}^{,2},frac{hbar^2}{2,m,a^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1567.png"
 ALT="\begin{displaymath}
E_{n,l} = z_{n,l}^{ 2} \frac{\hbar^2}{2 m a^2}.
\end{displaymath}">|; 

$key = q/{displaymath}u=int_{-infty}^{infty}rho(omega),domega,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img2555.png"
 ALT="\begin{displaymath}
u = \int_{-\infty}^{\infty} \rho(\omega) d\omega,
\end{displaymath}">|; 

$key = q/{displaymath}V(r)=-frac{e^2}{4pi,epsilon_0,r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1583.png"
 ALT="\begin{displaymath}
V(r) = - \frac{e^2}{4\pi \epsilon_0 r}.
\end{displaymath}">|; 

$key = q/R_{n,l}(r)=Z(rslasha),exp(-rslasha)slash(rslasha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1592.png"
 ALT="$R_{n,l}(r) = Z(r/a) \exp(-r/a)/(r/a)$">|; 

$key = q/n_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1669.png"
 ALT="$n_i$">|; 

$key = q/{displaymath}{{rm{Re}left[psi_0,{{rm{e}^{,{{rm{i},(k,x-omega,t)}right]=A,cos(k,x-omega,t+varphi)=psi(x,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="407" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img224.png"
 ALT="\begin{displaymath}
{\rm Re}\left[\psi_0 {\rm e}^{ {\rm i} (k x-\omega t)}\right] =A \cos(k x-\omega t+\varphi)=\psi(x,t).
\end{displaymath}">|; 

$key = q/a_0,r^2,|R_{n,l}(r)|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="$a_0 r^2 \vert R_{n,l}(r)\vert^{ 2}$">|; 

$key = q/displaystylepsi^{(1)}_{pm1,pm1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1984.png"
 ALT="$\displaystyle \psi^{(1)}_{\pm 1, \pm 1/2},$">|; 

$key = q/displaystylec_+(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1845.png"
 ALT="$\displaystyle c_+(t)$">|; 

$key = q/{displaymath}{{bf{J}={{bf{L}+{{bf{S}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1871.png"
 ALT="\begin{displaymath}
{\bf J} = {\bf L} + {\bf S}.
\end{displaymath}">|; 

$key = q/k_z^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1227.png"
 ALT="$k_z^{ 2}$">|; 

$key = q/{displaymath}E_{total}=N,bar{E}=frac{3}{5},N,E_F.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1260.png"
 ALT="\begin{displaymath}
E_{total} = N \bar{E} = \frac{3}{5} N E_F.
\end{displaymath}">|; 

$key = q/{displaymath}{{rm{i},hbar,frac{partialpsi_E(x,t,E_i)}{partialt}=E_i,psi_E(x,t,E_i).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="272" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img735.png"
 ALT="\begin{displaymath}
{\rm i} \hbar \frac{\partial \psi_E(x,t,E_i)}{\partial t} = E_i \psi_E(x,t,E_i).
\end{displaymath}">|; 

$key = q/mu=I,A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1821.png"
 ALT="$\mu=I A$">|; 

$key = q/-yslashsqrt{lambda-y^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="$-y/\sqrt {\lambda - y^2}$">|; 

$key = q/{displaymath}psi(x_1,x_2,ldots,x_n,t)=psi_E(x_1,x_2,ldots,x_N),{{rm{e}^{-{{rm{i},E,tslashhbar},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1106.png"
 ALT="\begin{displaymath}
\psi(x_1,x_2,\ldots, x_n, t) =\psi_E(x_1,x_2,\ldots, x_N) {\rm
e}^{-{\rm i} E t/\hbar},
\end{displaymath}">|; 

$key = q/{displaymath}P_{x,in,-infty:infty}(t)=int_{-infty}^{infty}|psi(x,t)|^{,2},dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img459.png"
 ALT="\begin{displaymath}
P_{x \in  -\infty:\infty}(t) = \int_{-\infty}^{\infty}\vert\psi(x,t)\vert^{ 2} dx.
\end{displaymath}">|; 

$key = q/y_l(k,r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2912.png"
 ALT="$y_l(k r)$">|; 

$key = q/displaystyle-frac{2}{X}int_X^infty{{rm{e}^{-2,x}left[{{rm{e}^{-X},(1+X+x)-{{rm{e}^X,(1-X+x)right]dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="438" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2816.png"
 ALT="$\displaystyle -\frac{2}{X}\int_X^\infty {\rm e}^{-2 x}\left[{\rm e}^{-X} (1+X+x)-
{\rm e}^X (1-X+x)\right] dx,$">|; 

$key = q/{displaymath}j(x,t)=frac{{{rm{i},hbar}{2,m}left(psi,frac{partialpsi^ast}{partialx}-psi^ast,frac{partialpsi}{partialx}right){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img482.png"
 ALT="\begin{displaymath}
j(x,t) = \frac{{\rm i} \hbar}{2 m}\left(\psi \frac{\parti...
...partial x} - \psi^\ast \frac{\partial\psi}{\partial x}\right)
\end{displaymath}">|; 

$key = q/{displaymath}L_x,x+L_y,y+L_z,z=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2202.png"
 ALT="\begin{displaymath}
L_x x+L_y y+L_z z = 0.
\end{displaymath}">|; 

$key = q/{displaymath}psi(x,t)propto{{rm{e}^{,{{rm{i},(k_0,x-omega_0,t)-(1+{{rm{i},beta_2ty{{rm{e}^{-(1+{{rm{i},beta_2),(y-y_0)^{,2}},dy,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="441" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img380.png"
 ALT="\begin{displaymath}
\psi(x,t)\propto {\rm e}^{ {\rm i} (k_0 x-\omega_0 t)-(1...
...ty}^\infty
{\rm e}^{-(1+{\rm i} \beta_2) (y-y_0)^{ 2}} dy,
\end{displaymath}">|; 

$key = q/g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img613.png"
 ALT="$g$">|; 

$key = q/m=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1572.png"
 ALT="$m=0$">|; 

$key = q/{displaymath}frac{Deltalambda}{lambda}=frac{DeltaE}{hbar,omega}=4.4times10^{-6}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2678.png"
 ALT="\begin{displaymath}
\frac{\Delta\lambda}{\lambda} = \frac{\Delta E}{\hbar \omega} =
4.4\times 10^{-6}.
\end{displaymath}">|; 

$key = q/displaystylefrac{d}{sqrt{2}}int_{-infty}^{infty}psi_m,(a_++a_-),psi_n,dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1046.png"
 ALT="$\displaystyle \frac{d}{\sqrt{2}}\int_{-\infty}^{\infty}\psi_m (a_+ + a_-) \psi_n dx$">|; 

$key = q/Z(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1599.png"
 ALT="$Z(z)$">|; 

$key = q/|c(p')|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img731.png"
 ALT="$\vert c(p')\vert^{ 2}$">|; 

$key = q/displaystylefrac{{{rm{e}^{-{{rm{i},E_0,tslashhbar}}{(2pi,sigma_t^{,2})^{1slash4}},{{rm{e}^{-(t-t_0)^{,2}slash4,sigma_t^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img644.png"
 ALT="$\displaystyle \frac{{\rm e}^{-{\rm i} E_0 t/\hbar}}{(2\pi \sigma_t^{ 2})^{1/4}} {\rm e}^{-(t-t_0)^{ 2}/4 \sigma_t^{ 2}},$">|; 

$key = q/8.5times10^{28},{{rm{m}^{-3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1301.png"
 ALT="$8.5\times 10^{28} {\rm m}^{-3}$">|; 

$key = q/{displaymath}langleV_{ee}rangle=-16,E_0int_{0}^infty{{rm{e}^{-2,r_2}left[1-{{rm{^{-2,r_2},(1+r_2)right]r_2,dr_2=-frac{5}{2},E_0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="486" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2752.png"
 ALT="\begin{displaymath}
\langle V_{ee}\rangle = -16 E_0\int_{0}^\infty
{\rm e}^{-2\...
...{\rm e}^{-2 r_2} (1+r_2)\right]r_2 dr_2=
-\frac{5}{2} E_0.
\end{displaymath}">|; 

$key = q/s_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img134.png"
 ALT="$s_0$">|; 

$key = q/2P_{3slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2332.png"
 ALT="$2P_{3/2}$">|; 

$key = q/{displaymath}omega_{n,n-1}=frac{E_n-E_{n-1}}{hbar}=omega_0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2630.png"
 ALT="\begin{displaymath}
\omega_{n,n-1} = \frac{E_n - E_{n-1}}{\hbar} = \omega_0.
\end{displaymath}">|; 

$key = q/displaystylesum_{n>1}langle1,0,0|z|n,1,0rangle,langlen,1,0|z|1,0,0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2228.png"
 ALT="$\displaystyle \sum_{n&gt;1}\langle 1,0,0\vert z\vert n,1,0\rangle \langle n,1,0\vert z\vert 1,0,0\rangle$">|; 

$key = q/{{bf{U};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2255.png"
 ALT="${\bf U}$">|; 

$key = q/H(x_1,x_2,ldots,x_N,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1087.png"
 ALT="$H(x_1,x_2,\ldots, x_N,t)$">|; 

$key = q/j_l=j_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img828.png"
 ALT="$j_l=j_r$">|; 

$key = q/displaystyle(cosphi,,sinphi,,0),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2591.png"
 ALT="$\displaystyle (\cos\phi,  \sin\phi, 0),$">|; 

$key = q/m_{s_1},m_{s_2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2069.png"
 ALT="$m_{s_1},m_{s_2}$">|; 

$key = q/Bsim6.4times10^{10},{{rm{N},{{rm{m}^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1270.png"
 ALT="$B\sim 6.4\times 10^{10} {\rm N} {\rm m}^{-2}$">|; 

$key = q/|psi(x,t)|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img316.png"
 ALT="$\vert\psi(x,t)\vert^{ 2}$">|; 

$key = q/{displaymath}zeta=frac{2,m_e,a,e^2}{4pi,epsilon_0,hbar^2}=2,sqrt{frac{E_0}{E}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1591.png"
 ALT="\begin{displaymath}
\zeta = \frac{2 m_e a e^2}{4\pi \epsilon_0 \hbar^2}=2 \sqrt{\frac{E_0}{E}}.
\end{displaymath}">|; 

$key = q/y_0=-{{rm{i},betaslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img342.png"
 ALT="$y_0 = - {\rm i} \beta/2$">|; 

$key = q/{displaymath}langleV_{ee}rangle=leftlanglepsileft|frac{e^2}{4pi,epsilon_0,|{{bf{{{bf{r}_2-{{bf{r}_1|},d^3{{bf{r}_1,d^3{{bf{r}_2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="504" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2728.png"
 ALT="\begin{displaymath}
\langle V_{ee}\rangle = \left\langle \psi\left\vert\frac{e^2...
...}{\vert{\bf r}_2- {\bf r}_1\vert} d^3{\bf r}_1 d^3{\bf r}_2.
\end{displaymath}">|; 

$key = q/S_x=-hbarslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2092.png"
 ALT="$S_x=-\hbar/2$">|; 

$key = q/{displaymath}sum_kfrac{(2,z)^k}{k!},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1610.png"
 ALT="\begin{displaymath}
\sum_k \frac{(2 z)^k}{k!},
\end{displaymath}">|; 

$key = q/displaystyle{{rm{i},frac{dc_2}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2415.png"
 ALT="$\displaystyle {\rm i} \frac{dc_2}{dt}$">|; 

$key = q/displaystyle+[l,(l+1)-m,(m+1)]^{1slash2},hbar^2,Y_{l,m+1},chi_-,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1942.png"
 ALT="$\displaystyle + [l (l+1)-m (m+1)]^{1/2} \hbar^2 Y_{l,m+1} \chi_-,$">|; 

$key = q/{displaymath}e_{mk}=langlem|H_1|krangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2153.png"
 ALT="\begin{displaymath}
e_{mk} =\langle m\vert H_1\vert k\rangle.
\end{displaymath}">|; 

$key = q/{{mitSigma}}otimes{{mitSigma}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img84.png"
 ALT="${{\mit\Sigma}}\otimes {{\mit\Sigma}}$">|; 

$key = q/{displaymath}oint(L_-,Y_{l,m})^ast,(L_-,Y_{l,m}),dOmega=ointY_{l,m}^{,ast},L_+,L_-,Y_{l,m},dOmegageq0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="424" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1415.png"
 ALT="\begin{displaymath}
\oint (L_- Y_{l,m})^\ast (L_- Y_{l,m}) d\Omega
=\oint Y_{l,m}^{ \ast} L_+ L_- Y_{l,m} d\Omega\geq 0
\end{displaymath}">|; 

$key = q/{displaymath}psi(x)=A,{{rm{e}^{,{{rm{i},q,x}+B,{{rm{e}^{-{{rm{i},q,x},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img834.png"
 ALT="\begin{displaymath}
\psi(x) = A {\rm e}^{ {\rm i} q x} +B {\rm e}^{-{\rm i} q x},
\end{displaymath}">|; 

$key = q/{displaymath}m_s=m_{s_1}+m_{s_2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2060.png"
 ALT="\begin{displaymath}
m_s = m_{s_1}+ m_{s_2}.
\end{displaymath}">|; 

$key = q/-{{rm{i},hbar,partialslashpartialx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img537.png"
 ALT="$-{\rm i} \hbar \partial/\partial x$">|; 

$key = q/{displaymath}E_c=frac{Z_1,Z_2,e^2}{4pi,epsilon_0,R}=1.44,Z_1^{2slash3},,{{rm{MeV}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img923.png"
 ALT="\begin{displaymath}
E_c = \frac{Z_1 Z_2 e^2}{4\pi \epsilon_0 R} = 1.44 Z_1^{2/3}  {\rm MeV}
\end{displaymath}">|; 

$key = q/{displaymath}exp(,{{rm{i},k,rcostheta)=sum_la_l,j_l(k,r),P_l(costheta),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="313" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2915.png"
 ALT="\begin{displaymath}
\exp( {\rm i} k r \cos\theta) = \sum_l a_l  j_l(k r)  P_l(\cos\theta),
\end{displaymath}">|; 

$key = q/langlef|mbox{{boldmath{{epsilon{}!cdot!{{bf{p},exp(,{{rm{i},{{bf{k}!cdot!{{bf{r})|irangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2682.png"
 ALT="$\langle f\vert\mbox{\boldmath$\epsilon$}\!\cdot\!{\bf p} \exp( {\rm i} {\bf k}\!\cdot\!{\bf r})\vert i\rangle$">|; 

$key = q/4,gamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2440.png"
 ALT="$4 \gamma$">|; 

$key = q/(m-1),hbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1377.png"
 ALT="$(m-1) \hbar$">|; 

$key = q/{displaymath}Y_{l,l-2}(theta,phi)simfrac{{{rm{e}^{,{{rm{i},(l-2),phi}}{(sinthetaintheta}frac{d}{dtheta}right)^2(sintheta)^{2,l}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1455.png"
 ALT="\begin{displaymath}
Y_{l,l-2}(\theta,\phi) \sim \frac{{\rm e}^{ {\rm i} (l-2)\...
...{1}{\sin\theta}\frac{d}{d\theta}\right)^2
(\sin\theta)^{2 l}.
\end{displaymath}">|; 

$key = q/displaystylepsi_0({{bf{r}_1),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2718.png"
 ALT="$\displaystyle \psi_0({\bf r}_1),$">|; 

$key = q/x_2equivy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1313.png"
 ALT="$x_2\equiv y$">|; 

$key = q/displaystylew_{irightarrowf}^{abs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2580.png"
 ALT="$\displaystyle w_{i\rightarrow f}^{abs}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter03slashfig01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="336" HEIGHT="283" BORDER="0"
 SRC="|."$dir".q|img176.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter03/fig01.eps}}
\end{figure}">|; 

$key = q/{displaymath}psi_0({{bf{r})=frac{4}{sqrt{2,pi},a_0^{,3slash2}},expleft(-frac{2,r}{a_0}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2721.png"
 ALT="\begin{displaymath}
\psi_0({\bf r}) = \frac{4}{\sqrt{2 \pi} a_0^{ 3/2}} \exp\left(-\frac{2 r}{a_0}\right).
\end{displaymath}">|; 

$key = q/{displaymath}P=frac{e^2,omega_0^{,2}}{6pi,epsilon_0,m_e,c^3},E,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2634.png"
 ALT="\begin{displaymath}
P= \frac{e^2 \omega_0^{ 2}}{6\pi \epsilon_0 m_e c^3} E,
\end{displaymath}">|; 

$key = q/{displaymath}{calE}=frac{A}{R^2}-frac{B}{R},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1281.png"
 ALT="\begin{displaymath}
{\cal E} = \frac{A}{R^2} - \frac{B}{R},
\end{displaymath}">|; 

$key = q/Ngg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img295.png"
 ALT="$N\gg 1$">|; 

$key = q/displaystylesqrt{frac{5}{16pi}},(3,cos^2theta-1),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1480.png"
 ALT="$\displaystyle \sqrt{\frac{5}{16\pi}} (3 \cos^2\theta - 1),$">|; 

$key = q/displaystylelanglem|Erangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2160.png"
 ALT="$\displaystyle \langle m\vert E\rangle$">|; 

$key = q/displaystyleC_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img941.png"
 ALT="$\displaystyle C_1$">|; 

$key = q/{displaymath}int_{-infty}^inftypsi_a^ast,psi_a',dx=c,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img674.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \psi_a^\ast \psi_a' dx = c,
\end{displaymath}">|; 

$key = q/{displaymath}P_{18}simeqfrac{N_{18}}{N_{{rm{students}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img92.png"
 ALT="\begin{displaymath}
P_{18} \simeq
\frac{N_{18}}{N_{\rm students}},
\end{displaymath}">|; 

$key = q/p^nomath_inline{dagnomath_inline{=p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img546.png"
 ALT="$p^\dag = p$">|; 

$key = q/{displaymath}[x,p]={{rm{i},hbar.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img624.png"
 ALT="\begin{displaymath}[x,p]= {\rm i} \hbar.
\end{displaymath}">|; 

$key = q/A_l=C_l,cosdelta_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2929.png"
 ALT="$A_l=C_l \cos\delta_l$">|; 

$key = q/displaystyle-{{rm{i},hbarint_{-infty}^{infty}left(frac{partialpsi^ast}{partialt}artialx}+psi^ast,frac{partial^2psi}{partialtpartialx}right)dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img510.png"
 ALT="$\displaystyle -{\rm i} \hbar\int_{-\infty}^{\infty}
\left(\frac{\partial\psi^\...
...}{\partial x} + \psi^\ast \frac{\partial^2\psi}{\partial t\partial x}\right)dx$">|; 

$key = q/{mitDelta}p_x=hbar,k,(sintheta-1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img422.png"
 ALT="${\mit\Delta}p_x = \hbar k (\sin\theta-1)$">|; 

$key = q/displaystylesigma_{{rm{total};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2936.png"
 ALT="$\displaystyle \sigma_{\rm total}$">|; 

$key = q/psi(x,y,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1210.png"
 ALT="$\psi(x,y,z)$">|; 

$key = q/{displaymath}S_-,chi_{s,m_{s,min}}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1761.png"
 ALT="\begin{displaymath}
S_- \chi_{s,m_{s min}}= 0.
\end{displaymath}">|; 

$key = q/-alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img423.png"
 ALT="$-\alpha$">|; 

$key = q/{displaymath}{calR}_l(r)=exp(,{{rm{i},delta_l),left[cosdelta_l,j_l(k,r)-sindelta_l,y_l(k,r)right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="373" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2944.png"
 ALT="\begin{displaymath}
{\cal R}_l(r) = \exp( {\rm i}  \delta_l) 
\left[\cos\delta_l  j_l(k r) -\sin\delta_l  y_l(k r)\right].
\end{displaymath}">|; 

$key = q/displaystyle(d_{if},sintheta,,0,,d_{if},costheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2589.png"
 ALT="$\displaystyle (d_{if} \sin\theta, 0, d_{if} \cos\theta),$">|; 

$key = q/exp(,{{rm{i},{{bf{k}!cdot!{{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2683.png"
 ALT="$\exp( {\rm i} {\bf k}\!\cdot\!{\bf r})$">|; 

$key = q/m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img297.png"
 ALT="$m$">|; 

$key = q/lambda_1=3,e,a_0,|{{bf{E}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2274.png"
 ALT="$\lambda_1=3 e a_0 \vert{\bf E}\vert$">|; 

$key = q/E<V(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img885.png"
 ALT="$E&lt;V(x)$">|; 

$key = q/displaystyle2,{mitDelta}k,(x-x_0-v_g,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img376.png"
 ALT="$\displaystyle 2 {\mit\Delta}k (x-x_0-v_g t),$">|; 

$key = q/a_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img765.png"
 ALT="$a_1$">|; 

$key = q/displaystyleS^2,psi^{(2)}_{l,s;j,m_j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1914.png"
 ALT="$\displaystyle S^2 \psi^{(2)}_{l,s;j,m_j}$">|; 

$key = q/displaystylefrac{2,m,(E+V_0)}{hbar^2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img956.png"
 ALT="$\displaystyle \frac{2 m (E+V_0)}{\hbar^2}.$">|; 

$key = q/{mitDelta}x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img333.png"
 ALT="${\mit\Delta} x$">|; 

$key = q/{displaymath}c_i=langlepsi_i|psi_arangleequivlanglei|arangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2107.png"
 ALT="\begin{displaymath}
c_i =\langle\psi_i\vert\psi_a\rangle\equiv \langle i\vert a\rangle.
\end{displaymath}">|; 

$key = q/displaystylepsi^{(2)}_{3slash2,pm3slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1983.png"
 ALT="$\displaystyle \psi^{(2)}_{3/2,\pm 3/2}$">|; 

$key = q/{displaymath}langlef|{{bf{p}|irangle=-{{rm{i},frac{m_e}{hbar},langlef|[{{bf{r},H={{rm{i},m_e,omega_{fi},langlef|{{bf{r}|irangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img2579.png"
 ALT="\begin{displaymath}
\langle f\vert{\bf p}\vert i\rangle = - {\rm i} \frac{m_e}{...
...\rm i} m_e \omega_{fi} \langle f\vert{\bf r}\vert i\rangle.
\end{displaymath}">|; 

$key = q/displaystyle-k_y^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1222.png"
 ALT="$\displaystyle - k_y^{ 2},$">|; 

$key = q/displaystyleD;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2807.png"
 ALT="$\displaystyle D$">|; 

$key = q/{displaymath}sigma_{{rm{total}simeqfrac{4pi}{k^2},sin^2delta_0=4pi,a^2left[frac{tan(k,'a)}{k',a}-1right]^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="353" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img3007.png"
 ALT="\begin{displaymath}
\sigma_{\rm total} \simeq \frac{4\pi}{k^2}  \sin^2\delta_0
=4\pi  a^2\left[\frac{\tan (k 'a)}{k' a} -1\right]^2.
\end{displaymath}">|; 

$key = q/{displaymath}chi^nomath_inline{dagnomath_inline{,chi'=(c_+^ast,c_-^ast)left(arra}c_+'c_-'array{right)=c_+^ast,c_+'+c_-^ast,c_-'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1792.png"
 ALT="\begin{displaymath}
\chi^\dag \chi' = (c_+^\ast, c_-^\ast)\left(\begin{array}{c}c_+' c_-'\end{array}\right) = c_+^\ast c_+' + c_-^\ast c_-'.
\end{displaymath}">|; 

$key = q/Y_{l,m}(theta,phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1366.png"
 ALT="$Y_{l,m}(\theta,\phi)$">|; 

$key = q/Hequiv{{rm{i},hbar,partialslashpartialt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img632.png"
 ALT="$H\equiv{\rm i} \hbar \partial/\partial t$">|; 

$key = q/psi^{(1)}_{m,m_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1972.png"
 ALT="$\psi^{(1)}_{m,m_s}$">|; 

$key = q/Z_2=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img910.png"
 ALT="$Z_2=2$">|; 

$key = q/displaystylex_ileft(x_i,p_j-[x_i,p_j]right)p_j-x_i,p_j,left(p_i,x_j+[x_j,p_i]right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1518.png"
 ALT="$\displaystyle x_i\left(x_i p_j - [x_i,p_j]\right) p_j
- x_i p_j \left(p_i x_j+[x_j,p_i]\right)$">|; 

$key = q/N-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img129.png"
 ALT="$N-1$">|; 

$key = q/displaystylefrac{e^2}{4piepsilon_0,m_e^{,2},c^2,r^3},{{bf{L}cdot{{bf{S},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2313.png"
 ALT="$\displaystyle \frac{e^2}{4\pi\epsilon_0 m_e^{ 2} c^2 r^3} {\bf L}\cdot {\bf S},$">|; 

$key = q/{displaymath}t_2simfrac{m,({mitDelta}x)^2}{hbar}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img394.png"
 ALT="\begin{displaymath}
t_2 \sim \frac{m ({\mit\Delta}x)^2}{\hbar}.
\end{displaymath}">|; 

$key = q/psi_c(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1169.png"
 ALT="$\psi_c(x)$">|; 

$key = q/{displaymath}r^2,frac{d^2R_l}{dr^2}+2,r,frac{dR_l}{dr}+[k^2,r^2-l,(l+1)]R_l=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="354" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img2911.png"
 ALT="\begin{displaymath}
r^2 \frac{d^2 R_l}{dr^2} + 2 r  \frac{dR_l}{dr} + [k^2  r^2 -
l (l+1)]R_l = 0.
\end{displaymath}">|; 

$key = q/df(x)slashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img528.png"
 ALT="$df(x)/dx$">|; 

$key = q/{displaymath}delta(x-x_0)=frac{1}{2pi,hbar}int_{-infty}^{infty}{{rm{e}^{+{{rm{i},p,(x-x_0)slashhbar},dp.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img584.png"
 ALT="\begin{displaymath}
\delta(x-x_0)= \frac{1}{2\pi \hbar}\int_{-\infty}^{\infty}{\rm e}^{+{\rm i} p (x-x_0)/\hbar} dp.
\end{displaymath}">|; 

$key = q/k_B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1257.png"
 ALT="$k_B$">|; 

$key = q/j=3slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1978.png"
 ALT="$j=3/2$">|; 

$key = q/|psi(x_1,x_2,ldots,x_N,t)|^2,dx_1,dx_2ldotsdx_N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1077.png"
 ALT="$\vert\psi(x_1,x_2,\ldots, x_N,t)\vert^2 dx_1 dx_2\ldots dx_N$">|; 

$key = q/(A,chi)^nomath_inline{dagnomath_inline{equivchi^nomath_inline{dagnomath_inline{,A^nomath_inline{dagnomath_inline{;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1734.png"
 ALT="$(A \chi)^\dag\equiv \chi^\dag A^\dag $">|; 

$key = q/F_+(X);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="$F_+(X)$">|; 

$key = q/O(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img539.png"
 ALT="$O(x)$">|; 

$key = q/|x|ggpi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2496.png"
 ALT="$\vert x\vert\gg \pi$">|; 

$key = q/{displaymath}mbox{{boldmath{{mu{}=-frac{e}{2,m_e},({{bf{L}+2,{{bf{S}){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img2340.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\mu$}= - \frac{e}{2 m_e} ({\bf L} + 2 {\bf S})
\end{displaymath}">|; 

$key = q/{displaymath}psi(x)=int_{-infty}^inftyc(x'),psi_x(x,x'),dx',{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img715.png"
 ALT="\begin{displaymath}
\psi(x) = \int_{-\infty}^\infty c(x') \psi_x(x,x') dx',
\end{displaymath}">|; 

$key = q/z=(1+{{rm{i},beta_2)^{1slash2},(y-y_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img383.png"
 ALT="$z=(1+{\rm i} \beta_2)^{1/2} (y-y_0)$">|; 

$key = q/displaystyleqleft(A,{{rm{e}^{,{{rm{i},q,a}-B,{{rm{e}^{-{{rm{i},q,a}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img844.png"
 ALT="$\displaystyle q\left(A {\rm e}^{ {\rm i} q a} -B  {\rm e}^{-{\rm i} q a}\right)$">|; 

$key = q/displaystylefrac{1}{sqrt{3},(2,a_0)^{3slash2}},frac{r}{a_0},expleft(-frac{r}{2,a_0}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1643.png"
 ALT="$\displaystyle \frac{1}{\sqrt{3} (2 a_0)^{3/2}} \frac{r}{a_0} 
\exp\left(-\frac{r}{2 a_0}\right),$">|; 

$key = q/x=-l-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1952.png"
 ALT="$x=-l-1$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter03slashfig08.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img410.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter03/fig08.eps}}
\end{figure}">|; 

$key = q/{displaymath}{{bf{r}cdot{{bf{p}=r,p_r=-{{rm{i},hbar,r,frac{partial}{partialr},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1527.png"
 ALT="\begin{displaymath}
{\bf r}\cdot{\bf p} = r p_r = -{\rm i} \hbar r \frac{\partial}{\partial r},
\end{displaymath}">|; 

$key = q/m_j=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2005.png"
 ALT="$m_j=1/2$">|; 

$key = q/-z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1785.png"
 ALT="$-z$">|; 

$key = q/S_2^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2031.png"
 ALT="$S_2^{ 2}$">|; 

$key = q/{displaymath}sigma^2_x=frac{1}{sqrt{2pi,sigma^2}}int_{-infty}^{infty}(x-x_0)^{,2},{{rm{e}^{-(x-x_0)^{,2}slash(2,sigma^2)},dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="362" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img499.png"
 ALT="\begin{displaymath}
\sigma^2_x = \frac{1}{\sqrt{2\pi \sigma^2}}\int_{-\infty}^{...
...ty}
(x-x_0)^{ 2} {\rm e}^{-(x-x_0)^{ 2}/(2 \sigma^2)} dx.
\end{displaymath}">|; 

$key = q/psi(x,E_a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1160.png"
 ALT="$\psi(x,E_a)$">|; 

$key = q/displaystyle-{{rm{i},hbarleft(cosphi,frac{partial}{partialtheta}-sinphi,cottheta,frac{partial}{partialphi}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1358.png"
 ALT="$\displaystyle - {\rm i} \hbar\left(\cos\phi \frac{\partial}{\partial\theta}
-\sin\phi \cot\theta \frac{\partial}{\partial\phi}\right),$">|; 

$key = q/D;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img262.png"
 ALT="$D$">|; 

$key = q/A^nomath_inline{dagnomath_inline{=A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1737.png"
 ALT="$A^\dag =A$">|; 

$key = q/{displaymath}Theta_{l,l}sim(sintheta)^l.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1441.png"
 ALT="\begin{displaymath}
\Theta_{l,l}\sim (\sin\theta)^l.
\end{displaymath}">|; 

$key = q/{displaymath}f(theta)=-frac{2,m,V_0}{hbar^2,mu}frac{1}{q^2+mu^2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img2884.png"
 ALT="\begin{displaymath}
f(\theta) = - \frac{2 m  V_0}{\hbar^2 \mu} \frac{1}{q^2 + \mu^2},
\end{displaymath}">|; 

$key = q/{displaymath}f({{bf{k}',{{bf{k})simeq-frac{m}{2pi,hbar^2}int!int!intexp(,{{rm{i}eta'),V(r'),r'^{,2},dr',sintheta',dtheta',dphi',{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="529" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img2877.png"
 ALT="\begin{displaymath}
f({\bf k}', {\bf k}) \simeq - \frac{m}{2\pi  \hbar^2} \int\...
...a')   V(r') r'^{ 2}  dr' \sin\theta'
 d\theta' d\phi',
\end{displaymath}">|; 

$key = q/P_2(t)=|c_2(t)|^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2428.png"
 ALT="$P_2(t)= \vert c_2(t)\vert^2$">|; 

$key = q/displaystylelanglen,l,m|L_z,z-z,L_z|n',l',m'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2185.png"
 ALT="$\displaystyle \langle n,l,m\vert L_z z-z L_z\vert n',l',m'\rangle$">|; 

$key = q/displaystylesin(alphaslash2),exp(+{{rm{i},Omega,tslash2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1848.png"
 ALT="$\displaystyle \sin(\alpha/2) \exp(+{\rm i} \Omega t/2).$">|; 

$key = q/{displaymath}S_pm=S_xpm{{rm{i},S_y.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1708.png"
 ALT="\begin{displaymath}
S_\pm = S_x \pm {\rm i} S_y.
\end{displaymath}">|; 

$key = q/{displaymath}bar{f}(k)=frac{1}{sqrt{2pi}}int_{-infty}^inftyf(x),{{rm{e}^{-{{rm{i},k,x},dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img324.png"
 ALT="\begin{displaymath}
\bar{f}(k) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(x) {\rm e}^{-{\rm i} k x} dx.
\end{displaymath}">|; 

$key = q/exp(2,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1611.png"
 ALT="$\exp(2 z)$">|; 

$key = q/{displaymath}-sleqm_sleqs.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1758.png"
 ALT="\begin{displaymath}
-s \leq m_s\leq s.
\end{displaymath}">|; 

$key = q/pi,a^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2981.png"
 ALT="$\pi  a^2$">|; 

$key = q/a_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1042.png"
 ALT="$a_-$">|; 

$key = q/{displaymath}d^2=|langlen,l,m|e,x|n',l',m'rangle|^2+|langlen,l,m|e,y|n',l',m'rangle|^2+|langlen,l,m|e,z|n',l',m'rangle|^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="607" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2637.png"
 ALT="\begin{displaymath}
d^2 = \vert\langle n,l,m\vert e x\vert n',l',m'\rangle\ver...
...2
+ \vert\langle n,l,m\vert e z\vert n',l',m'\rangle\vert^2
\end{displaymath}">|; 

$key = q/displaystyle{{bf{x}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2278.png"
 ALT="$\displaystyle {\bf x}_2$">|; 

$key = q/displaystyleE_2,psi_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2122.png"
 ALT="$\displaystyle E_2 \psi_2.$">|; 

$key = q/omega_{if};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2616.png"
 ALT="$\omega_{if}$">|; 

$key = q/{displaymath}R=beta^2slash(1+beta^2),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1062.png"
 ALT="\begin{displaymath}
R = \beta^2/(1+\beta^2),
\end{displaymath}">|; 

$key = q/|E_0|simalpha^2,m_e,c^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1624.png"
 ALT="$\vert E_0\vert\sim \alpha^2 m_e c^2$">|; 

$key = q/{displaymath}(H_0+H_1),psi=sum_mc_m,expleft(-{{rm{i},E_m,tslashhbarright)(E_m+H_1),psi_m.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="425" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2403.png"
 ALT="\begin{displaymath}
(H_0+H_1) \psi = \sum_m c_m \exp\left(-{\rm i} E_m t/\hbar\right)(E_m+H_1) \psi_m.
\end{displaymath}">|; 

$key = q/N=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2409.png"
 ALT="$N=2$">|; 

$key = q/{displaymath}{mitDelta}k=frac{1}{2,{mitDelta}x}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img345.png"
 ALT="\begin{displaymath}
{\mit\Delta} k = \frac{1}{2 {\mit\Delta} x}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{c_k}{c_{k-1}}=frac{2}{k},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1609.png"
 ALT="\begin{displaymath}
\frac{c_k}{c_{k-1}} = \frac{2}{k},
\end{displaymath}">|; 

$key = q/{displaymath}langlem|H_0+H_1|Erangle=E,langlem|Erangle,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2150.png"
 ALT="\begin{displaymath}
\langle m\vert H_0+H_1\vert E\rangle = E \langle m \vert E\rangle,
\end{displaymath}">|; 

$key = q/{displaymath}R_ast=frac{(9pi)^{2slash3}}{8},frac{hbar^2}{G,m_e,m_p^{,5slash3},M^{1slash3}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img1288.png"
 ALT="\begin{displaymath}
R_\ast = \frac{(9\pi)^{2/3}}{8} \frac{\hbar^2}{G m_e m_p^{ 5/3} 
M^{1/3}}.
\end{displaymath}">|; 

$key = q/|c_+|^2+|c_-|^2=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1837.png"
 ALT="$\vert c_+\vert^2+\vert c_-\vert^2=1$">|; 

$key = q/displaystylelanglen|Erangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2158.png"
 ALT="$\displaystyle \langle n\vert E\rangle$">|; 

$key = q/y_1(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="$y_1(z)$">|; 

$key = q/{displaymath}{{bf{B}=B_0,{{bf{e}_z+B_1,left[cos(omega,t),{{bf{e}_x+sin(omega,t),{{bf{e}_yright],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2443.png"
 ALT="\begin{displaymath}
{\bf B} = B_0 {\bf e}_z + B_1 \left[\cos(\omega t) {\bf e}_x + \sin(\omega t) {\bf e}_y\right],
\end{displaymath}">|; 

$key = q/displaystyleJ^2,Y_{l,m},chi_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1940.png"
 ALT="$\displaystyle J^2 Y_{l,m} \chi_+$">|; 

$key = q/{displaymath}[L_z,z]=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2183.png"
 ALT="\begin{displaymath}[L_z,z]= 0.
\end{displaymath}">|; 

$key = q/{displaymath}sum_{i}|iranglelanglei|equiv1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2118.png"
 ALT="\begin{displaymath}
\sum_{i} \vert i\rangle \langle i\vert \equiv 1,
\end{displaymath}">|; 

$key = q/psi_1+psi_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img763.png"
 ALT="$\psi_1+\psi_2$">|; 

$key = q/displaystyleL_z,Y_{l,m}(theta,phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1362.png"
 ALT="$\displaystyle L_z Y_{l,m}(\theta,\phi)$">|; 

$key = q/{displaymath}chi=left(array{{c}cosalphasinalpha,exp(,{{rm{i},beta)array{right){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1862.png"
 ALT="\begin{displaymath}
\chi =\left(\begin{array}{c}\cos\alpha \sin\alpha \exp( {\rm i} \beta)\end{array}\right)
\end{displaymath}">|; 

$key = q/mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1132.png"
 ALT="$\mu$">|; 

$key = q/c_1,chi_1+c_2,chi_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1723.png"
 ALT="$c_1 \chi_1+c_2 \chi_2$">|; 

$key = q/{displaymath}{mitDelta}uequivu-langleurangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img110.png"
 ALT="\begin{displaymath}
{\mit\Delta} u \equiv u- \langle u\rangle.
\end{displaymath}">|; 

$key = q/E_{nlm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2175.png"
 ALT="$E_{nlm}$">|; 

$key = q/m_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1419.png"
 ALT="$m_-$">|; 

$key = q/{displaymath}H=frac{p^2}{2,m}+V(r).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1494.png"
 ALT="\begin{displaymath}
H = \frac{p^2}{2 m} + V(r).
\end{displaymath}">|; 

$key = q/psi^{(2)}_{l,m_j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2358.png"
 ALT="$\psi^{(2)}_{l,m_j}$">|; 

$key = q/psi(x+lambda,t)=psi(x,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img154.png"
 ALT="$\psi(x+\lambda,t)=\psi(x,t)$">|; 

$key = q/displaystyles_1,(s_1+1),hbar^2,chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2040.png"
 ALT="$\displaystyle s_1 (s_1+1) \hbar^2
 \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)},$">|; 

$key = q/|psi|^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1183.png"
 ALT="$\vert\psi\vert^2$">|; 

$key = q/{displaymath}E=(1+X),{{rm{e}^{-X}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2817.png"
 ALT="\begin{displaymath}
E = (1+X) {\rm e}^{-X}.
\end{displaymath}">|; 

$key = q/{displaymath}dsimn^{-1slash3}simfrac{h}{sqrt{m_e,E}}simfrac{h}{p}simlambda,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1264.png"
 ALT="\begin{displaymath}
d\sim n^{-1/3}\sim \frac{h}{\sqrt{m_e E}}\sim \frac{h}{p}\sim \lambda,
\end{displaymath}">|; 

$key = q/s=1slash2pm1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2062.png"
 ALT="$s=1/2\pm 1/2$">|; 

$key = q/{displaymath}L^2=r^2,p^2-({{bf{r}cdot{{bf{p})^2+{{rm{i},hbar,{{bf{r}cdot{{bf{p}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1525.png"
 ALT="\begin{displaymath}
L^2 = r^2 p^2 - ({\bf r}\cdot{\bf p})^2 + {\rm i} \hbar {\bf r}\cdot{\bf p}.
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{(l+1slash2),n^3,a_0^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1660.png"
 ALT="$\displaystyle \frac{1}{(l+1/2) n^3 a_0^{ 2}},$">|; 

$key = q/V_0>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img813.png"
 ALT="$V_0&gt;0$">|; 

$key = q/displaystyle1.60,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img950.png"
 ALT="$\displaystyle 1.60,$">|; 

$key = q/r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img916.png"
 ALT="$r$">|; 

$key = q/1S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2658.png"
 ALT="$1S$">|; 

$key = q/displaystyle[y,p_z-z,p_y,z,p_x-x,p_z]=y,p_x,[p_z,z]+x,p_y,[z,p_z];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="399" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1316.png"
 ALT="$\displaystyle [y p_z-z p_y, z p_x-x p_z] = y p_x [p_z,z] + x p_y [z,p_z]$">|; 

$key = q/phi_1(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img768.png"
 ALT="$\phi_1(x)$">|; 

$key = q/{displaymath}int_{-infty}^{infty}|phi(p,t)|^{,2},dp=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img594.png"
 ALT="\begin{displaymath}
\int_{-\infty}^{\infty} \vert\phi(p,t)\vert^{ 2} dp = 1.
\end{displaymath}">|; 

$key = q/O^nomath_inline{dagnomath_inline{;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img544.png"
 ALT="$O^\dag $">|; 

$key = q/cosalpha_2,chi_++sinalpha_2,{{rm{e}^{,{{rm{i},beta_2},chi_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2094.png"
 ALT="$\cos\alpha_2 \chi_+
+ \sin\alpha_2 {\rm e}^{ {\rm i} \beta_2} \chi_-$">|; 

$key = q/x_1equivx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1311.png"
 ALT="$x_1\equiv x$">|; 

$key = q/{displaymath}T=sqrt{p^2,c^2+m^2,c^4}-m,c^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img2288.png"
 ALT="\begin{displaymath}
T = \sqrt{p^2 c^2+m^2 c^4} - m c^2.
\end{displaymath}">|; 

$key = q/{displaymath}a_l,j_l(k,r)=(l+1slash2)int_{-1}^1exp(,{{rm{i},k,r,mu),P_l(mu),dmu.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2918.png"
 ALT="\begin{displaymath}
a_l  j_l(k r) = (l+1/2)\int_{-1}^1 \exp( {\rm i} k r  \mu)  P_l(\mu)  d\mu.
\end{displaymath}">|; 

$key = q/2,E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1694.png"
 ALT="$2 E$">|; 

$key = q/dsigmaslashdOmega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2860.png"
 ALT="$d\sigma/d\Omega$">|; 

$key = q/E_a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1146.png"
 ALT="$E_a$">|; 

$key = q/{displaymath}L^2,Y_{l,m_-}=(L_+,L_-+L_z^{,2}-hbar,L_z),Y_{l,m_-},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="307" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1423.png"
 ALT="\begin{displaymath}
L^2 Y_{l,m_-} = (L_+ L_-+L_z^{ 2} - \hbar L_z) Y_{l,m_-},
\end{displaymath}">|; 

$key = q/displaystyleointY_{l,m}^{,ast},hbar^2left[l,(l+1)-m,(m+1)right]Y_{l,m},dOmega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1411.png"
 ALT="$\displaystyle \oint Y_{l,m}^{ \ast} \hbar^2\left[l (l+1) -m (m+1)\right]Y_{l,m} d\Omega$">|; 

$key = q/2,mu_B,B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2355.png"
 ALT="$2 \mu_B B$">|; 

$key = q/exp(,{{rm{i},k,r');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2899.png"
 ALT="$\exp( {\rm i} k  r')$">|; 

$key = q/{displaymath}S^2=S_x^{,2}+S_y^{,2}+S_z^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img1706.png"
 ALT="\begin{displaymath}
S^2 = S_x^{ 2} + S_y^{ 2}+ S_z^{ 2}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{d^2h}{dy^2}-2,y,frac{dh}{dy}+(epsilon-1),h=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1010.png"
 ALT="\begin{displaymath}
\frac{d^2h}{dy^2} - 2 y \frac{dh}{dy} + (\epsilon-1) h = 0.
\end{displaymath}">|; 

$key = q/omega>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img146.png"
 ALT="$\omega &gt;0$">|; 

$key = q/P_y(y)proptoI(y),dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img289.png"
 ALT="$P_y(y)\propto I(y)  dy$">|; 

$key = q/{displaymath}l,(l+1),Y_{l,m_-}=m_-,(m_--1),Y_{l,m_-},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1424.png"
 ALT="\begin{displaymath}
l (l+1) Y_{l,m_-} = m_- (m_- -1) Y_{l,m_-},
\end{displaymath}">|; 

$key = q/displaystyleR_{3,0}(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1644.png"
 ALT="$\displaystyle R_{3,0}(r)$">|; 

$key = q/{displaymath}H=hbar,omega,left(a_+,a_-+frac{1}{2}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1043.png"
 ALT="\begin{displaymath}
H = \hbar \omega \left(a_+ a_- + \frac{1}{2}\right),
\end{displaymath}">|; 

$key = q/{displaymath}P(t)={{rm{e}^{-w,t}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img143.png"
 ALT="\begin{displaymath}
P(t) = {\rm e}^{-w t}.
\end{displaymath}">|; 

$key = q/|E_0|slashhbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2617.png"
 ALT="$\vert E_0\vert/\hbar$">|; 

$key = q/{scriptstylesqrt{2slash3}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2019.png"
 ALT="${\scriptstyle\sqrt{2/3}}$">|; 

$key = q/1,1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2014.png"
 ALT="$1,1/2$">|; 

$key = q/{displaymath}frac{partialpsi}{partialx}=int_{-infty}^{infty}({{rm{i},k),bar{psi}(k),{{rm{e}^{,{{rm{i},(k,x-omega,t)},dk,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img430.png"
 ALT="\begin{displaymath}
\frac{\partial\psi}{\partial x} = \int_{-\infty}^{\infty}({\...
...,k) \bar{\psi}(k) {\rm e}^{ {\rm i} (k x-\omega t)} dk,
\end{displaymath}">|; 

$key = q/3,e,a_0,|{{bf{E}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2287.png"
 ALT="$3 e a_0 \vert{\bf E}\vert$">|; 

$key = q/{displaymath}psi(t)=sum_mc_m(t),expleft(-{{rm{i},E_m,tslashhbarright)psi_m,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2398.png"
 ALT="\begin{displaymath}
\psi(t) = \sum_m c_m(t) \exp\left(-{\rm i} E_m t/\hbar\right)\psi_m,
\end{displaymath}">|; 

$key = q/(1slash2),[x,p+(x,p)^nomath_inline{dagnomath_inline{]=(1slash2),(x,p+p^nomath_innline{,x^nomath_inline{dagnomath_inline{)=(1slash2),(x,p+p,x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="475" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img555.png"
 ALT="$(1/2) [x p + (x p)^\dag ]=(1/2) (x p+p^\dag x^\dag )=(1/2) (x p+p x)$">|; 

$key = q/k',aneqpislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3019.png"
 ALT="$k' a\neq \pi/2$">|; 

$key = q/{displaymath}{mitDelta}E=E_0left(frac{1}{n_f^{,2}}-frac{1}{n_i^{,2}}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1671.png"
 ALT="\begin{displaymath}
{\mit\Delta} E = E_0\left(\frac{1}{n_f^{ 2}}-\frac{1}{n_i^{ 2}}\right).
\end{displaymath}">|; 

$key = q/displaystyle(L_+,L_z+[L_z,L_+]),Y_{l,m}=(L_+,L_z+hbar,L_+),Y_{l,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1373.png"
 ALT="$\displaystyle (L_+ L_z + [L_z, L_+]) Y_{l,m}= (L_+ L_z + \hbar L_+) Y_{l,m}$">|; 

$key = q/{displaymath}langle1|H_1|2rangle=langle2|H_1|1rangle^ast=gamma,hbar,exp({{rm{i},omega,t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2411.png"
 ALT="\begin{displaymath}
\langle 1\vert H_1\vert 2\rangle = \langle 2\vert H_1\vert 1\rangle^\ast = \gamma \hbar \exp({\rm i} \omega t),
\end{displaymath}">|; 

$key = q/m>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1469.png"
 ALT="$m&gt;0$">|; 

$key = q/displaystyleleft(array{{cc}0,<col_mark>-{{rm{i}<row_mark>{{rm{i},<col_mark>0array{right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1812.png"
 ALT="$\displaystyle \left(\begin{array}{cc}0,&amp;-{\rm i}\\\\
{\rm i},&amp; 0\end{array}\right),$">|; 

$key = q/{displaymath}k,a+delta_0simeqfrac{k}{k'},tan(k',a).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img3005.png"
 ALT="\begin{displaymath}
k a + \delta_0 \simeq \frac{k}{k'} \tan (k' a).
\end{displaymath}">|; 

$key = q/displaystylesqrt{frac{1}{3}},psi^{(2)}_{3slash2,1slash2}-sqrt{frac{2}{3}},psi^{(2)}_{1slash2,1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1996.png"
 ALT="$\displaystyle \sqrt{\frac{1}{3}} \psi^{(2)}_{3/2,1/2} -
\sqrt{ \frac{2}{3}} \psi^{(2)}_{1/2,1/2},$">|; 

$key = q/{displaymath}langleS_yrangle=frac{hbar}{2},sinalpha,sin(Omega,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1855.png"
 ALT="\begin{displaymath}
\langle S_y\rangle = \frac{\hbar}{2} \sin\alpha \sin(\Omega t).
\end{displaymath}">|; 

$key = q/1,1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2074.png"
 ALT="$1, 1$">|; 

$key = q/{displaymath}{{bf{B}=-frac{{{bf{v}times{{bf{E}}{c^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2308.png"
 ALT="\begin{displaymath}
{\bf B} = -\frac{{\bf v}\times {\bf E}}{c^2}.
\end{displaymath}">|; 

$key = q/Phi_m(phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1400.png"
 ALT="$\Phi_m(\phi)$">|; 

$key = q/displaystyleL^2,Y_{l,m}(theta,phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1364.png"
 ALT="$\displaystyle L^2 Y_{l,m}(\theta,\phi)$">|; 

$key = q/k,x-omega,t+varphi=j,2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img163.png"
 ALT="$k x-\omega t+\varphi = j 2\pi$">|; 

$key = q/displaystylefrac{1}{sqrt{2}}left(chi^{(1)}_{-1slash2,1slash2}+chi^{(1)}_{1slash2,-1slash2}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2080.png"
 ALT="$\displaystyle \frac{1}{\sqrt{2}}\left(\chi^{(1)}_{-1/2,1/2}+ \chi^{(1)}_{1/2,-1/2}\right),$">|; 

$key = q/[J_z,L_z]=[J_z,S_z]=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1897.png"
 ALT="$[J_z,L_z]=[J_z,S_z]=0$">|; 

$key = q/displaystylelanglel,1slash2;j,m_j|H_1|l,1slash2;j,m_jrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2320.png"
 ALT="$\displaystyle \langle l,1/2;j,m_j\vert H_1\vert l,1/2;j,m_j\rangle$">|; 

$key = q/displaystylefrac{g,e,B_0}{2,m},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2457.png"
 ALT="$\displaystyle \frac{g e B_0}{2 m},$">|; 

$key = q/yllD;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img274.png"
 ALT="$y\ll D$">|; 

$key = q/displaystyleT,{{rm{e}^{,{{rm{i},k,a},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img843.png"
 ALT="$\displaystyle T {\rm e}^{ {\rm i} k a},$">|; 

$key = q/{displaymath}l,(l+1)geqm,(m+1).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1414.png"
 ALT="\begin{displaymath}
l (l+1) \geq m (m+1).
\end{displaymath}">|; 

$key = q/P_2(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2438.png"
 ALT="$P_2(t)$">|; 

$key = q/E_1,E_2,ldots,E_N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1164.png"
 ALT="$E_1, E_2, \ldots, E_N$">|; 

$key = q/displaystylefrac{1}{sqrt{2pi,hbar}}int_{-infty}^{infty}phi(p,t),{{rm{e}^{+{{rm{i},p,xslashhbar},dp,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img570.png"
 ALT="$\displaystyle \frac{1}{\sqrt{2\pi \hbar}}\int_{-\infty}^{\infty} \phi(p,t) {\rm e}^{+{\rm i} p x/\hbar} dp,$">|; 

$key = q/Phi_m(phi+2,pi)=Phi_m(phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1399.png"
 ALT="$\Phi_m(\phi+2 \pi)= \Phi_m(\phi)$">|; 

$key = q/n'=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1172.png"
 ALT="$n'=3$">|; 

$key = q/{displaymath}frac{dV(x)}{dx}=frac{dV(langlexrangle)}{dlanglexrangle}+frac{dV^2(lngle)}{dlanglexrangle^3},(x-langlexrangle)^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="528" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img518.png"
 ALT="\begin{displaymath}
\frac{dV(x)}{dx} = \frac{dV(\langle x\rangle)}{d\langle x\ra...
...e x\rangle)}{d\langle x\rangle^3} (x-\langle x\rangle)^{ 2}.
\end{displaymath}">|; 

$key = q/scriptstyle{1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2016.png"
 ALT="$\scriptstyle{1}$">|; 

$key = q/{displaymath}a=sqrt{frac{hbar^2}{2,m_e,(-E)}}=sqrt{frac{E_0}{E}},a_0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="65" BORDER="0"
 SRC="|."$dir".q|img1587.png"
 ALT="\begin{displaymath}
a = \sqrt{\frac{\hbar^2}{2 m_e (-E)}}=\sqrt{\frac{E_0}{E}} a_0,
\end{displaymath}">|; 

$key = q/{displaymath}P_{irightarrowf}^{abs}(t)=frac{pi,e^2,rho(omega_{fi})}{epsilon_0,hbrm{i},{{bf{k}!cdot!{{bf{r})|irangleright|^{,2}t.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="383" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img2562.png"
 ALT="\begin{displaymath}
P_{i\rightarrow f}^{abs}(t) =\frac{\pi e^2 \rho(\omega_{fi...
... i} {\bf k}\!\cdot\!{\bf r})\vert i\rangle\right\vert^{ 2}t.
\end{displaymath}">|; 

$key = q/displaystyle{{rm{i},frac{dc_1}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2413.png"
 ALT="$\displaystyle {\rm i} \frac{dc_1}{dt}$">|; 

$key = q/{displaymath}hbar,omega=E_0slash4-E_0=-frac{3}{4},E_0=10.2,{{rm{eV}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="301" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img2669.png"
 ALT="\begin{displaymath}
\hbar \omega = E_0/4 - E_0 = -\frac{3}{4} E_0 = 10.2 {\rm eV}.
\end{displaymath}">|; 

$key = q/sin^2(thetaslash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1860.png"
 ALT="$\sin^2(\theta/2)$">|; 

$key = q/displaystyleE_1+frac{|e_{12}|^2}{E_1-E_2}+cdots,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2140.png"
 ALT="$\displaystyle E_1 + \frac{\vert e_{12}\vert^2}{E_1-E_2}+ \cdots,$">|; 

$key = q/displaystyleR_{2,1}(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1642.png"
 ALT="$\displaystyle R_{2,1}(r)$">|; 

$key = q/langlexrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img490.png"
 ALT="$\langle x\rangle$">|; 

$key = q/p_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1190.png"
 ALT="$p_y$">|; 

$key = q/{displaymath}langleUrangle(Z)=2,(Z-2)left(frac{e^2}{4pi,epsilon_0}right)leftlanglefrac{1}{r}rightrangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2765.png"
 ALT="\begin{displaymath}
\langle U\rangle(Z) = 2 (Z-2)\left(\frac{e^2}{4\pi \epsilon_0}\right)\left\langle\frac{1}{r}\right\rangle.
\end{displaymath}">|; 

$key = q/L_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1318.png"
 ALT="$L_y$">|; 

$key = q/H_1(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2463.png"
 ALT="$H_1(t)$">|; 

$key = q/x=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img697.png"
 ALT="$x=1$">|; 

$key = q/displaystyleS_+,S_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1711.png"
 ALT="$\displaystyle S_+ S_-$">|; 

$key = q/{displaymath}psi(x)=int_{-infty}^inftyc(p'),psi_p(x,p'),dp',{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img728.png"
 ALT="\begin{displaymath}
\psi(x) = \int_{-\infty}^\infty c(p') \psi_p(x,p') dp',
\end{displaymath}">|; 

$key = q/{displaymath}|mbox{{boldmath{{epsilon{}!cdot!{{bf{d}_{if}|^{,2}=d_{if}^2,sin^2theta,cos^2phi.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img2592.png"
 ALT="\begin{displaymath}
\vert\mbox{\boldmath$\epsilon$}\!\cdot\!{\bf d}_{if}\vert^{ 2} = d_{if}^2 \sin^2\theta \cos^2\phi.
\end{displaymath}">|; 

$key = q/{displaymath}tan(k,a+delta_0)=frac{k}{k'},tan(k',a){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img3000.png"
 ALT="\begin{displaymath}
\tan(k a+\delta_0) = \frac{k}{k'}  \tan( k' a)
\end{displaymath}">|; 

$key = q/(nabla^2+k^2),u_0=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2849.png"
 ALT="$(\nabla^2+k^2) u_0 = 0$">|; 

$key = q/langleurangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img100.png"
 ALT="$\langle u\rangle$">|; 

$key = q/{displaymath}S_z,(S_-,chi_{s,m_s})=(m_s-1),hbar,(S_-,chi_{s,m_s}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="301" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1747.png"
 ALT="\begin{displaymath}
S_z (S_- \chi_{s,m_s}) = (m_s-1) \hbar (S_- \chi_{s,m_s}).
\end{displaymath}">|; 

$key = q/{displaymath}psi(x_2,x_1,t)={{rm{e}^{,{{rm{i},varphi},psi(x_1,x_2,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1138.png"
 ALT="\begin{displaymath}
\psi(x_2,x_1,t) = {\rm e}^{ {\rm i} \varphi} \psi(x_1,x_2,t).
\end{displaymath}">|; 

$key = q/x=x';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img710.png"
 ALT="$x=x'$">|; 

$key = q/displaystyle-k_z^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1224.png"
 ALT="$\displaystyle - k_z^{ 2},$">|; 

$key = q/{displaymath}{mitDelta}E,{mitDelta}tstackrel{_{{normalsize{>}}{_{{normalsize{sim}}hbar,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img635.png"
 ALT="\begin{displaymath}
{\mit\Delta}E {\mit\Delta} t\stackrel {_{\normalsize &gt;}}{_{\normalsize\sim}}\hbar,
\end{displaymath}">|; 

$key = q/2,l+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1664.png"
 ALT="$2 l+1$">|; 

$key = q/{rmsinc}(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2492.png"
 ALT="${\rm sinc}(x)$">|; 

$key = q/{displaymath}frac{d^2psi}{dx^2}=frac{2,m}{hbar^2}left[V(x)-Eright]psi.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img787.png"
 ALT="\begin{displaymath}
\frac{d^2 \psi}{d x^2} = \frac{2 m}{\hbar^2}
\left[V(x)-E\right]\psi.
\end{displaymath}">|; 

$key = q/{displaymath}P_{irightarrowf}^{abs}(t)=1-expleft(-w_{irightarrowf}^{abs},tright).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img2571.png"
 ALT="\begin{displaymath}
P_{i\rightarrow f}^{abs}(t) = 1 - \exp\left(-w_{i\rightarrow f}^{abs} t\right).
\end{displaymath}">|; 

$key = q/{displaymath}I({{bf{r}_2)=frac{pi}{r_2}left[1-{{rm{e}^{-2,r_2},(1+r_2)right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img2750.png"
 ALT="\begin{displaymath}
I({\bf r}_2) = \frac{\pi}{r_2}\left[1-{\rm e}^{-2 r_2} (1+r_2)\right].
\end{displaymath}">|; 

$key = q/psi(x_1,x_2,ldots,x_N,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1074.png"
 ALT="$\psi(x_1,x_2,\ldots, x_N,t)$">|; 

$key = q/lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="$\lambda $">|; 

$key = q/{displaymath}DeltaE=frac{mu_0,g_p,e^2}{8pi,m_p,m_e}leftlanglefrac{3,({{bf{S}_pcd},langle{{bf{S}_pcdot{{bf{S}_erangle,|psi(0)|^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="592" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2373.png"
 ALT="\begin{displaymath}
\Delta E = \frac{\mu_0 g_p e^2}{8\pi m_p m_e}\left\langl...
...} \langle{\bf S}_p\cdot{\bf S}_e\rangle \vert\psi(0)\vert^2.
\end{displaymath}">|; 

$key = q/{displaymath}J^2=J_x^{,2}+J_y^{,2}+J_z^{,2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img1881.png"
 ALT="\begin{displaymath}
J^2 =J_x^{ 2}+J_y^{ 2}+J_z^{ 2},
\end{displaymath}">|; 

$key = q/displaystyleL_+,L_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1336.png"
 ALT="$\displaystyle L_+ L_-$">|; 

$key = q/s_{1z},s_{2z}=pm1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2059.png"
 ALT="$s_{1z}, s_{2z}=\pm 1/2$">|; 

$key = q/psi({{bf{r})rightarrowpsi_0({{bf{r})=sqrt{n},exp(,{{rm{i},{{bf{k}cdot{{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2874.png"
 ALT="$\psi({\bf r})\rightarrow\psi_0({\bf r}) = \sqrt{n} \exp( {\rm i} 
{\bf k}\cdot{\bf r})$">|; 

$key = q/{displaymath}E_0=-frac{m_e,e^4}{2,(4pi,epsilon_0)^2,hbar^2}=-frac{e^2}{8pi,epsilon_0,a_0}=-13.6,{{rm{eV},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="365" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1622.png"
 ALT="\begin{displaymath}
E_0 = -\frac{m_e e^4}{2 (4\pi \epsilon_0)^2 \hbar^2} = - \frac{e^2}{8\pi \epsilon_0 a_0}
= -13.6 {\rm eV},
\end{displaymath}">|; 

$key = q/|y|rightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1002.png"
 ALT="$\vert y\vert\rightarrow\infty$">|; 

$key = q/{displaymath}Y_{l,m}(theta,phi)simfrac{{{rm{e}^{,{{rm{i},m,phi}}{(sintheta)^m}leeta}frac{d}{dtheta}right)^{l-m}(sintheta)^{2,l}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1457.png"
 ALT="\begin{displaymath}
Y_{l,m}(\theta,\phi)\sim \frac{{\rm e}^{ {\rm i} m \phi}}...
...\sin\theta}\frac{d}{d\theta}\right)^{l-m}
(\sin\theta)^{2 l}.
\end{displaymath}">|; 

$key = q/S_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1699.png"
 ALT="$S_y$">|; 

$key = q/x_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img266.png"
 ALT="$x_2$">|; 

$key = q/x_2+dx_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1076.png"
 ALT="$x_2+dx_2$">|; 

$key = q/j_l(k,r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1561.png"
 ALT="$j_l(k r)$">|; 

$key = q/{displaymath}P(XotimesY)=lim_{{mitOmega}({mitSigma})rightarrowinfty}frac{{mitOmeega}({{mitSigma}}otimes{{mitSigma}})}=P(X),P(Y).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="383" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img87.png"
 ALT="\begin{displaymath}
P(X\otimes Y) =\lim_{{\mit\Omega}({\mit\Sigma})\rightarrow\i...
...it\Omega}({{\mit\Sigma}}\otimes {{\mit\Sigma}})}= P(X)  P(Y).
\end{displaymath}">|; 

$key = q/{displaymath}psi,frac{partialpsi^ast}{partialt}=-frac{{{rm{i},hbar}{2,m},psi,frast}{partialx^2}+frac{{{rm{i}}{hbar},V,|psi|^{,2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img473.png"
 ALT="\begin{displaymath}
\psi \frac{\partial\psi^\ast}{\partial t}= -\frac{{\rm i} ...
...\partial x^2} + \frac{{\rm i}}{\hbar} V \vert\psi\vert^{ 2}
\end{displaymath}">|; 

$key = q/displaystylepsi(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img627.png"
 ALT="$\displaystyle \psi(x)$">|; 

$key = q/z=r,costheta+{{rm{i},rsintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img209.png"
 ALT="$z= r \cos\theta+ {\rm i} r\sin\theta$">|; 

$key = q/n=1,2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="$n=1,2$">|; 

$key = q/{displaymath}E_0=frac{2,hbar^2}{m,a^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img977.png"
 ALT="\begin{displaymath}
E_0 = \frac{2 \hbar^2}{m a^2}.
\end{displaymath}">|; 

$key = q/phi({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2526.png"
 ALT="$\phi({\bf r})$">|; 

$key = q/{displaymath}(nabla^2+k^2),psi=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2907.png"
 ALT="\begin{displaymath}
(\nabla^2 + k^2) \psi = 0.
\end{displaymath}">|; 

$key = q/{displaymath}{calR}_0(r)=B,frac{sinh(kappa,r)}{r},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2997.png"
 ALT="\begin{displaymath}
{\cal R}_0(r) = B  \frac{\sinh(\kappa  r)}{r},
\end{displaymath}">|; 

$key = q/chi';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1730.png"
 ALT="$\chi'$">|; 

$key = q/{displaymath}psi(r,theta,phi)=R(r),Y_{l,m}(theta,phi).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1530.png"
 ALT="\begin{displaymath}
\psi(r,\theta,\phi) = R(r) Y_{l,m}(\theta,\phi).
\end{displaymath}">|; 

$key = q/{displaymath}H,psi=E,psi,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1493.png"
 ALT="\begin{displaymath}
H \psi = E \psi,
\end{displaymath}">|; 

$key = q/displaystylechi(E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img645.png"
 ALT="$\displaystyle \chi(E)$">|; 

$key = q/displaystyleint_0^pi{{rm{e}^{,(x^2+X^2-2,x,X,costheta)^{1slash2}},sintheta,dtheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2793.png"
 ALT="$\displaystyle \int_0^\pi {\rm e}^{ (x^2+X^2-2 x X \cos\theta)^{1/2}} \sin\theta d\theta$">|; 

$key = q/x_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img330.png"
 ALT="$x_0$">|; 

$key = q/tandelta_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2969.png"
 ALT="$\tan\delta_l$">|; 

$key = q/R^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1294.png"
 ALT="$R^{-2}$">|; 

$key = q/{displaymath}-j,-j+1,cdots,j-1,j.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1888.png"
 ALT="\begin{displaymath}
-j, -j+1,\cdots, j-1, j.
\end{displaymath}">|; 

$key = q/{displaymath}|psi_0|^{,2}=frac{1}{(2pi,sigma^2)^{1slash2}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img467.png"
 ALT="\begin{displaymath}
\vert\psi_0\vert^{ 2} = \frac{1}{(2\pi \sigma^2)^{1/2}}.
\end{displaymath}">|; 

$key = q/S_{2z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2033.png"
 ALT="$S_{2z}$">|; 

$key = q/{displaymath}l,(l+1)geqm,(m-1).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1416.png"
 ALT="\begin{displaymath}
l (l+1) \geq m (m-1).
\end{displaymath}">|; 

$key = q/sigma_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img493.png"
 ALT="$\sigma_x$">|; 

$key = q/displaystylecotdelta_l(E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3023.png"
 ALT="$\displaystyle \cot \delta_l(E)$">|; 

$key = q/displaystyleL^2-L_z^{,2}+hbar,L_z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1338.png"
 ALT="$\displaystyle L^2 - L_z^{ 2} + \hbar L_z.$">|; 

$key = q/{displaymath}sigma_x^{,2}=sigma^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img502.png"
 ALT="\begin{displaymath}
\sigma_x^{ 2} = \sigma^2.
\end{displaymath}">|; 

$key = q/{displaymath}psi(x,t)proptofrac{expleft[,{{rm{i},(k_0,x-omega_0,t)-(x-x_0-v_g,t)},2,alpha,({mitDelta}k)^{,2},tright]^{1slash2}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="593" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img385.png"
 ALT="\begin{displaymath}
\psi(x,t)\propto\frac{\exp\left[ {\rm i} (k_0 x-\omega_0\...
...[1+{\rm i} 2 \alpha ({\mit\Delta}k)^{ 2} t\right]^{1/2}},
\end{displaymath}">|; 

$key = q/p_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1191.png"
 ALT="$p_z$">|; 

$key = q/sigma_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img749.png"
 ALT="$\sigma_p$">|; 

$key = q/displaystylew_{irightarrowf}^{stm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2582.png"
 ALT="$\displaystyle w_{i\rightarrow f}^{stm}$">|; 

$key = q/displaystyle-frac{(2,l-1)!!}{(k,r)^{l+1}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2975.png"
 ALT="$\displaystyle -\frac{(2 l-1)!!}{(k r)^{l+1}},$">|; 

$key = q/j=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1979.png"
 ALT="$j=1/2$">|; 

$key = q/{displaymath}sigma^2(t)=({mitDelta}x)^{,2}+frac{alpha^2,t^2}{4,({mitDelta}x)^{,2}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img386.png"
 ALT="\begin{displaymath}
\sigma^2(t) = ({\mit\Delta}x)^{ 2} + \frac{\alpha^2 t^2}{4 ({\mit\Delta}x)^{ 2}}.
\end{displaymath}">|; 

$key = q/{displaymath}int_{-infty}^infty|c(x')|^{,2},dx'=int_{-infty}^infty|psi(x')|^{,2},dx'=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="313" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img720.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \vert c(x')\vert^{ 2} dx'= \int_{-\infty}^\infty \vert\psi(x')\vert^{ 2} dx' =1.
\end{displaymath}">|; 

$key = q/displaystyleE_1';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2139.png"
 ALT="$\displaystyle E_1'$">|; 

$key = q/dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1633.png"
 ALT="$dV$">|; 

$key = q/x=l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1951.png"
 ALT="$x=l$">|; 

$key = q/{mitDelta}k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img352.png"
 ALT="${\mit\Delta} k$">|; 

$key = q/displaystyleleft(array{{c}1slashsqrt{2}<row_mark>[0.5ex]-1slashsqrt{2}array{right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2279.png"
 ALT="$\displaystyle \left(\begin{array}{c}
1/\sqrt{2}  [0.5ex]
-1/\sqrt{2}\end{array}\right).$">|; 

$key = q/c_1(0)=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2419.png"
 ALT="$c_1(0)=1$">|; 

$key = q/{displaymath}(L_pm)^nomath_inline{dagnomath_inline{=L_mp,[0.5ex]{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" BORDER="0"
 SRC="|."$dir".q|img1335.png"
 ALT="\begin{displaymath}
(L_\pm)^\dag = L_\mp, [0.5ex]
\end{displaymath}">|; 

$key = q/{displaymath}frac{partialpsi^ast}{partialt},psi+psi^{ast},frac{partialpsi}{partitialx}-psi,frac{partialpsi^ast}{partialx}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="586" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img477.png"
 ALT="\begin{displaymath}
\frac{\partial\psi^\ast}{\partial t}  \psi+ \psi^{\ast} \f...
...artial x} - \psi \frac{\partial\psi^\ast}{\partial x}\right).
\end{displaymath}">|; 

$key = q/textstylefrac{d^2phi(k_0)}{dk^2}=-alpha,t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img366.png"
 ALT="$\textstyle \frac{d^2\phi(k_0)}{dk^2} = - \alpha t,$">|; 

$key = q/displaystyle2,{{rm{i},hbar,(L_y,x-L_x,y+{{rm{i},hbar,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2192.png"
 ALT="$\displaystyle 2 {\rm i} \hbar (L_y x -L_x y + {\rm i} \hbar z)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter03slashfig04.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="378" HEIGHT="341" BORDER="0"
 SRC="|."$dir".q|img254.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter03/fig04.eps}}
\end{figure}">|; 

$key = q/{displaymath}m_j=m+m_s.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1910.png"
 ALT="\begin{displaymath}
m_j = m+m_s.
\end{displaymath}">|; 

$key = q/displaystylehbar^2,left[l,(l+1)-m,(m+1)right],ointY_{l,m}^{,ast},Y_{l,m},dOmega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1412.png"
 ALT="$\displaystyle \hbar^2 \left[l (l+1) -m (m+1)\right] \oint Y_{l,m}^{ \ast} Y_{l,m} d\Omega$">|; 

$key = q/|c_i|^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img698.png"
 ALT="$\vert c_i\vert^2$">|; 

$key = q/{displaymath}int_{-infty}^{infty}delta(x),dx=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img577.png"
 ALT="\begin{displaymath}
\int_{-\infty}^{\infty}\delta(x) dx = 1.
\end{displaymath}">|; 

$key = q/+infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img193.png"
 ALT="$+\infty$">|; 

$key = q/{displaymath}-frac{y}{sqrt{lambda-y^2}}=tany.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img990.png"
 ALT="\begin{displaymath}
-\frac{y}{\sqrt{\lambda-y^2}} = \tan y.
\end{displaymath}">|; 

$key = q/displaystyleZ(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1232.png"
 ALT="$\displaystyle Z(z)$">|; 

$key = q/displaystylej_l(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1547.png"
 ALT="$\displaystyle j_l(z)$">|; 

$key = q/j,m_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1971.png"
 ALT="$j, m_j$">|; 

$key = q/c_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1724.png"
 ALT="$c_1$">|; 

$key = q/{calE};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img899.png"
 ALT="${\cal E}$">|; 

$key = q/W(<E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1053.png"
 ALT="$W (&lt;E)$">|; 

$key = q/j=l-1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1954.png"
 ALT="$j=l-1/2$">|; 

$key = q/b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img456.png"
 ALT="$b$">|; 

$key = q/displaystyleY_{2,pm1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1481.png"
 ALT="$\displaystyle Y_{2,\pm 1}$">|; 

$key = q/{displaymath}psi(x)=sqrt{frac{2,a^3}{pi}},frac{1}{x^2+a^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img750.png"
 ALT="\begin{displaymath}
\psi(x) = \sqrt{\frac{2 a^3}{\pi}} \frac{1}{x^2+a^2}.
\end{displaymath}">|; 

$key = q/displaystylesqrt{frac{2}{3}},psi^{(1)}_{0,1slash2}+sqrt{frac{1}{3}},psi^{(1)}_{1,-1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1986.png"
 ALT="$\displaystyle \sqrt{\frac{2}{3}} \psi^{(1)}_{0,1/2} +
\sqrt{ \frac{1}{3}} \psi^{(1)}_{1,-1/2},$">|; 

$key = q/x=a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img836.png"
 ALT="$x=a$">|; 

$key = q/x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="$x$">|; 

$key = q/{displaymath}q,a=n,pi,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img854.png"
 ALT="\begin{displaymath}
q a = n \pi,
\end{displaymath}">|; 

$key = q/A_2=4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img911.png"
 ALT="$A_2=4$">|; 

$key = q/hbar=hslash2pi=1.0546times10^{-34},{{rm{J,s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img256.png"
 ALT="$\hbar = h/2\pi = 1.0546\times 10^{-34} {\rm J s}$">|; 

$key = q/-A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img157.png"
 ALT="$-A$">|; 

$key = q/displaystylefrac{l_y,pi}{a},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1237.png"
 ALT="$\displaystyle \frac{l_y \pi}{a},$">|; 

$key = q/displaystylesqrt{n+1},psi_{n+1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1038.png"
 ALT="$\displaystyle \sqrt{n+1} \psi_{n+1},$">|; 

$key = q/{displaymath}O(c,f(x))=c,O(f(x)),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img532.png"
 ALT="\begin{displaymath}
O(c f(x)) =c O(f(x)),
\end{displaymath}">|; 

$key = q/m,hbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1367.png"
 ALT="$m \hbar$">|; 

$key = q/{displaymath}E=4,E_0+4,E_0=8,E_0=-108.8,{{rm{eV}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="301" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img2724.png"
 ALT="\begin{displaymath}
E = 4 E_0 + 4 E_0 = 8 E_0 = -108.8 {\rm eV}.
\end{displaymath}">|; 

$key = q/psi_n(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img806.png"
 ALT="$\psi_n(x)$">|; 

$key = q/displaystyleE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2809.png"
 ALT="$\displaystyle E$">|; 

$key = q/p=hbar,k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img348.png"
 ALT="$p = \hbar k$">|; 

$key = q/{displaymath}psi(0)=psi(a)=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img794.png"
 ALT="\begin{displaymath}
\psi(0) = \psi(a) = 0.
\end{displaymath}">|; 

$key = q/omega=pmomega_{fi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2503.png"
 ALT="$\omega=\pm\omega_{fi}$">|; 

$key = q/{calO}(epsilon^3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2167.png"
 ALT="${\cal O}(\epsilon^3)$">|; 

$key = q/{displaymath}w_{irightarrowf}^{stm}=frac{pi,e^2,rho(omega_{if})}{epsilon_0,hbar^{rm{i},{{bf{k}!cdot!{{bf{r})|frangleright|^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="357" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img2575.png"
 ALT="\begin{displaymath}
w_{i\rightarrow f}^{stm} =
\frac{\pi e^2 \rho(\omega_{if})...
...m i} {\bf k}\!\cdot\!{\bf r})\vert f\rangle\right\vert^{ 2}.
\end{displaymath}">|; 

$key = q/{displaymath}int_{-infty}^{infty}psi^ast(x,t),psi(x,t),dx=frac{1}{2pi,hbar}int_{,{{rm{e}^{+{{rm{i},(p-p'),xslashhbar},dx,dp,dp'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="638" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img600.png"
 ALT="\begin{displaymath}
\int_{-\infty}^{\infty} \psi^\ast(x,t) \psi(x,t) dx = \fra...
... \phi(p,t) {\rm e}^{+{\rm i} (p-p') x/\hbar} dx dp dp'.
\end{displaymath}">|; 

$key = q/epsilon=mu_B,B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="$\epsilon = \mu _B B$">|; 

$key = q/m_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1759.png"
 ALT="$m_s$">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{Chapter12slashfig02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="517" HEIGHT="400" BORDER="0"
 SRC="|."$dir".q|img2359.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{Chapter12/fig02.eps}}
\end{figure}">|; 

$key = q/{displaymath}int_{-infty}^infty|f|^{,2},dx=int_{-infty}^inftyf^ast,f,dx=int_{-inArangle),psi]^{,ast},[(A-langleArangle),psi],dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="523" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img607.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \vert f\vert^{ 2} dx= \int_{-\infty}...
...e A\rangle) \psi]^{ \ast} [(A-\langle A\rangle) \psi] dx.
\end{displaymath}">|; 

$key = q/displaystyleomega_{21};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2456.png"
 ALT="$\displaystyle \omega_{21}$">|; 

$key = q/displaystyleJ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2796.png"
 ALT="$\displaystyle J$">|; 

$key = q/c_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2390.png"
 ALT="$c_m$">|; 

$key = q/{displaymath}E=h,nu.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img247.png"
 ALT="\begin{displaymath}
E = h \nu.
\end{displaymath}">|; 

$key = q/displaystyleint{{rm{e}^{-beta,x},x^2,dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2748.png"
 ALT="$\displaystyle \int{\rm e}^{-\beta x} x^2 dx$">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{Chapter12slashfig01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="491" HEIGHT="400" BORDER="0"
 SRC="|."$dir".q|img2338.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{Chapter12/fig01.eps}}
\end{figure}">|; 

$key = q/N_ntimesN_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2254.png"
 ALT="$N_n\times N_n$">|; 

$key = q/{displaymath}Omegasimeqfrac{e,B}{m_e}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1856.png"
 ALT="\begin{displaymath}
\Omega \simeq \frac{e B}{m_e}.
\end{displaymath}">|; 

$key = q/displaystylecostheta,frac{partial}{partialr}-frac{sintheta}{r},frac{partial}{partialtheta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1356.png"
 ALT="$\displaystyle \cos\theta \frac{\partial}{\partial r} -\frac{\sin\theta}{r} \frac{\partial}{\partial \theta}.$">|; 

$key = q/{displaymath}m'=m.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img2187.png"
 ALT="\begin{displaymath}
m' = m.
\end{displaymath}">|; 

$key = q/{displaymath}psi(x,0)=int_{-infty}^{infty}bar{psi}(k),{{rm{e}^{,{{rm{i},k,x},dk.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img335.png"
 ALT="\begin{displaymath}
\psi(x,0) = \int_{-\infty}^{\infty} \bar{\psi}(k) {\rm e}^{ {\rm i} k x} dk.
\end{displaymath}">|; 

$key = q/{displaymath}psi(x)=T,{{rm{e}^{,{{rm{i},k,x}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img823.png"
 ALT="\begin{displaymath}
\psi(x) = T {\rm e}^{ {\rm i} k x}.
\end{displaymath}">|; 

$key = q/s_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2036.png"
 ALT="$s_2$">|; 

$key = q/displaystylefrac{dphi(k_0)}{dk}=x-v_g,t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img364.png"
 ALT="$\displaystyle \frac{d\phi(k_0)}{dk} = x - v_g t,$">|; 

$key = q/{displaymath}alpha<frac{16}{3},4piepsilon_0,a_0,sum_{n>1}|langle1,0,0|z|n,1,0rangle|^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2226.png"
 ALT="\begin{displaymath}
\alpha &lt; \frac{16}{3} 4\pi\epsilon_0 a_0 \sum_{n&gt;1}
\vert\langle 1,0,0\vert z\vert n,1,0\rangle\vert^2.
\end{displaymath}">|; 

$key = q/varphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img162.png"
 ALT="$\varphi$">|; 

$key = q/displaystyle[L_x,L_z]+{{rm{i},[L_y,L_z]=-{{rm{i},hbar,L_y-hbar,L_x=-hbar,L_+,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1342.png"
 ALT="$\displaystyle [L_x,L_z] + {\rm i} [L_y,L_z] = -{\rm i} \hbar L_y - \hbar L_x= - \hbar L_+,$">|; 

$key = q/displaystylefrac{1}{sqrt{2pi,hbar}}int_{-infty}^inftypsi(x,t),{{rm{e}^{-{{rm{i},p,xslashhbar},dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img572.png"
 ALT="$\displaystyle \frac{1}{\sqrt{2\pi \hbar}}\int_{-\infty}^\infty \psi(x,t) {\rm e}^{-{\rm i} p x/\hbar} dx,$">|; 

$key = q/displaystyleS_2^{,2},chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2041.png"
 ALT="$\displaystyle S_2^{ 2}  \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}$">|; 

$key = q/{displaymath}langlex^nrangle_{a,b}=int_{-infty}^inftypsi^ast(x,E_{a,b}),x^n,psi(x,E_{a,b}),dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img1153.png"
 ALT="\begin{displaymath}
\langle x^n\rangle_{a,b} = \int_{-\infty}^\infty\psi^\ast(x,E_{a,b}) x^n \psi(x,E_{a,b}) dx.
\end{displaymath}">|; 

$key = q/displaystylem_j,hbar,psi^{(2)}_{l,s;j,m_j}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1919.png"
 ALT="$\displaystyle m_j \hbar \psi^{(2)}_{l,s;j,m_j}.$">|; 

$key = q/b_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img704.png"
 ALT="$b_i$">|; 

$key = q/i=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2126.png"
 ALT="$i=1$">|; 

$key = q/displaystylek_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1238.png"
 ALT="$\displaystyle k_z$">|; 

$key = q/{displaymath}psi^{(2)}_{l+1slash2,m+1slash2}=left(frac{l+m+1}{2,l+1}right)^{1sla{2,l+1}right)^{1slash2}psi^{(1)}_{m+1,-1slash2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="508" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1955.png"
 ALT="\begin{displaymath}
\psi^{(2)}_{l+1/2,m+1/2} = \left(\frac{l+m+1}{2 l+1}\right)...
... + \left(\frac{l-m}{2 l+1}\right)^{1/2}\psi^{(1)}_{m+1,-1/2},
\end{displaymath}">|; 

$key = q/e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1825.png"
 ALT="$e$">|; 

$key = q/{displaymath}mbox{{boldmath{{mu{}=-frac{g,e}{2,m_e},{{bf{S},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img1826.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\mu$}= -\frac{g e}{2 m_e} {\bf S},
\end{displaymath}">|; 

$key = q/xgeqa;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img792.png"
 ALT="$x\geq a$">|; 

$key = q/{displaymath}frac{dsigma}{dOmega}=frac{sin^2k,a}{k^2}simeqa^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2979.png"
 ALT="\begin{displaymath}
\frac{d\sigma}{d\Omega} = \frac{\sin^2 k a}{k^2} \simeq a^2
\end{displaymath}">|; 

$key = q/displaystylefrac{pi}{3,epsilon_0,hbar^2},d^{,2}_{if},rho(omega_{if}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2598.png"
 ALT="$\displaystyle \frac{\pi}{3 \epsilon_0 \hbar^2} d^{ 2}_{if} \rho(\omega_{if}),$">|; 

$key = q/displaystylefrac{domega(k_0)}{dk},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img370.png"
 ALT="$\displaystyle \frac{d\omega(k_0)}{dk},$">|; 

$key = q/t+dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img934.png"
 ALT="$t+dt$">|; 

$key = q/displaystylec_{l,m}^+,Y_{l,m+1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1379.png"
 ALT="$\displaystyle c_{l,m}^+ Y_{l,m+1},$">|; 

$key = q/chi_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1782.png"
 ALT="$\chi_+$">|; 

$key = q/{displaymath}leftlanglefrac{1}{r}rightrangle=frac{Z}{a_0}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2768.png"
 ALT="\begin{displaymath}
\left\langle \frac{1}{r}\right\rangle = \frac{Z}{a_0}.
\end{displaymath}">|; 

$key = q/psi^{(2)}_{j,m_j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1977.png"
 ALT="$\psi^{(2)}_{j,m_j}$">|; 

$key = q/2S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2264.png"
 ALT="$2S$">|; 

$key = q/tany=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img992.png"
 ALT="$\tan y =0$">|; 

$key = q/omega+domega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2557.png"
 ALT="$\omega+d\omega$">|; 

$key = q/displaystylec_1(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2423.png"
 ALT="$\displaystyle c_1(t)$">|; 

$key = q/{displaymath}psi(x,t)=A,exp[-a,(m,x^2slashhbar+{{rm{i},t)],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img455.png"
 ALT="\begin{displaymath}
\psi(x,t) = A \exp[-a (m x^2/\hbar + {\rm i}  t)],
\end{displaymath}">|; 

$key = q/{displaymath}T=frac{p^2}{2,m}left[1-frac{1}{4}left(frac{p}{m,c}right)^2+{calO}left(frac{p}{m,c}right)^4right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2290.png"
 ALT="\begin{displaymath}
T = \frac{p^2}{2 m}\left[1- \frac{1}{4}\left(\frac{p}{m c}\right)^2+
{\cal O}\left(\frac{p}{m c}\right)^4\right].
\end{displaymath}">|; 

$key = q/c_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2397.png"
 ALT="$c_n$">|; 

$key = q/3S_{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2333.png"
 ALT="$3S_{1/2}$">|; 

$key = q/{displaymath}P=-frac{partialE_{total}}{partialV}=frac{2}{5},n,E_F,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1261.png"
 ALT="\begin{displaymath}
P = -\frac{\partial E_{total}}{\partial V} = \frac{2}{5} n E_F,
\end{displaymath}">|; 

$key = q/aslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1051.png"
 ALT="$a/2$">|; 

$key = q/{displaymath}{{rm{i},hbar,frac{dc_n}{dt}=sum_mH_{nm},exp(,{{rm{i},omega_{nm},t),c_m,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="272" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2466.png"
 ALT="\begin{displaymath}
{\rm i} \hbar \frac{d c_n}{dt} = \sum_m H_{nm} \exp( {\rm i} \omega_{nm} t) c_m,
\end{displaymath}">|; 

$key = q/L^3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1297.png"
 ALT="$L^3$">|; 

$key = q/lambda_2=-3,e,a_0,|{{bf{E}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2275.png"
 ALT="$\lambda_2=-3 e a_0 \vert{\bf E}\vert$">|; 

$key = q/displaystylem';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2651.png"
 ALT="$\displaystyle m'$">|; 

$key = q/sqrt{2,m,|V_0|,a^2slashhbar^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3013.png"
 ALT="$\sqrt{2 m  \vert V_0\vert a^2/\hbar^2}$">|; 

$key = q/displaystyleint_{-infty}^inftyP(u),u,du,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img125.png"
 ALT="$\displaystyle \int_{-\infty}^\infty P(u) u du,$">|; 

$key = q/|y|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1015.png"
 ALT="$\vert y\vert$">|; 

$key = q/langlen,l,m|x_-|n',l',m'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2646.png"
 ALT="$\langle n,l,m\vert x_-\vert n',l',m'\rangle$">|; 

$key = q/{displaymath}psi=A,exp(-beta,r),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1688.png"
 ALT="\begin{displaymath}
\psi = A \exp(-\beta r),
\end{displaymath}">|; 

$key = q/displaystyleH_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2310.png"
 ALT="$\displaystyle H_1$">|; 

$key = q/{displaymath}v_g=frac{p}{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img393.png"
 ALT="\begin{displaymath}
v_g = \frac{p}{m}.
\end{displaymath}">|; 

$key = q/P_{irightarrowf}^{abs}ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2563.png"
 ALT="$P_{i\rightarrow f}^{abs}\ll 1$">|; 

$key = q/{displaymath}frac{e_{mk}}{E_m-E_k}sim{calO}(epsilon){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img2154.png"
 ALT="\begin{displaymath}
\frac{e_{mk}}{E_m-E_k} \sim {\cal O}(\epsilon)
\end{displaymath}">|; 

$key = q/{displaymath}d^2=frac{2^{15}}{3^{10}},(e,a_0)^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img2667.png"
 ALT="\begin{displaymath}
d^2 = \frac{2^{15}}{3^{10}} (e a_0)^2
\end{displaymath}">|; 

$key = q/m,1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1963.png"
 ALT="$m, 1/2$">|; 

$key = q/{displaymath}[L^2,[L^2,z]]=2,hbar^2,(L^2,z+z,L^2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2203.png"
 ALT="\begin{displaymath}[L^2,[L^2,z]] = 2 \hbar^2 (L^2 z+z L^2).
\end{displaymath}">|; 

$key = q/(1slash2),hbar,omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1025.png"
 ALT="$(1/2) \hbar \omega$">|; 

$key = q/cosalpha_1,chi_++sinalpha_1,{{rm{e}^{,{{rm{i},beta_1},chi_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2093.png"
 ALT="$\cos\alpha_1 \chi_+
+ \sin\alpha_1 {\rm e}^{ {\rm i} \beta_1} \chi_-$">|; 

$key = q/{displaymath}k_n=frac{n,pi}{a},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img801.png"
 ALT="\begin{displaymath}
k_n = \frac{n \pi}{a},
\end{displaymath}">|; 

$key = q/l,1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1957.png"
 ALT="$l,1/2$">|; 

$key = q/t^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2505.png"
 ALT="$t^2$">|; 

$key = q/{displaymath}w_{2Prightarrow1S}=frac{omega^{,3},d^2}{3pi,epsilon_0,hbar,c^3},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2671.png"
 ALT="\begin{displaymath}
w_{2P\rightarrow 1S} = \frac{\omega^{ 3} d^2}{3\pi \epsilon_0 \hbar c^3},
\end{displaymath}">|; 

$key = q/{mitOmega}(XmidY);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.png"
 ALT="${\mit\Omega}(X \mid Y)$">|; 

$key = q/{displaymath}int_0^pifrac{1}{sqrt{r_1^{,2}+r_2^{,2}-2,r_1,r_2,costheta_1}},sinthfrac{dmu}{sqrt{r_1^{,2}+r_2^{,2}-2,r_1,r_2,mu}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="532" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img2740.png"
 ALT="\begin{displaymath}
\int_0^\pi\frac{1}{\sqrt{r_1^{ 2}+r_2^{ 2}-2 r_1 r_2 \c...
...1}^1 \frac{d\mu}{\sqrt{r_1^{ 2}+r_2^{ 2}-2 r_1 r_2 \mu}}.
\end{displaymath}">|; 

$key = q/displaystylea,int_{-infty}^inftypsi_a^ast,psi_{a'},dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img667.png"
 ALT="$\displaystyle a \int_{-\infty}^\infty\psi_a^\ast \psi_{a'} dx,$">|; 

$key = q/w_{irightarrowf}^{spn};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2602.png"
 ALT="$w_{i\rightarrow f}^{spn}$">|; 

$key = q/displaystyleL_y,[L_y,L_x]+[L_y,L_x],L_y+L_z,[L_z,L_x]+[L_z,L_x],L_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="410" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1329.png"
 ALT="$\displaystyle L_y [L_y,L_x] + [L_y,L_x] L_y + L_z [L_z,L_x] + [L_z, L_x] L_z$">|; 

$key = q/epsilonll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img926.png"
 ALT="$\epsilon\ll 1$">|; 

$key = q/displaystyle-{{rm{i},hbar,frac{partial}{partialy},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1196.png"
 ALT="$\displaystyle -{\rm i} \hbar \frac{\partial}{\partial y},$">|; 

$key = q/displaystylesintheta,cosphi,frac{partial}{partialr}+frac{costheta,cosphi}{r},fraialtheta}-frac{sinphi}{r,sintheta},frac{partial}{partialphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="362" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1352.png"
 ALT="$\displaystyle \sin\theta \cos\phi \frac{\partial}{\partial r} + \frac{\cos\th...
...partial\theta} - \frac{\sin\phi}{r \sin\theta} \frac{\partial}{\partial\phi},$">|; 

$key = q/displaystylelangle1,0,0|y|2,1,pm1rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2663.png"
 ALT="$\displaystyle \langle 1,0,0\vert y\vert 2,1,\pm 1\rangle$">|; 

$key = q/{displaymath}I({{bf{r}_2)=4pileft(frac{1}{r_2}int_0^{r_2}{{rm{e}^{-2,r_1},r_1^{,+int_{r_2}^infty{{rm{e}^{-2,r_1},r_1,dr_1right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="413" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img2745.png"
 ALT="\begin{displaymath}
I({\bf r}_2) = 4\pi\left(\frac{1}{r_2}\int_0^{r_2}
{\rm e}^{...
...,dr_1 + \int_{r_2}^\infty
{\rm e}^{-2 r_1} r_1 dr_1\right).
\end{displaymath}">|; 

$key = q/y_0={{rm{i},betaslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img381.png"
 ALT="$y_0={\rm i} \beta/2$">|; 

$key = q/|psi(x,t)|^2,dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1069.png"
 ALT="$\vert\psi(x,t)\vert^2 dx$">|; 

$key = q/exp(-Eslashk_B,T);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2609.png"
 ALT="$\exp(-E/k_B T)$">|; 

$key = q/x=x_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img332.png"
 ALT="$x=x_0$">|; 

$key = q/{displaymath}frac{sqrt{lambda-y^2}}{y}=tany,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img978.png"
 ALT="\begin{displaymath}
\frac{\sqrt{\lambda-y^2}}{y} = \tan y,
\end{displaymath}">|; 

$key = q/z,z^ast=x^2+y^2=|z|^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img204.png"
 ALT="$z z^\ast = x^2 + y^2=\vert z\vert^2$">|; 

$key = q/{displaymath}|psi(x,t)|^{,2}proptosigma^{-1}(t)expleft[-frac{(x-x_0-v_g,t)^{,2}}{2,sigma^{,2}(t)}right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="354" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img387.png"
 ALT="\begin{displaymath}
\vert\psi(x,t)\vert^{ 2} \propto \sigma^{-1}(t)\exp\left[-\frac{(x-x_0-v_g t)^{ 2}}{2 \sigma^{ 2}(t)}\right].
\end{displaymath}">|; 

$key = q/E>V_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img833.png"
 ALT="$E&gt; V_0$">|; 

$key = q/displaystylek_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1236.png"
 ALT="$\displaystyle k_y$">|; 

$key = q/dVslashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img517.png"
 ALT="$dV/dx$">|; 

$key = q/L;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img753.png"
 ALT="$L$">|; 

$key = q/A_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img800.png"
 ALT="$A_n$">|; 

$key = q/{displaymath}frac{d^2c_2}{dt^2}+{{rm{i},(omega-omega_{21}),frac{dc_2}{dt}+gamma^2,c_2=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img2418.png"
 ALT="\begin{displaymath}
\frac{d^2 c_2}{dt^2} + {\rm i} (\omega-\omega_{21}) \frac{dc_2}{dt}+\gamma^2 c_2=0.
\end{displaymath}">|; 

$key = q/{displaymath}Esimeqfrac{1}{2},(E_1+E_2)pmfrac{1}{2},(E_1-E_2)(1+2,epsilon^2+cdots).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img2138.png"
 ALT="\begin{displaymath}
E \simeq \frac{1}{2} (E_1+E_2) \pm \frac{1}{2} (E_1-E_2)(1+2 \epsilon^2+\cdots).
\end{displaymath}">|; 

$key = q/{displaymath}frac{partialpsi}{partialt}=int_{-infty}^{infty}(-{{rm{i},omega),bar{psi}(k),{{rm{e}^{,{{rm{i},(k,x-omega,t)},dk.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img432.png"
 ALT="\begin{displaymath}
\frac{\partial \psi}{\partial t} = \int_{-\infty}^{\infty}(-...
...ga) \bar{\psi}(k) {\rm e}^{ {\rm i} (k x-\omega t)} dk.
\end{displaymath}">|; 

$key = q/{scriptstyle-sqrt{2slash3}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2024.png"
 ALT="${\scriptstyle - \sqrt{2/3}}$">|; 

$key = q/{displaymath}l'=lpm1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img2214.png"
 ALT="\begin{displaymath}
l' = l\pm 1.
\end{displaymath}">|; 

$key = q/{{rm{i}^2=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img191.png"
 ALT="${\rm i}^2 = -1$">|; 

$key = q/{displaymath}{mitDelta}xsimeqfrac{lambda}{2,alpha}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img416.png"
 ALT="\begin{displaymath}
{\mit\Delta} x\simeq \frac{\lambda}{2 \alpha}.
\end{displaymath}">|; 

$key = q/displaystylepsi^{(2)}_{3slash2,-1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1991.png"
 ALT="$\displaystyle \psi^{(2)}_{3/2,-1/2}$">|; 

$key = q/+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img152.png"
 ALT="$+1$">|; 

$key = q/displaystyle[sigma_x,sigma_y];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1802.png"
 ALT="$\displaystyle [\sigma_x, \sigma_y]$">|; 

$key = q/{{bf{A};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2532.png"
 ALT="${\bf A}$">|; 

$key = q/displaystyleint_{-infty}^{infty}psi^ast,x,psi,dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img534.png"
 ALT="$\displaystyle \int_{-\infty}^{\infty}\psi^\ast x \psi dx,$">|; 

$key = q/theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="$\theta $">|; 

$key = q/(k,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img419.png"
 ALT="$(k,0)$">|; 

$key = q/displaystyle+3,e,a_0,|{{bf{E}|,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2284.png"
 ALT="$\displaystyle +3 e a_0 \vert{\bf E}\vert,$">|; 

$key = q/displaystyle-frac{1}{2,m_e,c^2},langlen,l,m|(E_n-V)^2|n,l,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2298.png"
 ALT="$\displaystyle -\frac{1}{2 m_e c^2} \langle n,l,m\vert(E_n -V)^2\vert n,l,m\rangle$">|; 

$key = q/{displaymath}R_ast=frac{2,A}{B},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1287.png"
 ALT="\begin{displaymath}
R_\ast = \frac{2 A}{B},
\end{displaymath}">|; 

$key = q/delta(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img574.png"
 ALT="$\delta(x)$">|; 

$key = q/{displaymath}psi(x)=B,cos(q,x),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img970.png"
 ALT="\begin{displaymath}
\psi(x) = B \cos(q x),
\end{displaymath}">|; 

$key = q/partialx_islashpartialx_j=delta_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1081.png"
 ALT="$\partial x_i/\partial x_j=\delta_{ij}$">|; 

$key = q/{displaymath}sum_{l''=1,N_n}langlen,l',m|H_1|n,l'',mrangle,langlen,l'',m|n,l^{(1ngle=lambda_{nl},langlen,l',m|n,l^{(1)},mrangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="563" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2252.png"
 ALT="\begin{displaymath}
\sum_{l''=1,N_n}\langle n,l',m\vert H_1\vert n,l'',m\rangle\...
...rangle
= \lambda_{nl} \langle n,l',m\vert n,l^{(1)},m\rangle.
\end{displaymath}">|; 

$key = q/displaystylesqrt{2},frac{2^7}{3^5},a_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2666.png"
 ALT="$\displaystyle \sqrt{2}  \frac{2^7}{3^5} a_0,$">|; 

$key = q/{displaymath}W_{irightarrowf}=N_i,(w_{irightarrowf}^{spn}+w_{irightarrowf}^{stm}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img2601.png"
 ALT="\begin{displaymath}
W_{i\rightarrow f} = N_i (w_{i\rightarrow f}^{spn} + w_{i\rightarrow f}^{stm}),
\end{displaymath}">|; 

$key = q/displaystylepmfrac{2^7}{3^5},a_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2662.png"
 ALT="$\displaystyle \pm \frac{2^7}{3^5} a_0,$">|; 

$key = q/m_s>s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1773.png"
 ALT="$m_s&gt;s$">|; 

$key = q/-E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1695.png"
 ALT="$-E$">|; 

$key = q/displaystyle-frac{{{rm{e}^{-beta,x}}{beta^3},(2+2,beta,x+beta^2,x^2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2749.png"
 ALT="$\displaystyle - \frac{{\rm e}^{-\beta x}}{\beta^3} (2+2 \beta x+\beta^2 x^2),$">|; 

$key = q/{displaymath}V(r)=left{array{{lcl}-V_0&mbox{hspace{1cm}}&mbox{forrleqa}0&&mbox{forr>a}array{right.,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1681.png"
 ALT="\begin{displaymath}
V(r) = \left\{
\begin{array}{lcl}
-V_0&amp;\mbox{\hspace{1cm}}&amp;\mbox{for $r\leq a$}\\\\
0&amp;&amp;\mbox{for $r&gt;a$}
\end{array}\right. ,
\end{displaymath}">|; 

$key = q/n>n';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2628.png"
 ALT="$n&gt;n'$">|; 

$key = q/{displaymath}A_{ij}=int_{-infty}^inftyint_{-infty}^inftyint_{-infty}^inftypsi_i^ast,A,psi_j,dx,dy,dz.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img2113.png"
 ALT="\begin{displaymath}
A_{ij} = \int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty\psi_i^\ast A \psi_j dx dy dz.
\end{displaymath}">|; 

$key = q/displaystylev_g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img369.png"
 ALT="$\displaystyle v_g$">|; 

$key = q/E_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img782.png"
 ALT="$E_1$">|; 

$key = q/{displaymath}P_y(y)equivlim_{Nrightarrowinfty}left[frac{n(y)}{N}right]proptoI(y),{mitDelta}y,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img282.png"
 ALT="\begin{displaymath}
P_y(y) \equiv \lim_{N\rightarrow\infty}\left[\frac{n(y)}{N}\right] \propto I(y) {\mit\Delta}y,
\end{displaymath}">|; 

$key = q/chi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1719.png"
 ALT="$\chi$">|; 

$key = q/h(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1009.png"
 ALT="$h(y)$">|; 

$key = q/j_l'(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2948.png"
 ALT="$j_l'(x)$">|; 

$key = q/k-k_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img359.png"
 ALT="$k-k_0$">|; 

$key = q/|x|leqpi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="$\vert x\vert\leq \pi $">|; 

$key = q/g=(B-langleBrangle),psi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img610.png"
 ALT="$g=(B-\langle B\rangle) \psi$">|; 

$key = q/bar{f}(k);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img325.png"
 ALT="$\bar{f}(k)$">|; 

$key = q/J_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1878.png"
 ALT="$J_x$">|; 

$key = q/displaystylep_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1195.png"
 ALT="$\displaystyle p_y$">|; 

$key = q/displaystyle2,{{rm{i},hbar,(L_y,x-y,L_x)=2,{{rm{i},hbar,(x,L_y-L_x,y),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="317" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2193.png"
 ALT="$\displaystyle 2 {\rm i} \hbar (L_y x - y L_x) = 2 {\rm i} \hbar (x L_y-L_x y),$">|; 

$key = q/displaystyle[L_y,L_z];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1321.png"
 ALT="$\displaystyle [L_y, L_z]$">|; 

$key = q/displaystylem_j,hbar,psi_{j,m_j},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1884.png"
 ALT="$\displaystyle m_j \hbar \psi_{j,m_j},$">|; 

$key = q/{displaymath}tau=frac{ln2}{nu,|T|^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img938.png"
 ALT="\begin{displaymath}
\tau = \frac{\ln 2}{\nu \vert T\vert^2}.
\end{displaymath}">|; 

$key = q/k,agg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2982.png"
 ALT="$k a\gg 1$">|; 

$key = q/{displaymath}{{bf{B}=B_0,cos(omega,t),{{bf{e}_z,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1863.png"
 ALT="\begin{displaymath}
{\bf B} = B_0 \cos(\omega t) {\bf e}_z,
\end{displaymath}">|; 

$key = q/displaystylechi^{(1)}_{-1slash2,-1.2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2078.png"
 ALT="$\displaystyle \chi^{(1)}_{-1/2,-1.2},$">|; 

$key = q/displaystylefrac{partial}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1351.png"
 ALT="$\displaystyle \frac{\partial}{\partial x}$">|; 

$key = q/bar{psi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img229.png"
 ALT="$\bar{\psi}$">|; 

$key = q/{displaymath}O(f(x))rightarrowg(x).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img525.png"
 ALT="\begin{displaymath}
O(f(x))\rightarrow g(x).
\end{displaymath}">|; 

$key = q/displaystyledelta_{ll'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1393.png"
 ALT="$\displaystyle \delta_{ll'},$">|; 

$key = q/1slash6times1slash6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img88.png"
 ALT="$1/6 \times 1/6$">|; 

$key = q/n';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2624.png"
 ALT="$n'$">|; 

$key = q/|c_-|^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1819.png"
 ALT="$\vert c_-\vert^2$">|; 

$key = q/displaystylefrac{sinz}{z},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1552.png"
 ALT="$\displaystyle \frac{\sin z}{z},$">|; 

$key = q/{displaymath}{mitDelta}x,{mitDelta}k=frac{1}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img353.png"
 ALT="\begin{displaymath}
{\mit\Delta}x {\mit\Delta} k = \frac{1}{2}.
\end{displaymath}">|; 

$key = q/{displaymath}langlei|jrangle=delta_{ij},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img2127.png"
 ALT="\begin{displaymath}
\langle i\vert j\rangle = \delta_{ij},
\end{displaymath}">|; 

$key = q/{displaymath}langleprangle=-frac{{{rm{i},hbar}{2}int_{-infty}^{infty}left(psi^asy}^{infty}psi^ast,frac{partialpsi}{partialx},dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="477" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img507.png"
 ALT="\begin{displaymath}
\langle p\rangle = - \frac{{\rm i} \hbar}{2}\int_{-\infty}^...
...nfty}^{\infty}
\psi^\ast \frac{\partial\psi}{\partial x} dx,
\end{displaymath}">|; 

$key = q/psi(-x)=-psi(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img960.png"
 ALT="$\psi(-x) =-\psi(x)$">|; 

$key = q/{displaymath}psi({{bf{r})simeqsqrt{n}sum_lC_lleft[frac{{{rm{e}^{,{{rm{i},(k,r-l,h2+delta_l)}}{2,{{rm{i},k,r}right]P_l(costheta),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="478" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img2931.png"
 ALT="\begin{displaymath}
\psi({\bf r}) \simeq \sqrt{n} \sum_l C_l\left[
\frac{{\rm e}...
...\pi/2+ \delta_l)} }{2 {\rm i} k r} \right] P_l(\cos\theta),
\end{displaymath}">|; 

$key = q/cos(theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img148.png"
 ALT="$\cos(\theta)$">|; 

$key = q/|phi(p,t)|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img593.png"
 ALT="$\vert\phi(p,t)\vert^{ 2}$">|; 

$key = q/displaystyle{calR}_0(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2964.png"
 ALT="$\displaystyle {\cal R}_0(r)$">|; 

$key = q/{displaymath}sum_i|c_i|^2=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img684.png"
 ALT="\begin{displaymath}
\sum_i \vert c_i\vert^2 =1.
\end{displaymath}">|; 

$key = q/P(s),ds;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img135.png"
 ALT="$P(s) ds$">|; 

$key = q/displaystyle-{{rm{i},hbarleft(-sinphi,frac{partial}{partialtheta}-cosphi,cottheta,frac{partial}{partialphi}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1357.png"
 ALT="$\displaystyle - {\rm i} \hbar\left(-\sin\phi \frac{\partial}{\partial\theta}
-\cos\phi \cot\theta \frac{\partial}{\partial\phi}\right),$">|; 

$key = q/n=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="$n=1$">|; 

$key = q/p';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img721.png"
 ALT="$p'$">|; 

$key = q/{displaymath}(E_m-E+e_{mm}),langlem|Erangle+sum_{kneqm}e_{mk},langlek|Erangle=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2152.png"
 ALT="\begin{displaymath}
(E_m-E+e_{mm}) \langle m\vert E\rangle + \sum_{k\neq m}
e_{mk} \langle k\vert E\rangle = 0,
\end{displaymath}">|; 

$key = q/{displaymath}lambda_p=2.9times10^{-11},[E({{rm{eV})]^{-1slash2},{{rm{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img302.png"
 ALT="\begin{displaymath}
\lambda_p = 2.9\times 10^{-11} [E({\rm eV})]^{-1/2} {\rm m}.
\end{displaymath}">|; 

$key = q/{displaymath}langleArangle=chi^nomath_inline{dagnomath_inline{,A,chi.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1850.png"
 ALT="\begin{displaymath}
\langle A \rangle = \chi^\dag A \chi.
\end{displaymath}">|; 

$key = q/{displaymath}H=-mbox{{boldmath{{mu{}cdot{{bf{B}=H_0+H_1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2446.png"
 ALT="\begin{displaymath}
H = -\mbox{\boldmath$\mu$}\cdot{\bf B} = H_0 + H_1,
\end{displaymath}">|; 

$key = q/2Srightarrow1S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2688.png"
 ALT="$2S\rightarrow 1S$">|; 

$key = q/displaystyleint_{-infty}^{infty}left[left({{rm{i},hbar,frac{partialpsi}{partialtx}left({{rm{i},hbar,frac{partialpsi}{partialt}right)right]dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img511.png"
 ALT="$\displaystyle \int_{-\infty}^{\infty}\left[ \left({\rm i} \hbar \frac{\partia...
...artial x}\left({\rm i} \hbar \frac{\partial\psi}{\partial t}\right)\right]dx,$">|; 

$key = q/{displaymath}delta(p-p_0)=frac{1}{2pi,hbar}int_{-infty}^{infty}{{rm{e}^{+{{rm{i},(p-p_0),xslashhbar},dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img585.png"
 ALT="\begin{displaymath}
\delta(p-p_0)= \frac{1}{2\pi \hbar}\int_{-\infty}^{\infty}{\rm e}^{+{\rm i} (p-p_0) x/\hbar} dx.
\end{displaymath}">|; 

$key = q/E_{n10}=E_{n00};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2222.png"
 ALT="$E_{n10}=E_{n00}$">|; 

$key = q/1slash(E_n-E_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2240.png"
 ALT="$1/(E_n-E_n)$">|; 

$key = q/f({{bf{k}',{{bf{k});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2879.png"
 ALT="$f({\bf k}', {\bf k})$">|; 

$key = q/{displaymath}langlepsi|H|psiranglegeqE_0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2702.png"
 ALT="\begin{displaymath}
\langle \psi\vert H\vert\psi\rangle \geq E_0.
\end{displaymath}">|; 

$key = q/f(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img326.png"
 ALT="$f(x)$">|; 

$key = q/{displaymath}langlen,l,m|[L_z,x_-]+hbar,x_-|n',l',m'rangle=hbar,(m-m'+1),langlen,l,m|x_-|n',l',m'rangle=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="581" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2643.png"
 ALT="\begin{displaymath}
\langle n,l,m\vert[L_z,x_-]+\hbar x_-\vert n',l',m'\rangle...
...,(m-m'+1) 
\langle n,l,m\vert x_-\vert n',l',m'\rangle = 0.
\end{displaymath}">|; 

$key = q/{displaymath}frac{2,m}{hbar^2}frac{|V_0|}{mu^2}geq2.7,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2901.png"
 ALT="\begin{displaymath}
\frac{2 m}{\hbar^2} \frac{\vert V_0\vert} {\mu^2} \geq 2.7,
\end{displaymath}">|; 

$key = q/{displaymath}omega=k,c,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img235.png"
 ALT="\begin{displaymath}
\omega = k c,
\end{displaymath}">|; 

$key = q/(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img418.png"
 ALT="$(x,y)$">|; 

$key = q/{displaymath}int_{-infty}^inftypsi_x^ast(x,x'),psi_x(x,x''),dx=delta(x'-x'').{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img713.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \psi_x^\ast(x,x') \psi_x(x,x'') dx = \delta(x'-x'').
\end{displaymath}">|; 

$key = q/{displaymath}P(y)propto|psi(y,t)|^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img292.png"
 ALT="\begin{displaymath}
P(y) \propto \vert\psi(y,t)\vert^{ 2}.
\end{displaymath}">|; 

$key = q/{displaymath}L^2=L_x^{,2}+L_y^{,2}+L_z^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img1325.png"
 ALT="\begin{displaymath}
L^2 = L_x^{ 2} + L_y^{ 2} + L_z^{ 2}.
\end{displaymath}">|; 

$key = q/{displaymath}D=2int_0^inftyint_0^pifrac{{{rm{e}^{-2,x}}{(x^2+X^2-2,x,X,costheta)^{1slash2}},x^2,dx,sintheta,dtheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="443" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2811.png"
 ALT="\begin{displaymath}
D = 2\int_0^\infty\int_0^\pi \frac{{\rm e}^{-2 x}}{(x^2+X^2-2 x X \cos\theta)^{1/2}} x^2 dx \sin\theta d\theta,
\end{displaymath}">|; 

$key = q/displaystyle2,{{rm{i},sigma_y.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1807.png"
 ALT="$\displaystyle 2 {\rm i} \sigma_y.$">|; 

$key = q/S_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1798.png"
 ALT="$S_i$">|; 

$key = q/r<a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2952.png"
 ALT="$r&lt;a$">|; 

$key = q/{scriptstyle1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2026.png"
 ALT="${\scriptstyle 1}$">|; 

$key = q/n,l=1,0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="$n,l=1,0$">|; 

$key = q/m,m_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1965.png"
 ALT="$m, m_s$">|; 

$key = q/displaystylebeta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img375.png"
 ALT="$\displaystyle \beta_1$">|; 

$key = q/displaystylex_i,x_i,p_j,p_j-x_i,p_i,p_j,x_j-2,{{rm{i},hbar,x_i,p_i.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1520.png"
 ALT="$\displaystyle x_i x_i p_j p_j -x_i p_i p_j x_j - 2 {\rm i} \hbar x_i p_i.$">|; 

$key = q/{displaymath}A,(B,psi_i)=a_i,(B,psi_i).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img700.png"
 ALT="\begin{displaymath}
A (B \psi_i) = a_i (B \psi_i).
\end{displaymath}">|; 

$key = q/z^{k-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1597.png"
 ALT="$z^{k-2}$">|; 

$key = q/{displaymath}{{rm{Average~Age}simeqfrac{N_{18}times18+N_{19}times19+N_{20}times20+cdots}{N_{18}+N_{19}+N_{20}cdots},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="442" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img90.png"
 ALT="\begin{displaymath}
{\rm Average&nbsp;Age} \simeq \frac{N_{18}\times 18 + N_{19}\times 19 +N_{20}
\times 20+\cdots}
{N_{18}+N_{19}+N_{20}\cdots},
\end{displaymath}">|; 

$key = q/{displaymath}W_{frightarrowi}=N_f,w^{abs}_{frightarrowi},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img2604.png"
 ALT="\begin{displaymath}
W_{f\rightarrow i} = N_f w^{abs}_{f\rightarrow i},
\end{displaymath}">|; 

$key = q/displaystylej,(j+1),hbar^{,2},psi_{j,m_j},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1886.png"
 ALT="$\displaystyle j (j+1) \hbar^{ 2} \psi_{j,m_j},$">|; 

$key = q/chi_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1721.png"
 ALT="$\chi_1$">|; 

$key = q/E=hbar^2,k^2slash2,m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2841.png"
 ALT="$E = \hbar^2 k^2/2 m$">|; 

$key = q/N(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img939.png"
 ALT="$N(0)$">|; 

$key = q/c^ast,chi^nomath_inline{dagnomath_inline{;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1727.png"
 ALT="$c^\ast \chi^\dag $">|; 

$key = q/x>a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img822.png"
 ALT="$x&gt;a$">|; 

$key = q/1slashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1663.png"
 ALT="$1/r$">|; 

$key = q/displaystylec_{l,m}^-,Y_{l,m-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1381.png"
 ALT="$\displaystyle c_{l,m}^- Y_{l,m-1},$">|; 

$key = q/{displaymath}|psi_2|^{,2}=|psi_1|^{,2},expleft(-2,int_{x_1}^{x_2}|k(x')|,dx'right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img893.png"
 ALT="\begin{displaymath}
\vert\psi_2\vert^{ 2} = \vert\psi_1\vert^{ 2} \exp\left(-2 \int_{x_1}^{x_2} \vert k(x')\vert dx'\right).
\end{displaymath}">|; 

$key = q/R^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1293.png"
 ALT="$R^{-1}$">|; 

$key = q/{displaymath}langlexrangle=frac{x_0}{sqrt{pi}}int_{-infty}^{infty}{{rm{e}^{-y^2}rt{pi}},int_{-infty}^{infty}y,{{rm{e}^{-y^2},dy.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img496.png"
 ALT="\begin{displaymath}
\langle x\rangle = \frac{x_0}{\sqrt{\pi}}\int_{-\infty}^{\in...
...ma}{\sqrt{\pi}} \int_{-\infty}^{\infty}y {\rm e}^{-y^2} dy.
\end{displaymath}">|; 

$key = q/g_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2363.png"
 ALT="$g_p$">|; 

$key = q/{displaymath}langlepsi_n|psi_mrangle=delta_{nm}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2692.png"
 ALT="\begin{displaymath}
\langle \psi_n\vert\psi_m\rangle = \delta_{nm}.
\end{displaymath}">|; 

$key = q/displaystyle[S_+,S_z];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1715.png"
 ALT="$\displaystyle [S_+,S_z]$">|; 

$key = q/{displaymath}psi({{bf{r})=sqrt{n},sum_{l=0}^infty{{rm{i}^l,(2,l+1),{calR}_l(r),P_l(costheta),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img2943.png"
 ALT="\begin{displaymath}
\psi({\bf r}) = \sqrt{n}  \sum_{l=0}^\infty
{\rm i}^l  (2 l+1)   {\cal R}_l(r)  P_l(\cos\theta),
\end{displaymath}">|; 

$key = q/{displaymath}langleprangle=-mint_{-infty}^{infty}x,frac{partialj}{partialx},dx=mint_{-infty}^{infty}j,dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img506.png"
 ALT="\begin{displaymath}
\langle p\rangle = -m\int_{-\infty}^{\infty} x \frac{\partial j}{\partial x} dx
= m\int_{-\infty}^{\infty}j dx,
\end{displaymath}">|; 

$key = q/displaystyle{{bf{d}_{if};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2588.png"
 ALT="$\displaystyle {\bf d}_{if}$">|; 

$key = q/{displaymath}epsilon=frac{|e_{12}|}{|E_1-E_2|}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2137.png"
 ALT="\begin{displaymath}
\epsilon = \frac{\vert e_{12}\vert}{\vert E_1-E_2\vert}.
\end{displaymath}">|; 

$key = q/(3slash2),k_B,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1256.png"
 ALT="$(3/2) k_B T$">|; 

$key = q/1slash2,-1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2020.png"
 ALT="$1/2, -1/2$">|; 

$key = q/chi=S_-,chi_{s,m_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1755.png"
 ALT="$\chi=S_- \chi_{s,m_s}$">|; 

$key = q/displaystylefrac{psi_{200}+psi_{210}}{sqrt{2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2280.png"
 ALT="$\displaystyle \frac{\psi_{200} + \psi_{210}}{\sqrt{2}},$">|; 

$key = q/displaystyler,sintheta,sinphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1348.png"
 ALT="$\displaystyle r \sin\theta \sin\phi,$">|; 

$key = q/{displaymath}chi_+^nomath_inline{dagnomath_inline{,chi_-=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1787.png"
 ALT="\begin{displaymath}
\chi_+^\dag \chi_- = 0.
\end{displaymath}">|; 

$key = q/{displaymath}I=intleft[|psi_0({{bf{r}_1)|^2+|psi_0({{bf{r}_2)|^2pm2,psi_0({{bf{r}_1),psi({{bf{r}_2)right]d^3{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="395" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img2784.png"
 ALT="\begin{displaymath}
I = \int\left[\vert\psi_0({\bf r}_1)\vert^2 + \vert\psi_0({\...
...2 \pm
2 \psi_0({\bf r}_1) \psi({\bf r}_2)\right] d^3{\bf r}.
\end{displaymath}">|; 

$key = q/A^nomath_inline{dagnomath_inline{;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1735.png"
 ALT="$A^\dag $">|; 

$key = q/{displaymath}H_i,psi_{E_i}=E_i,psi_{E_i},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1104.png"
 ALT="\begin{displaymath}
H_i \psi_{E_i} =E_i \psi_{E_i},
\end{displaymath}">|; 

$key = q/{displaymath}left(-frac{d^2}{dy^2}+y^2right)psi_n=(2n+1),psi_n,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1031.png"
 ALT="\begin{displaymath}
\left(-\frac{d^2}{d y^2}+y^2\right)\psi_n = (2n+1) \psi_n,
\end{displaymath}">|; 

$key = q/chi_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1784.png"
 ALT="$\chi_-$">|; 

$key = q/displaystyle[S_z,S_x];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1704.png"
 ALT="$\displaystyle [S_z,S_x]$">|; 

$key = q/L_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1334.png"
 ALT="$L_-$">|; 

$key = q/dj_l(x)slashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2949.png"
 ALT="$dj_l(x)/dx$">|; 

$key = q/-l,-l+1,cdots,l-1,l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1665.png"
 ALT="$-l,-l+1, \cdots, l-1, l$">|; 

$key = q/displaystyleL_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1306.png"
 ALT="$\displaystyle L_y$">|; 

$key = q/displaystylechi^{(2)}_{1,-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2077.png"
 ALT="$\displaystyle \chi^{(2)}_{1,-1}$">|; 

$key = q/|{{bf{v}'|=|{{bf{v}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2870.png"
 ALT="$\vert{\bf v}'\vert=\vert{\bf v}\vert$">|; 

$key = q/{displaymath}g=2left(1+frac{alpha}{2pi}+cdotsright)=2.0023192,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="281" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1828.png"
 ALT="\begin{displaymath}
g = 2\left(1+\frac{\alpha}{2\pi}+\cdots\right) = 2.0023192,
\end{displaymath}">|; 

$key = q/|bar{psi}(k)|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img347.png"
 ALT="$\vert\bar{\psi}(k)\vert^{ 2}$">|; 

$key = q/displaystyle[L_+,L_z];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1341.png"
 ALT="$\displaystyle [L_+,L_z]$">|; 

$key = q/L_pm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1891.png"
 ALT="$L_\pm$">|; 

$key = q/{displaymath}[L_i,S_j]=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1869.png"
 ALT="\begin{displaymath}[L_i, S_j]= 0,
\end{displaymath}">|; 

$key = q/{mitDelta}v=hbarslash(m,{mitDelta}x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img405.png"
 ALT="${\mit\Delta}v = \hbar/(m {\mit\Delta}x)$">|; 

$key = q/displaystyle{{rm{Re}left[A,{{rm{e}^{,{{rm{i},varphi},{{rm{e}^{,{{rm{i},(k,x-omega,t)}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img221.png"
 ALT="$\displaystyle {\rm Re}\left[A {\rm e}^{ {\rm i} \varphi} {\rm e}^{ {\rm i} (k x-\omega t)}\right]$">|; 

$key = q/{displaymath}frac{dlangleprangle}{dt}=-frac{dV(langlexrangle)}{dlanglexrangle}-f{2},frac{dV^3(langlexrangle)}{dlanglexrangle^3},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img519.png"
 ALT="\begin{displaymath}
\frac{d\langle p\rangle}{dt} = - \frac{dV(\langle x\rangle)}...
...{ 2}}{2} \frac{dV^3(\langle x\rangle)}{d\langle x\rangle^3},
\end{displaymath}">|; 

$key = q/y={{rm{Im}(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img199.png"
 ALT="$y={\rm Im}(z)$">|; 

$key = q/dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img287.png"
 ALT="$dy$">|; 

$key = q/{displaymath}langleOrangle=int_{-infty}^{infty}phi^ast(p,t),O(p),phi(p,t),dp.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="269" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img588.png"
 ALT="\begin{displaymath}
\langle O\rangle = \int_{-\infty}^{\infty}\phi^\ast(p,t) O(p) \phi(p,t) dp.
\end{displaymath}">|; 

$key = q/{displaymath}c_{i+2}=frac{(2,i-epsilon+1)}{(i+1),(i+2)},c_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1013.png"
 ALT="\begin{displaymath}
c_{i+2} = \frac{(2 i-\epsilon+1)}{(i+1) (i+2)} c_i.
\end{displaymath}">|; 

$key = q/{displaymath}c_n=int_0^apsi_n(x),psi(x,0),dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img811.png"
 ALT="\begin{displaymath}
c_n = \int_0^a\psi_n(x) \psi(x,0) dx.
\end{displaymath}">|; 

$key = q/10^{-14};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img454.png"
 ALT="$10^{-14}$">|; 

$key = q/x=x_1-x_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1133.png"
 ALT="$x=x_1-x_2$">|; 

$key = q/{displaymath}c_k,left[k,(k-1)-l,(l+1)right]=c_{k-1},left[2,(k-1)-zetaright].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="374" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1598.png"
 ALT="\begin{displaymath}
c_k \left[k (k-1)-l (l+1)\right] = c_{k-1} \left[2 (k-1) - \zeta\right].
\end{displaymath}">|; 

$key = q/psi^{(2)}_{l+1slash2,m+1slash2}equivpsi^{(2)}_{l,1slash2;l+1slash2,m+1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1958.png"
 ALT="$\psi^{(2)}_{l+1/2,m+1/2}\equiv \psi^{(2)}_{l,1/2;l+1/2,m+1/2}$">|; 

$key = q/{displaymath}{{bf{j}=n,{{bf{v},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2863.png"
 ALT="\begin{displaymath}
{\bf j} = n {\bf v},
\end{displaymath}">|; 

$key = q/{displaymath}langleurangleequivsum_{i=1}^{M}P(u_i),u_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img101.png"
 ALT="\begin{displaymath}
\langle u\rangle \equiv \sum_{i=1}^{M} P(u_i)  u_i.
\end{displaymath}">|; 

$key = q/displaystyle-frac{1}{x,X}left[{{rm{e}^{-(x+X)},(1+x+X)-{{rm{e}^{-|x-X|},(1+|x-X|)right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="434" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2795.png"
 ALT="$\displaystyle - \frac{1}{x X}\left[{\rm e}^{-(x+X)} (1+x+X) - {\rm e}^{-\vert x-X\vert} (1+\vert x-X\vert)\right].$">|; 

$key = q/V(r)=(1slash2),m,omega^2,r^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1684.png"
 ALT="$V(r)=(1/2) m \omega^2 r^2$">|; 

$key = q/{displaymath}langle{mitDelta}urangle=leftlangle(u-langleurangle)rightrangle=langleurangle-langleurangle=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img111.png"
 ALT="\begin{displaymath}
\langle {\mit\Delta} u\rangle = \left\langle(u-\langle u\rangle)\right\rangle = \langle u\rangle-\langle u\rangle = 0.
\end{displaymath}">|; 

$key = q/{displaymath}J^2=L^2+S^2+2,L_z,S_z+L_+,S_-+L_-,S_+.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="336" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1890.png"
 ALT="\begin{displaymath}
J^2 = L^2+S^2 +2 L_z S_z+ L_+ S_-+L_- S_+.
\end{displaymath}">|; 

$key = q/psi(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img639.png"
 ALT="$\psi(t)$">|; 

$key = q/{displaymath}V_{ee}=frac{e^2}{4pi,epsilon_0},frac{1}{|{{bf{r}_2-{{bf{r}_1|}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2759.png"
 ALT="\begin{displaymath}
V_{ee} = \frac{e^2}{4\pi \epsilon_0} \frac{1}{\vert{\bf r}_2-{\bf r}_1\vert}
\end{displaymath}">|; 

$key = q/R_{n,l}(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1542.png"
 ALT="$R_{n,l}(r)$">|; 

$key = q/{displaymath}Y_{l,l-2}simL_-,Y_{l,l-1}sim{{rm{e}^{-{{rm{i},phi}left(-frac{partia{sintheta}frac{d}{dtheta}right)(sintheta)^{2,l},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="603" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1453.png"
 ALT="\begin{displaymath}
Y_{l,l-2}\sim L_- Y_{l,l-1}\sim {\rm e}^{-{\rm i} \phi}\le...
...ac{1}{\sin\theta}\frac{d}{d\theta}\right)
(\sin\theta)^{2 l},
\end{displaymath}">|; 

$key = q/-1,-1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2009.png"
 ALT="$-1, -1/2$">|; 

$key = q/3slash2,1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2022.png"
 ALT="$3/2, 1/2$">|; 

$key = q/{displaymath}{mitOmega}(XmidY)={mitOmega}(X)+{mitOmega}(Y){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img72.png"
 ALT="\begin{displaymath}
{\mit\Omega}(X\mid Y) = {\mit\Omega}(X) + {\mit\Omega}(Y)
\end{displaymath}">|; 

$key = q/{displaymath}x,frac{d}{dx}neqfrac{d}{dx},x,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img530.png"
 ALT="\begin{displaymath}
x \frac{d}{dx} \neq \frac{d}{dx} x,
\end{displaymath}">|; 

$key = q/{displaymath}frac{dlangleprangle}{dt}=int_{-infty}^{infty}left[-frac{hbar^2}{2,minfty}V(x),frac{partial|psi|^{,2}}{partialx},dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="596" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img512.png"
 ALT="\begin{displaymath}
\frac{d\langle p\rangle}{dt} = \int_{-\infty}^{\infty}
\left...
...y} V(x) \frac{\partial \vert\psi\vert^{ 2}}{\partial x} dx.
\end{displaymath}">|; 

$key = q/mbox{{boldmath{{mu{}_e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2370.png"
 ALT="$\mbox{\boldmath$\mu$}_e$">|; 

$key = q/{{bf{L}=m_e,{{bf{r}times{{bf{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2314.png"
 ALT="${\bf L} = m_e {\bf r}\times {\bf v}$">|; 

$key = q/0,-1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2011.png"
 ALT="$0,-1/2$">|; 

$key = q/{{bf{k}=(k_x,,k_y,,k_z)=k,{{bf{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img182.png"
 ALT="${\bf k} = (k_x, k_y, k_z)=k {\bf n}$">|; 

$key = q/A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img133.png"
 ALT="$A$">|; 

$key = q/x_2=Wslashe,{calE};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img904.png"
 ALT="$x_2 = W / e {\cal E}$">|; 

$key = q/psi^{(1)}_{l,s;m,m_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1898.png"
 ALT="$\psi^{(1)}_{l,s;m,m_s}$">|; 

$key = q/{displaymath}int_{-infty}^inftypsi_n,x,psi_{n'},dx=sqrt{frac{hbar}{2,m_e,omega_0n},delta_{n,n'+1}+sqrt{n'},delta_{n,n'-1}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="445" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2627.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \psi_n x \psi_{n'} dx =\sqrt{\frac...
...sqrt{n} \delta_{n,n'+1} + \sqrt{n'} \delta_{n,n'-1}\right).
\end{displaymath}">|; 

$key = q/displaystyle+[l,(l+1)-m,(m+1)]^{1slash2},hbar^2,Y_{l,m},chi_+.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1945.png"
 ALT="$\displaystyle + [l (l+1)-m (m+1)]^{1/2} \hbar^2 Y_{l,m} \chi_+.$">|; 

$key = q/{displaymath}L_pm=hbar,{{rm{e}^{pm{{rm{i},phi}left(pmfrac{partial}{partialtheta}{rm{i},cottheta,frac{partial}{partialphi}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1361.png"
 ALT="\begin{displaymath}
L_\pm = \hbar {\rm e}^{\pm{\rm i} \phi}\left(\pm\frac{\par...
...a} +{\rm i} \cot\theta \frac{\partial}{\partial\phi}\right).
\end{displaymath}">|; 

$key = q/psi=phi_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img784.png"
 ALT="$\psi=\phi_1$">|; 

$key = q/0,1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2012.png"
 ALT="$0,1/2$">|; 

$key = q/n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img988.png"
 ALT="$n$">|; 

$key = q/displaystylepmfrac{1}{2},hbar,chi_pm,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1779.png"
 ALT="$\displaystyle \pm \frac{1}{2} \hbar \chi_\pm,$">|; 

$key = q/displaystyleS^2,chi_{s,m_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1744.png"
 ALT="$\displaystyle S^2 \chi_{s,m_s}$">|; 

$key = q/displaystylem_j,hbar,psi^{(1)}_{l,s;m,m_s}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1909.png"
 ALT="$\displaystyle m_j \hbar \psi^{(1)}_{l,s;m,m_s}.$">|; 

$key = q/epsilon_{123}=epsilon_{231}=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1505.png"
 ALT="$\epsilon_{123}=\epsilon_{231}=1$">|; 

$key = q/psi(theta,phi)=L_+,Y_{l,m}(theta,phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1404.png"
 ALT="$\psi(\theta,\phi) = L_+ Y_{l,m}(\theta,\phi)$">|; 

$key = q/r=sqrt{x^2+y^2+z^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1491.png"
 ALT="$r=\sqrt{x^2+y^2+z^2}$">|; 

$key = q/{displaymath}c_{i+2}simeqfrac{2}{i},c_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1014.png"
 ALT="\begin{displaymath}
c_{i+2} \simeq \frac{2}{i} c_i.
\end{displaymath}">|; 

$key = q/{displaymath}mu=frac{m_1,m_2}{m_1+m_2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img1119.png"
 ALT="\begin{displaymath}
\mu = \frac{m_1 m_2}{m_1+m_2}
\end{displaymath}">|; 

$key = q/displaystyler,sintheta,cosphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1346.png"
 ALT="$\displaystyle r \sin\theta \cos\phi,$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter08slashfig01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="367" HEIGHT="326" BORDER="0"
 SRC="|."$dir".q|img1461.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter08/fig01.eps}}
\end{figure}">|; 

$key = q/{displaymath}psi(x,y,z,t)=A,cos({{bf{k}cdot{{bf{r}-omega,t+varphi),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img181.png"
 ALT="\begin{displaymath}
\psi(x,y,z,t)=A \cos({\bf k}\cdot{\bf r}-\omega t+\varphi),
\end{displaymath}">|; 

$key = q/N_{{rm{students};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img93.png"
 ALT="$N_{\rm students}$">|; 

$key = q/displaystylek,(1-R);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img840.png"
 ALT="$\displaystyle k (1-R)$">|; 

$key = q/x,p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img551.png"
 ALT="$x p$">|; 

$key = q/x=r,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img207.png"
 ALT="$x=r \cos\theta$">|; 

$key = q/P(XotimesY);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img82.png"
 ALT="$P(X\otimes Y)$">|; 

$key = q/omega_{21}=(E_2-E_1)slashhbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2417.png"
 ALT="$\omega_{21}=(E_2-E_1)/\hbar$">|; 

$key = q/{displaymath}[L_-,L_z]=hbar,L_-.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1343.png"
 ALT="\begin{displaymath}[L_-,L_z]= \hbar L_-.
\end{displaymath}">|; 

$key = q/hbarslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1858.png"
 ALT="$\hbar/2$">|; 

$key = q/{displaymath}p=hbar,k,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img305.png"
 ALT="\begin{displaymath}
p = \hbar k,
\end{displaymath}">|; 

$key = q/{displaymath}q=frac{(2,j-1),pi}{a}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img987.png"
 ALT="\begin{displaymath}
q = \frac{(2 j-1) \pi}{a}.
\end{displaymath}">|; 

$key = q/z+dz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1176.png"
 ALT="$z+dz$">|; 

$key = q/sim;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1398.png"
 ALT="$\sim$">|; 

$key = q/{displaymath}j_{y}=frac{{{rm{i},hbar}{2,m}left(psi,frac{partialpsi^ast}{partialy}-psi^ast,frac{partialpsi}{partialy}right){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1186.png"
 ALT="\begin{displaymath}
j_{y} =\frac{{\rm i} \hbar}{2 m}\left(\psi \frac{\partial...
...partial y} - \psi^\ast \frac{\partial\psi}{\partial y}\right)
\end{displaymath}">|; 

$key = q/|psi(x)|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img346.png"
 ALT="$\vert\psi(x)\vert^{ 2}$">|; 

$key = q/psi_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2401.png"
 ALT="$\psi_m$">|; 

$key = q/lambda=2pislashk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img153.png"
 ALT="$\lambda=2\pi/k$">|; 

$key = q/2P;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2657.png"
 ALT="$2P$">|; 

$key = q/P(Y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.png"
 ALT="$P(Y)$">|; 

$key = q/{displaymath}J^2,psi^{(2)}_{l,1slash2;j,m+1slash2}=j,(j+1),hbar^2,psi^{(2)}_{l,1slash2;j,m+1slash2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="337" HEIGHT="37" BORDER="0"
 SRC="|."$dir".q|img1930.png"
 ALT="\begin{displaymath}
J^2 \psi^{(2)}_{l,1/2;j,m+1/2}= j (j+1) \hbar^2 \psi^{(2)}_{l,1/2;j,m+1/2},
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{Chapter03slashfig06.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="226" BORDER="0"
 SRC="|."$dir".q|img278.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{Chapter03/fig06.eps}}
\end{figure}">|; 

$key = q/{displaymath}-frac{hbar^2}{2,m}frac{partial^2}{partialx^2}rightarrow-frac{hbar^2}{2,m}frac{partial^2}{partialx^2}+V(x).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img441.png"
 ALT="\begin{displaymath}
-\frac{\hbar^2}{2 m}\frac{\partial^2}{\partial x^2}\rightarrow
-\frac{\hbar^2}{2 m}\frac{\partial^2}{\partial x^2} + V(x).
\end{displaymath}">|; 

$key = q/partialP(x,t)slashpartialt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img760.png"
 ALT="$\partial P(x,t)/\partial t$">|; 

$key = q/{displaymath}s,(s+1)=m_{s,min},(m_{s,min}-1).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1764.png"
 ALT="\begin{displaymath}
s (s+1) = m_{s min} (m_{s min}-1).
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{sqrt{N!}}left|array{{cccc}psi(x_1,E_1)<col_mark>psi(x_2,E_1)si(x_2,E_N)<col_mark>ldots<col_mark>psi(x_N,E_N)array{right|.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="402" HEIGHT="131" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1163.png"
 ALT="$\displaystyle \frac{1}{\sqrt{N!}} \left\vert
\begin{array}{cccc}
\psi(x_1,E_1)&amp;...
...0.5ex]
\psi(x_1,E_N)&amp;\psi(x_2,E_N)&amp;\ldots&amp;\psi(x_N,E_N)
\end{array}\right\vert.$">|; 

$key = q/{{bf{S}_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2362.png"
 ALT="${\bf S}_p$">|; 

$key = q/{displaymath}E_n=frac{n^2,pi^2,hbar^2}{2,m,a^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img803.png"
 ALT="\begin{displaymath}
E_n = \frac{n^2 \pi^2 \hbar^2}{2 m a^2}.
\end{displaymath}">|; 

$key = q/(r_1,,theta_1,,phi_1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2735.png"
 ALT="$(r_1, \theta_1, \phi_1)$">|; 

$key = q/{displaymath}m_-=-l.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1426.png"
 ALT="\begin{displaymath}
m_- = - l.
\end{displaymath}">|; 

$key = q/{displaymath}mbox{{boldmath{{mu{}=-frac{e}{m_e},{{bf{S}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img2309.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\mu$}= - \frac{e}{m_e} {\bf S}
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{2pi,hbar}int_{-infty}^{infty}int_{-infty}^{infty}int_{-inftyartialp}right){{rm{e}^{+{{rm{i},(p-p'),xslashhbar},dx,dp,dp'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="531" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img596.png"
 ALT="$\displaystyle \frac{1}{2\pi \hbar}\int_{-\infty}^{\infty}\int_{-\infty}^{\inft...
...c{\partial}{\partial p}\right){\rm e}^{+{\rm i} (p-p') x/\hbar} dx dp dp'.$">|; 

$key = q/tau_{ex};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$\tau _{ex}$">|; 

$key = q/{displaymath}psi(x_1,x_2,ldots,x_N,t)=psi_E(x_1,x_2,ldots,x_N),{{rm{e}^{-{{rm{i},E,tslashhbar},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="391" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1088.png"
 ALT="\begin{displaymath}
\psi(x_1,x_2,\ldots, x_N,t) = \psi_E(x_1,x_2,\ldots, x_N) {\rm e}^{-{\rm i} E t/\hbar},
\end{displaymath}">|; 

$key = q/f=(A-langleArangle),psi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img605.png"
 ALT="$f=(A-\langle A\rangle) \psi$">|; 

$key = q/{displaymath}v_p=frac{omega}{k}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img310.png"
 ALT="\begin{displaymath}
v_p = \frac{\omega}{k}.
\end{displaymath}">|; 

$key = q/{displaymath}DeltaE_{l,1slash2;j,m_j}=E_n,frac{alpha^2}{n^2}left(frac{n}{j+1slash2}-frac{3}{4}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2325.png"
 ALT="\begin{displaymath}
\Delta E_{l,1/2;j,m_j}= E_n \frac{\alpha^2}{n^2}\left(\frac{n}{j+1/2}-\frac{3}{4}\right).
\end{displaymath}">|; 

$key = q/{displaymath}H=-frac{hbar^2}{2,m_e}left(nabla_1^{,2}+nabla_2^{,2}right)-frac{e^2ac{2}{r_2}-frac{1}{|{{bf{r_2}-{{bf{r_1}|}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="453" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2709.png"
 ALT="\begin{displaymath}
H = -\frac{\hbar^2}{2 m_e}\left(\nabla_1^{ 2} + \nabla_2^{...
...\frac{2}{r_2}-
\frac{1}{\vert{\bf r_2}-{\bf r_1}\vert}\right),
\end{displaymath}">|; 

$key = q/{displaymath}psi({{bf{r}_1,{{bf{r}_2)=psi_1({{bf{r}_1),psi_2({{bf{r}_2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2712.png"
 ALT="\begin{displaymath}
\psi({\bf r}_1,{\bf r}_2) = \psi_1({\bf r}_1) \psi_2({\bf r}_2).
\end{displaymath}">|; 

$key = q/V(x)=W;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1055.png"
 ALT="$V(x)=W$">|; 

$key = q/{displaymath}DeltaE_{nlm}=e^2,|{{bf{E}|^2sum_{n',l'=lpm1}frac{|langlen,l,m|z|n',l',mrangle|^2}{E_{nlm}-E_{n'l'm}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="364" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img2215.png"
 ALT="\begin{displaymath}
\Delta E_{nlm} = e^2 \vert{\bf E}\vert^2\sum_{n',l'=l\pm 1}...
...e n,l,m\vert z\vert n',l',m\rangle\vert^2}{E_{nlm}-E_{n'l'm}}.
\end{displaymath}">|; 

$key = q/Eslashp=c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img259.png"
 ALT="$E/p=c$">|; 

$key = q/j=lpm1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2061.png"
 ALT="$j=l\pm 1/2$">|; 

$key = q/displaystyle{{rm{i},hbar,S_x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1703.png"
 ALT="$\displaystyle {\rm i} \hbar S_x,$">|; 

$key = q/n_f=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1679.png"
 ALT="$n_f=2$">|; 

$key = q/{displaymath}langleprangle=m,frac{dlanglexrangle}{dt}=m,frac{d}{dt}int_{-infty}^}^{infty}x,frac{partial|psi|^{,2}}{partialt},dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="452" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img504.png"
 ALT="\begin{displaymath}
\langle p \rangle = m \frac{d\langle x\rangle}{dt} = m \f...
...\infty}x \frac{\partial\vert\psi\vert^{ 2}}{\partial t} dx.
\end{displaymath}">|; 

$key = q/{displaymath}-{{rm{i},hbar,frac{partialpsi_p(x,p')}{partialx}=p',psi_p(x,p').{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img722.png"
 ALT="\begin{displaymath}
-{\rm i} \hbar \frac{\partial \psi_p(x,p')}{\partial x} = p' \psi_p(x,p').
\end{displaymath}">|; 

$key = q/B_l=-C_l,sindelta_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2930.png"
 ALT="$B_l=-C_l \sin\delta_l$">|; 

$key = q/{displaymath}ointY_{l',m'}^{,ast},Y_{l,m},dOmega=delta_{ll'},delta_{mm'},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1471.png"
 ALT="\begin{displaymath}
\oint Y_{l',m'}^{ \ast} Y_{l,m} d\Omega = \delta_{ll'} \delta_{mm'},
\end{displaymath}">|; 

$key = q/alpha=1slash137;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2673.png"
 ALT="$\alpha=1/137$">|; 

$key = q/displaystyleleft[frac{gamma^2}{gamma^2+(omega-omega_{21})^2slash4}right]sin^2(Omega,t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2432.png"
 ALT="$\displaystyle \left[\frac{\gamma^2}{\gamma^2 + (\omega-\omega_{21})^2/4}\right]
\sin^2(\Omega t).$">|; 

$key = q/displaystyle{{rm{i},hbar,(-L_x,y-y,L_x+L_y,x+x,L_y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2191.png"
 ALT="$\displaystyle {\rm i} \hbar (-L_x y-y L_x+L_y x+x L_y)$">|; 

$key = q/x';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img706.png"
 ALT="$x'$">|; 

$key = q/E_0=A_0,omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2552.png"
 ALT="$E_0=A_0 \omega$">|; 

$key = q/displaystylebeta_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img377.png"
 ALT="$\displaystyle \beta_2$">|; 

$key = q/2580,AA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img447.png"
 ALT="$2580 \AA$">|; 

$key = q/+x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img486.png"
 ALT="$+x$">|; 

$key = q/{displaymath}p=hbar,k{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img260.png"
 ALT="\begin{displaymath}
p = \hbar k
\end{displaymath}">|; 

$key = q/{displaymath}P_{l,m}(u)=(-1)^{l+m},frac{(l+m)!}{(l-m)!},frac{(1-u^2)^{-mslash2}}{2^l,l!}left(frac{d}{du}right)^{l-m}(1-u^2)^l{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="482" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1466.png"
 ALT="\begin{displaymath}
P_{l,m}(u) = (-1)^{l+m} \frac{(l+m)!}{(l-m)!} \frac{(1-u^2)^{-m/2}}{2^l l!}\left(\frac{d}{du}\right)^{l-m} (1-u^2)^l
\end{displaymath}">|; 

$key = q/(2slash3),mu_B,B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2356.png"
 ALT="$(2/3) \mu_B B$">|; 

$key = q/Y_{l,m_-};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1420.png"
 ALT="$Y_{l,m_-}$">|; 

$key = q/A^{ast,ast}=A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img475.png"
 ALT="$A^{\ast \ast}=A$">|; 

$key = q/displaystylesqrt{frac{hbar}{2,m,omega}}left(sqrt{m},delta_{m,n+1}+sqrt{n},delta_{m,n-1}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="289" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1047.png"
 ALT="$\displaystyle \sqrt{\frac{\hbar}{2 m \omega}}\left(\sqrt{m} \delta_{m,n+1} + \sqrt{n} \delta_{m,n-1}\right).$">|; 

$key = q/f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img411.png"
 ALT="$f$">|; 

$key = q/{{rm{i}=sqrt{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img228.png"
 ALT="${\rm i} = \sqrt{-1}$">|; 

$key = q/{displaymath}w_{irightarrowf}^{abs}equivfrac{dP^{abs}_{irightarrowf}}{dt}=frac{p{{rm{i},{{bf{k}!cdot!{{bf{r})|irangleright|^{,2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="426" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img2564.png"
 ALT="\begin{displaymath}
w_{i\rightarrow f}^{abs} \equiv \frac{dP^{abs}_{i\rightarrow...
...rm i} {\bf k}\!\cdot\!{\bf r})\vert i\rangle\right\vert^{ 2}
\end{displaymath}">|; 

$key = q/Z(z)rightarrowexp(2,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1612.png"
 ALT="$Z(z)\rightarrow
\exp(2 z)$">|; 

$key = q/{displaymath}langlexrangle=frac{1}{sqrt{2pi,sigma^2}}int_{-infty}^{infty}x,{{rm{e}^{-(x-x_0)^{,2}slash(2,sigma^2)},dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img495.png"
 ALT="\begin{displaymath}
\langle x\rangle = \frac{1}{\sqrt{2\pi \sigma^2}}\int_{-\infty}^{\infty}
x {\rm e}^{-(x-x_0)^{ 2}/(2 \sigma^2)} dx.
\end{displaymath}">|; 

$key = q/{displaymath}frac{dpsi(x)}{dx}={{rm{i},k(x),psi(x),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img877.png"
 ALT="\begin{displaymath}
\frac{d\psi(x)}{dx} = {\rm i} k(x) \psi(x),
\end{displaymath}">|; 

$key = q/{displaymath}psi({{bf{r})simeqsqrt{n}sum_lC_l,frac{sin(k,r-l,pislash2+delta_l)}{k,r},P_l(costheta),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="378" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2927.png"
 ALT="\begin{displaymath}
\psi ({\bf r} ) \simeq \sqrt{n} \sum_l C_l 
\frac{\sin(k r - l \pi/2+ \delta_l)}{k r}  P_l(\cos\theta),
\end{displaymath}">|; 

$key = q/n=4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1171.png"
 ALT="$n=4$">|; 

$key = q/displaystyleleft(array{{c}1slashsqrt{2}<row_mark>[0.5ex]1slashsqrt{2}array{right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2277.png"
 ALT="$\displaystyle \left(\begin{array}{c}
1/\sqrt{2}  [0.5ex]
1/\sqrt{2}\end{array}\right),$">|; 

$key = q/{displaymath}H_1=-frac{g,e,B_1}{4,m}left[exp(,{{rm{i},omega,t),S_-+exp(-{{rm{i},omega,t),S_+right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="373" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2450.png"
 ALT="\begin{displaymath}
H_1 = - \frac{g e B_1}{4 m}\left[\exp( {\rm i} \omega t) S_- +
\exp(-{\rm i} \omega t) S_+\right],
\end{displaymath}">|; 

$key = q/displaystyle(3,phi_1-2,phi_2)leftslashsqrt{13}.right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img775.png"
 ALT="$\displaystyle (3 \phi_1-2 \phi_2) \left/ \sqrt{13}.\right.$">|; 

$key = q/r=sqrt{x^2+y^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img202.png"
 ALT="$r=\sqrt{x^2+y^2}$">|; 

$key = q/{displaymath}frac{d^2R_{n,l}}{dr^2}+frac{2}{r}frac{dR_{n,l}}{dr}+left(k^2-frac{l,(l+1)}{r^2}right)R_{n,l}=0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1543.png"
 ALT="\begin{displaymath}
\frac{d^2 R_{n,l}}{dr^2} + \frac{2}{r}\frac{dR_{n,l}}{dr} + \left(k^2
- \frac{l (l+1)}{r^2}\right) R_{n,l} = 0
\end{displaymath}">|; 

$key = q/{displaymath}psi(x,0)=sqrt{2slasha},sin(3pi,xslasha).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img1049.png"
 ALT="\begin{displaymath}
\psi(x,0) = \sqrt{2/a} \sin(3\pi x/a).
\end{displaymath}">|; 

$key = q/c(p')=phi(p');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img729.png"
 ALT="$c(p') = \phi(p')$">|; 

$key = q/R_{n,l}(a)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1562.png"
 ALT="$R_{n,l}(a)=0$">|; 

$key = q/psi_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img218.png"
 ALT="$\psi_0$">|; 

$key = q/displaystyle[l,(l+1)-m,(m+1)]^{1slash2},hbar,Y_{l,m+1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1389.png"
 ALT="$\displaystyle [l (l+1)-m (m+1)]^{1/2} \hbar Y_{l,m+1},$">|; 

$key = q/{displaymath}|R|^{,2}=frac{(k^2-q^2)^{,2},sin^2(q,a)}{4,k^2,q^2+(k^2-q^2)^{,2},sin^2(q,a)},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="294" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img846.png"
 ALT="\begin{displaymath}
\vert R\vert^{ 2} = \frac{(k^2-q^2)^{ 2} \sin^2(q a)}{4 k^2 q^2 + (k^2-q^2)^{ 2} \sin^2(q a)},
\end{displaymath}">|; 

$key = q/y=q,aslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img975.png"
 ALT="$y= q a/2$">|; 

$key = q/B_1llB_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2445.png"
 ALT="$B_1\ll B_0$">|; 

$key = q/textstylesimeq;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2972.png"
 ALT="$\textstyle \simeq$">|; 

$key = q/{displaymath}{{rm{i},hbar,frac{partialchi}{partialt}=H,chi,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1835.png"
 ALT="\begin{displaymath}
{\rm i} \hbar \frac{\partial\chi}{\partial t} = H \chi,
\end{displaymath}">|; 

$key = q/{displaymath}langleHrangle=E_0+4,A^2,(DpmE),E_0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2806.png"
 ALT="\begin{displaymath}
\langle H\rangle = E_0 + 4 A^2 (D\pm E) E_0,
\end{displaymath}">|; 

$key = q/{displaymath}sigma_{{rm{total}simeqfrac{4pi}{k^2}sum_{l=0}^{l_{{rm{max}}(2,l+1),sin^2delta_l.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img2984.png"
 ALT="\begin{displaymath}
\sigma_{\rm total} \simeq \frac{4\pi}{k^2} \sum_{l=0}^{l_{\rm max}}
(2 l+1) \sin^2\delta_l.
\end{displaymath}">|; 

$key = q/{displaymath}p_i=-{{rm{i},hbar,frac{partial}{partialx_i}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1495.png"
 ALT="\begin{displaymath}
p_i = -{\rm i} \hbar \frac{\partial}{\partial x_i}
\end{displaymath}">|; 

$key = q/hat{{bf{r}_{1,2}=2,{{bf{r}_{1,2}slasha_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2730.png"
 ALT="$\hat{\bf r}_{1,2} = 2  {\bf r}_{1,2}/a_0$">|; 

$key = q/|x|stackrel{_{{normalsize{>}}{_{{normalsize{sim}}pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2502.png"
 ALT="$\vert x\vert\stackrel {_{\normalsize &gt;}}{_{\normalsize\sim}}\pi$">|; 

$key = q/displaystylesigma_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1813.png"
 ALT="$\displaystyle \sigma_z$">|; 

$key = q/displaystyle{{rm{i},k,T,{{rm{e}^{,{{rm{i},k,a}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img861.png"
 ALT="$\displaystyle {\rm i} k T {\rm e}^{ {\rm i} k a}.$">|; 

$key = q/rslasha_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="$r/a_0$">|; 

$key = q/{displaymath}sigma_{{rm{total}=intfrac{dsigma}{dOmega},dOmega=4pi,a^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2980.png"
 ALT="\begin{displaymath}
\sigma_{\rm total} = \int\frac{d\sigma}{d\Omega} d\Omega = 4\pi  a^2
\end{displaymath}">|; 

$key = q/{displaymath}V(r)=left{array{{lcl}0&mbox{hspace{1cm}}&mbox{for{0leqrleqa{}[0.5ex]infty&&mbox{otherwise}array{right..{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img1537.png"
 ALT="\begin{displaymath}
V(r) = \left\{\begin{array}{lcl}
0&amp;\mbox{\hspace{1cm}}&amp;\mbox...
...leq a$} [0.5ex]
\infty&amp;&amp;\mbox{otherwise}
\end{array}\right..
\end{displaymath}">|; 

$key = q/3D_{5slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2337.png"
 ALT="$3D_{5/2}$">|; 

$key = q/{displaymath}[J^2,L_z]neq0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1894.png"
 ALT="\begin{displaymath}[J^2,L_z]\neq 0.
\end{displaymath}">|; 

$key = q/displaystyleint_{-infty}^inftypsi_a^ast,(A,psi_{a'}),dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img668.png"
 ALT="$\displaystyle \int_{-\infty}^\infty \psi_a^\ast (A \psi_{a'}) dx$">|; 

$key = q/{displaymath}P_{irightarrowf}(t)=|c_f(t)|^2=left|-frac{i}{hbar}int_0^tH_{fi}(t'),exp(,{{rm{i},omega_{fi},t'),dt'right|^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="429" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img2479.png"
 ALT="\begin{displaymath}
P_{i\rightarrow f}(t) =\vert c_f(t)\vert^2 = \left\vert -\fr...
...}(t') \exp( {\rm i} \omega_{fi} t') dt'\right\vert^{ 2}.
\end{displaymath}">|; 

$key = q/{displaymath}langle(x_1-x_2)^2rangle_{boson}=langle(x_1-x_2)^2rangle_{dist}-2,|langlexrangle_{ab}|^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="371" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1154.png"
 ALT="\begin{displaymath}
\langle (x_1-x_2)^2\rangle_{boson} = \langle (x_1-x_2)^2\rangle_{dist} -
2 \vert\langle x\rangle_{ab}\vert^2,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter13slashfig01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="337" HEIGHT="326" BORDER="0"
 SRC="|."$dir".q|img2494.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter13/fig01.eps}}
\end{figure}">|; 

$key = q/+2,e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2706.png"
 ALT="$+2 e$">|; 

$key = q/kequiv|{{bf{k}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img187.png"
 ALT="$k\equiv \vert{\bf k}\vert$">|; 

$key = q/{displaymath}psi(x)=A,{{rm{e}^{,q,x}+B,{{rm{e}^{-q,x},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img856.png"
 ALT="\begin{displaymath}
\psi(x) = A {\rm e}^{ q x} + B {\rm e}^{-q x},
\end{displaymath}">|; 

$key = q/{displaymath}E_0<E_1<E_2<cdots,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2691.png"
 ALT="\begin{displaymath}
E_0 &lt; E_1 &lt; E_2 &lt; \cdots,
\end{displaymath}">|; 

$key = q/xleq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img791.png"
 ALT="$x\leq 0$">|; 

$key = q/{displaymath}u({{bf{r})=u_0({{bf{r})-intfrac{{{rm{e}^{,{{rm{i},k,|{{bf{r}-{{bf{ri,|{{bf{r}-{{bf{r}'|},rho({{bf{r}'),d^3{{bf{r}'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img2847.png"
 ALT="\begin{displaymath}
u({\bf r}) = u_0({\bf r}) - \int \frac{{\rm e}^{ {\rm i} k...
...\pi \vert{\bf r}-{\bf r}'\vert} \rho({\bf r}') d^3{\bf r}'.
\end{displaymath}">|; 

$key = q/R_{n,l}(r)simeqZ(rslasha)slash(rslasha)simeq(rslasha)^l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1606.png"
 ALT="$R_{n,l}(r)\simeq Z(r/a)/(r/a)\simeq (r/a)^l$">|; 

$key = q/|psi_E|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img739.png"
 ALT="$\vert\psi_E\vert^{ 2}$">|; 

$key = q/displaystyle2,{{rm{i},hbar,left(L_y,[L^2,x]-L_x,[L^2,y]+{{rm{i},hbar,[L^2,z]right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="325" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2196.png"
 ALT="$\displaystyle 2 {\rm i} \hbar \left(L_y [L^2,x] - L_x [L^2,y] + {\rm i} \hbar [L^2,z]\right)$">|; 

$key = q/n'=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2654.png"
 ALT="$n'=1$">|; 

$key = q/m_s=pm1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1776.png"
 ALT="$m_s=\pm 1/2$">|; 

$key = q/displaystylefrac{dlangleprangle}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img509.png"
 ALT="$\displaystyle \frac{d\langle p\rangle}{dt}$">|; 

$key = q/{displaymath}P_{irightarrowf}^{abs}(t)=frac{t^2}{hbar^2},|V_{fi}^nomath_inline{drm{sinc}^2left[(omega-omega_{fi}),tslash2right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2518.png"
 ALT="\begin{displaymath}
P_{i\rightarrow f}^{abs}(t) = \frac{t^2}{\hbar^2} 
\vert V_...
...ert^{ 2} {\rm sinc}^2\left[(\omega-\omega_{fi}) t/2\right].
\end{displaymath}">|; 

$key = q/V(x_1-x_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1111.png"
 ALT="$V(x_1-x_2)$">|; 

$key = q/exp[{{rm{i},(k,x-omega,t)];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img321.png"
 ALT="$\exp[{\rm i} (k x-\omega t)]$">|; 

$key = q/p_1=p_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1202.png"
 ALT="$p_1=p_x$">|; 

$key = q/displaystylesum_n,c_n^{,ast},c_m,E_m,langlepsi_n|psi_mrangle=sum_nE_n,|c_n|^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2697.png"
 ALT="$\displaystyle \sum_n c_n^{ \ast} c_m E_m \langle \psi_n\vert\psi_m\rangle=
\sum_n E_n \vert c_n\vert^{ 2},$">|; 

$key = q/A_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2111.png"
 ALT="$A_{ij}$">|; 

$key = q/{displaymath}x=x_0+v_g,t.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img390.png"
 ALT="\begin{displaymath}
x = x_0+v_g t.
\end{displaymath}">|; 

$key = q/{displaymath}H_1,psi_{nlm}^{(1)}=lambda_{nl},psi_{nlm}^{(1)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2248.png"
 ALT="\begin{displaymath}
H_1 \psi_{nlm}^{(1)} = \lambda_{nl} \psi_{nlm}^{(1)}.
\end{displaymath}">|; 

$key = q/{displaymath}psi(x,0)=sqrt{1slasha},sin(pi,xslasha),[1+2,cos(pi,xslasha)].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img1050.png"
 ALT="\begin{displaymath}
\psi(x,0) = \sqrt{1/a} \sin(\pi x/a) [1+2 \cos(\pi x/a)].
\end{displaymath}">|; 

$key = q/{displaymath}langleS_zrangle=frac{hbar}{2}left(c_+^ast,c_-^astright)left(array{{&-1array{right)left(array{{c}c_+c_-array{right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="310" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1851.png"
 ALT="\begin{displaymath}
\langle S_z\rangle= \frac{\hbar}{2}\left(c_+^\ast, c_-^\ast\...
...rray}\right)\left(\begin{array}{c}c_+ c_-\end{array}\right),
\end{displaymath}">|; 

$key = q/displaystyleR_{2,0}(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1640.png"
 ALT="$\displaystyle R_{2,0}(r)$">|; 

$key = q/omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img160.png"
 ALT="$\omega$">|; 

$key = q/chi(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1865.png"
 ALT="$\chi(t)$">|; 

$key = q/y+dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1175.png"
 ALT="$y+dx$">|; 

$key = q/{displaymath}psi(x)={{rm{e}^{,{{rm{i},k,x}+R,{{rm{e}^{-{{rm{i},k,x}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img816.png"
 ALT="\begin{displaymath}
\psi(x) = {\rm e}^{ {\rm i} k x} + R {\rm e}^{-{\rm i} k x}.
\end{displaymath}">|; 

$key = q/{displaymath}a_-,a_+,psi_n=(n+1),psi_n.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1036.png"
 ALT="\begin{displaymath}
a_- a_+ \psi_n = (n+1) \psi_n.
\end{displaymath}">|; 

$key = q/mneqk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2155.png"
 ALT="$m\neq k$">|; 

$key = q/{displaymath}{{rm{i},hbar,frac{partialpsi(t)}{partialt}=H(t),psi(t)=[H_0+H_1(t)],psi(t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2402.png"
 ALT="\begin{displaymath}
{\rm i} \hbar \frac{\partial\psi(t)}{\partial t} = H(t) \psi(t)
= [H_0+H_1(t)] \psi(t).
\end{displaymath}">|; 

$key = q/displaystylefrac{pi}{3,epsilon_0,hbar^2},d^{,2}_{if},rho(omega_{fi}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2597.png"
 ALT="$\displaystyle \frac{\pi}{3 \epsilon_0 \hbar^2} d^{ 2}_{if} \rho(\omega_{fi}),$">|; 

$key = q/{displaymath}H=-mbox{{boldmath{{mu{}cdot{{bf{B}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1832.png"
 ALT="\begin{displaymath}
H = - \mbox{\boldmath$\mu$}\cdot {\bf B}.
\end{displaymath}">|; 

$key = q/m_{s,min};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1760.png"
 ALT="$m_{s min}$">|; 

$key = q/{displaymath}DeltaE_{j=l-1slash2}=mu_B,B,frac{2,l}{2,l+1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2354.png"
 ALT="\begin{displaymath}
\Delta E_{j=l-1/2} = \mu_B B \frac{2 l}{2 l+1}.
\end{displaymath}">|; 

$key = q/{displaymath}langlen|mrangle=delta_{nm}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2393.png"
 ALT="\begin{displaymath}
\langle n\vert m\rangle =\delta_{nm}.
\end{displaymath}">|; 

$key = q/sigma_x,sigma_p=hbarslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img626.png"
 ALT="$\sigma_x \sigma_p=\hbar/2$">|; 

$key = q/-hbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1488.png"
 ALT="$-\hbar$">|; 

$key = q/c_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img677.png"
 ALT="$c_i$">|; 

$key = q/{displaymath}frac{hbar^2}{2,m},frac{d^2psi_i}{dx^2}=left[V(x)-E_iright]psi_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img740.png"
 ALT="\begin{displaymath}
\frac{\hbar^2}{2 m} \frac{d^2 \psi_i}{d x^2} =
\left[V(x)-E_i\right]\psi_i.
\end{displaymath}">|; 

$key = q/m'=mpm1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2648.png"
 ALT="$m'=m\pm 1$">|; 

$key = q/kllmu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2898.png"
 ALT="$k\ll \mu$">|; 

$key = q/c^{(1)}_n(0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2476.png"
 ALT="$c^{(1)}_n(0)=0$">|; 

$key = q/|T|^{,2}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img869.png"
 ALT="$\vert T\vert^{ 2}=0$">|; 

$key = q/{displaymath}E=frac{(E_1+E_2)pmsqrt{(E_1-E_2)^2+4,|e_{12}|^2}}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img2136.png"
 ALT="\begin{displaymath}
E = \frac{(E_1+E_2)\pm\sqrt{(E_1-E_2)^2 + 4 \vert e_{12}\vert^2}}
{2}.
\end{displaymath}">|; 

$key = q/{displaymath}L_-,Y_{l,m_-}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1421.png"
 ALT="\begin{displaymath}
L_- Y_{l,m_-} = 0.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter03slashfig09.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="341" BORDER="0"
 SRC="|."$dir".q|img443.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter03/fig09.eps}}
\end{figure}">|; 

$key = q/{displaymath}E_{total}=-F_pm(Rslasha_0),E_0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2821.png"
 ALT="\begin{displaymath}
E_{total} = - F_\pm(R/a_0) E_0,
\end{displaymath}">|; 

$key = q/dOmega=sintheta,dtheta,dphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1371.png"
 ALT="$d\Omega = \sin\theta d\theta d\phi$">|; 

$key = q/psi_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1089.png"
 ALT="$\psi_E$">|; 

$key = q/omega=-omega_{fi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2509.png"
 ALT="$\omega=-\omega_{fi}$">|; 

$key = q/{displaymath}psi(x,t)=sum_ic_i,psi_i(x),{{rm{e}^{-{{rm{i},E_i,tslashhbar},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img742.png"
 ALT="\begin{displaymath}
\psi(x,t) = \sum_i c_i \psi_i(x) {\rm e}^{-{\rm i} E_i t/\hbar},
\end{displaymath}">|; 

$key = q/{displaymath}H=H_1(Z)+H_2(Z)+V_{ee}+U(Z),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2756.png"
 ALT="\begin{displaymath}
H = H_1(Z) + H_2(Z) + V_{ee} + U(Z),
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.8in{centerline{epsffile{Chapter05slashfig04.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="303" BORDER="0"
 SRC="|."$dir".q|img867.png"
 ALT="\begin{figure}
\epsfysize =2.8in
\centerline{\epsffile{Chapter05/fig04.eps}}
\end{figure}">|; 

$key = q/(1slash2),[O(x,p,H)+O^nomath_inline{dagnomath_inline{(x,p,H)];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img563.png"
 ALT="$(1/2) [O(x,p,H)+ O^\dag (x,p,H)]$">|; 

$key = q/displaystyle[L^2,[L^2,z]];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2194.png"
 ALT="$\displaystyle [L^2,[L^2,z]]$">|; 

$key = q/tgg2pislashomega_{fi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2560.png"
 ALT="$t\gg 2\pi/\omega_{fi}$">|; 

$key = q/displaystylephi(p,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img571.png"
 ALT="$\displaystyle \phi(p,t)$">|; 

$key = q/psi_{nlm}^{(1)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2247.png"
 ALT="$\psi_{nlm}^{(1)}$">|; 

$key = q/{{rm{i};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="10" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img190.png"
 ALT="${\rm i}$">|; 

$key = q/l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="$l$">|; 

$key = q/displaystyley_0(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1555.png"
 ALT="$\displaystyle y_0(z)$">|; 

$key = q/{displaymath}z^ast=int_{-infty}^inftypsi^ast,B,A,psi,dx-langleArangle,langleBrangle.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img619.png"
 ALT="\begin{displaymath}
z ^\ast=\int_{-\infty}^\infty \psi^\ast B A \psi dx -\langle A\rangle \langle B\rangle.
\end{displaymath}">|; 

$key = q/sigma_E,sigma_t>hbarslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img648.png"
 ALT="$\sigma_E \sigma_t&gt;\hbar/2$">|; 

$key = q/displaystyleJ_z,psi_{j,m_j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1883.png"
 ALT="$\displaystyle J_z \psi_{j,m_j}$">|; 

$key = q/theta={{rm{arg}(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img215.png"
 ALT="$\theta={\rm arg}(z)$">|; 

$key = q/psi_{a'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img659.png"
 ALT="$\psi_{a'}$">|; 

$key = q/{displaymath}H=frac{p^2}{2,m}+frac{1}{2},K,x^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img995.png"
 ALT="\begin{displaymath}
H = \frac{p^2}{2 m} + \frac{1}{2} K x^2,
\end{displaymath}">|; 

$key = q/{displaymath}psi({{bf{r})simeqsqrt{n}left[{{rm{e}^{,{{rm{i},{{bf{k}cdot{{bf{r}}+{e^{,{{rm{i},k,r}}{r},f({{bf{k},{{bf{k}')right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2858.png"
 ALT="\begin{displaymath}
\psi({\bf r}) \simeq \sqrt{n}\left[{\rm e}^{ {\rm i} {\bf ...
...
+ \frac{e^{ {\rm i} k r}}{r} f({\bf k}, {\bf k}')\right],
\end{displaymath}">|; 

$key = q/0leqxleq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img695.png"
 ALT="$0\leq x\leq 1$">|; 

$key = q/{displaymath}frac{d^2psi}{dy^2}-(y^2-epsilon),psi=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1001.png"
 ALT="\begin{displaymath}
\frac{d^2\psi}{dy^2} - (y^2-\epsilon) \psi = 0.
\end{displaymath}">|; 

$key = q/x_1,x_2,ldotsx_N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1079.png"
 ALT="$x_1 x_2 \ldots x_N$">|; 

$key = q/{displaymath}frac{partial|psi|^{,2}}{partialt}+frac{partialj}{partialx}=0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img505.png"
 ALT="\begin{displaymath}
\frac{\partial\vert\psi\vert^{ 2}}{\partial t} + \frac{\partial j}{\partial x} = 0
\end{displaymath}">|; 

$key = q/{displaymath}I({{bf{r}_2)=int_0^inftyint_0^piint_0^{2pi}frac{{{rm{e}^{-2,r_1}}{sta_1}},r_1^{,2},dr_1,sintheta_1,dtheta_1,dphi_1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="528" HEIGHT="62" BORDER="0"
 SRC="|."$dir".q|img2737.png"
 ALT="\begin{displaymath}
I({\bf r}_2) = \int_0^\infty\int_0^\pi\int_0^{2\pi}
\frac{{\...
...theta_1}} 
r_1^{ 2} dr_1 \sin\theta_1 d\theta_1 d\phi_1,
\end{displaymath}">|; 

$key = q/displaystyle-frac{cos(z-l,pislash2)}{z}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2914.png"
 ALT="$\displaystyle - \frac{\cos(z-l \pi/2)}{z}.$">|; 

$key = q/{displaymath}{{rm{i},hbar,frac{partialpsi}{partialt}=-frac{hbar^2}{2,m}frac{partial^2psi}{partialx^2}+V(x),psi.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img442.png"
 ALT="\begin{displaymath}
{\rm i} \hbar \frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2 m}\frac{\partial^2\psi}{\partial x^2} + V(x) \psi.
\end{displaymath}">|; 

$key = q/{displaymath}psi(x,t)=A,cos(-k,x-omega,t+varphi)=A,cos(k,x+omega,t-varphi),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="452" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img169.png"
 ALT="\begin{displaymath}
\psi(x,t) = A \cos(-k x-\omega t+\varphi)=A \cos(k x+\omega t-\varphi),
\end{displaymath}">|; 

$key = q/displaystyleS^2-S_z^{,2}-hbar,S_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1714.png"
 ALT="$\displaystyle S^2-S_z^{ 2}-\hbar S_z,$">|; 

$key = q/c_2(0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2420.png"
 ALT="$c_2(0)=0$">|; 

$key = q/epsilon_{ijk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1502.png"
 ALT="$\epsilon_{ijk}$">|; 

$key = q/x_2=y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1200.png"
 ALT="$x_2=y$">|; 

$key = q/p_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1310.png"
 ALT="$p_i$">|; 

$key = q/phi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1401.png"
 ALT="$\phi=0$">|; 

$key = q/m_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1110.png"
 ALT="$m_2$">|; 

$key = q/displaystylej_1(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1553.png"
 ALT="$\displaystyle j_1(z)$">|; 

$key = q/{displaymath}omega=frac{hbar,k^2}{2,m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img309.png"
 ALT="\begin{displaymath}
\omega = \frac{\hbar k^2}{2 m}.
\end{displaymath}">|; 

$key = q/displaystyle-frac{1}{8,m_e^{,3},c^2},langlen,l,m|p^2,p^2|n,l,mrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2294.png"
 ALT="$\displaystyle - \frac{1}{8 m_e^{ 3} c^2} 
\langle n,l,m\vert p^2 p^2\vert n,l,m\rangle.$">|; 

$key = q/{displaymath}frac{2,m}{hbar^2}frac{|V_0|}{mu^2}ll1{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2900.png"
 ALT="\begin{displaymath}
\frac{2 m}{\hbar^2} \frac{\vert V_0\vert}{\mu^2} \ll 1
\end{displaymath}">|; 

$key = q/x={{rm{Re}(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img198.png"
 ALT="$x={\rm Re}(z)$">|; 

$key = q/l-1slash2,m+1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1969.png"
 ALT="$l-1/2, m+1/2$">|; 

$key = q/displaystylem_s,hbar,chi_{s_1,s_2;s,m_s}^{(2)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2057.png"
 ALT="$\displaystyle m_s \hbar
 \chi_{s_1,s_2;s,m_s}^{(2)}.$">|; 

$key = q/V=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2990.png"
 ALT="$V=0$">|; 

$key = q/{displaymath}psi(x)=psi_1,expleft(-int_{x_1}^x|k(x')|,dx'right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img888.png"
 ALT="\begin{displaymath}
\psi(x) = \psi_1 \exp\left(-\int_{x_1}^x \vert k(x')\vert dx'\right),
\end{displaymath}">|; 

$key = q/E_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img783.png"
 ALT="$E_2$">|; 

$key = q/displaystyle[L_x^{,2},z]+[L_y^{,2},z];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2189.png"
 ALT="$\displaystyle [L_x^{ 2},z] + [L_y^{ 2},z]$">|; 

$key = q/{displaymath}sum_m{{rm{i},hbar,frac{dc_m}{dt},expleft(-{{rm{i},E_m,tslashhbarrig,expleft(-{{rm{i},E_m,tslashhbarright)H_1,psi_m.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="502" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2405.png"
 ALT="\begin{displaymath}
\sum_m {\rm i} \hbar \frac{dc_m}{dt} \exp\left(-{\rm i} ...
...\sum_m c_m exp\left(-{\rm i} E_m t/\hbar\right)H_1 \psi_m.
\end{displaymath}">|; 

$key = q/k_y^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1226.png"
 ALT="$k_y^{ 2}$">|; 

$key = q/{displaymath}V(x_1,x_2,ldots,x_N,t)=sum_{i=1,N}V(x_i,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1092.png"
 ALT="\begin{displaymath}
V(x_1,x_2,\ldots, x_N,t) = \sum_{i=1,N} V(x_i,t).
\end{displaymath}">|; 

$key = q/{displaymath}int_{-infty}^{infty}|psi(x,t)|^{,2},dx=1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img461.png"
 ALT="\begin{displaymath}
\int_{-\infty}^{\infty}\vert\psi(x,t)\vert^{ 2} dx = 1,
\end{displaymath}">|; 

$key = q/phi(p,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img586.png"
 ALT="$\phi(p,t)$">|; 

$key = q/{displaymath}chi=A,left(array{{c}1-2,{{rm{i}2array{right){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1861.png"
 ALT="\begin{displaymath}
\chi = A \left(\begin{array}{c}1-2 {\rm i} 2\end{array}\right)
\end{displaymath}">|; 

$key = q/displaystyle(u_1+u_2)leftslashsqrt{2},right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img778.png"
 ALT="$\displaystyle (u_1+u_2)\left/\sqrt{2},\right.$">|; 

$key = q/{displaymath}c_i=int_{-infty}^inftypsi_i^ast,psi,dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img683.png"
 ALT="\begin{displaymath}
c_i = \int_{-\infty}^\infty \psi_i^\ast \psi dx.
\end{displaymath}">|; 

$key = q/X=Rslasha_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2790.png"
 ALT="$X=R/a_0$">|; 

$key = q/J;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2819.png"
 ALT="$J$">|; 

$key = q/displaystylesin^2(gamma,t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2435.png"
 ALT="$\displaystyle \sin^2(\gamma t).$">|; 

$key = q/m=0,,pm1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2656.png"
 ALT="$m=0, \pm 1$">|; 

$key = q/{displaymath}chi^nomath_inline{dagnomath_inline{,chi=|c_+|^2+|c_-|^2=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1816.png"
 ALT="\begin{displaymath}
\chi^\dag \chi = \vert c_+\vert^2 + \vert c_-\vert^2 =1.
\end{displaymath}">|; 

$key = q/displaystylechi^{(2)}_{1,0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2079.png"
 ALT="$\displaystyle \chi^{(2)}_{1,0}$">|; 

$key = q/{displaymath}V(r)=frac{V_0,exp(-mu,r)}{mu,r},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2882.png"
 ALT="\begin{displaymath}
V(r) = \frac{V_0 \exp(-\mu  r)}{\mu  r},
\end{displaymath}">|; 

$key = q/alpha=e^2slash(2,epsilon_0,h,c)simeq1slash137;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1829.png"
 ALT="$\alpha= e^2/(2 \epsilon_0 h c) \simeq 1/137$">|; 

$key = q/{displaymath}L_-,L_+=L^2-L_z^{,2}-hbar,L_z,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1339.png"
 ALT="\begin{displaymath}
L_- L_+ = L^2 - L_z^{ 2} -\hbar L_z,
\end{displaymath}">|; 

$key = q/{displaymath}H=frac{p_x^{,2}+p_y^{,2}+p_z^{,2}}{2,m}+V(x,y,z,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1205.png"
 ALT="\begin{displaymath}
H = \frac{p_x^{ 2}+ p_y^{ 2}+p_z^{ 2}}{2 m} + V(x,y,z,t).
\end{displaymath}">|; 

$key = q/psi(x_1,x_2,t)=psi_{x'}(x',t),psi_0equivpsi(x_1-x_2,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="337" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1130.png"
 ALT="$\psi(x_1,x_2,t) = \psi_{x'}(x',t) \psi_0\equiv \psi(x_1-x_2,t)$">|; 

$key = q/mbox{{boldmath{{epsilon{}!cdot!{{bf{k}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2542.png"
 ALT="$\mbox{\boldmath$\epsilon$}\!\cdot\!{\bf k} = 0$">|; 

$key = q/{displaymath}int_0^{infty}R_{n',l}^ast,R_{n,l},r^2,dr=delta_{nn'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img1637.png"
 ALT="\begin{displaymath}
\int_0^{\infty} R_{n',l}^\ast R_{n,l} r^2 dr = \delta_{nn'}.
\end{displaymath}">|; 

$key = q/{displaymath}psi_0({{bf{r})=sqrt{n},exp(,{{rm{i},k,rcostheta)=sqrt{n},sum_l{{rm{i}^{,l},(2,l+1),j_l(k,r),P_l(costheta).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="514" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2922.png"
 ALT="\begin{displaymath}
\psi_0({\bf r}) = \sqrt{n} \exp( {\rm i} k r \cos\theta)...
... \sum_l {\rm i}^{ l} (2 l+1)  j_l(k r)  P_l(\cos\theta).
\end{displaymath}">|; 

$key = q/psi(x,t+T)=psi(x,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img156.png"
 ALT="$\psi(x,t+T)=\psi(x,t)$">|; 

$key = q/2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img149.png"
 ALT="$2\pi$">|; 

$key = q/{displaymath}chi(E)=frac{1}{sqrt{2pi,hbar}}int_{-infty}^{infty}psi(t),{{rm{e}^{+{{rm{i},E,tslashhbar},dt.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img642.png"
 ALT="\begin{displaymath}
\chi(E) = \frac{1}{\sqrt{2\pi \hbar}}\int_{-\infty}^{\infty}\psi(t) {\rm e}^{+{\rm i} E t/\hbar} dt.
\end{displaymath}">|; 

$key = q/j(x,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img485.png"
 ALT="$j(x,t)$">|; 

$key = q/{displaymath}P_{irightarrowf}^{abs}(t+dt)-P_{irightarrowf}^{abs}(t)=left[1-P_{irrrowf}^{abs}(t)right],w_{irightarrowf}^{abs},dt:{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="391" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img2566.png"
 ALT="\begin{displaymath}
P_{i\rightarrow f}^{abs}(t+dt) - P_{i\rightarrow f}^{abs}(t)...
...i\rightarrow f}^{abs}(t)\right] w_{i\rightarrow f}^{abs} dt:
\end{displaymath}">|; 

$key = q/displaystylefrac{4,sqrt{2}}{9,(3,a_0)^{3slash2}},frac{r}{a_0}left(1-frac{r}{6,a_0}right),expleft(-frac{r}{3,a_0}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="322" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1647.png"
 ALT="$\displaystyle \frac{4 \sqrt{2}}{9 (3 a_0)^{3/2}} \frac{r}{a_0}
\left(1-\frac{r}{6 a_0}\right) \exp\left(-\frac{r}{3 a_0}\right),$">|; 

$key = q/{displaymath}V(x)=left{array{{lcl}-V_0&mbox{hspace{1cm}}&mbox{for{-aslash2leqxleqaslash2{}[0.5ex]0&&mbox{otherwise}array{right.,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="375" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img952.png"
 ALT="\begin{displaymath}
V(x) = \left\{\begin{array}{lcl}
-V_0&amp;\mbox{\hspace{1cm}}&amp;\m...
... x\leq a/2$} [0.5ex]
0&amp;&amp;\mbox{otherwise}
\end{array}\right.,
\end{displaymath}">|; 

$key = q/+e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1580.png"
 ALT="$+e$">|; 

$key = q/{displaymath}j=frac{{{rm{i},hbar}{2,m}left(psi,frac{partialpsi^ast}{partialx}-psi^ast,frac{partialpsi}{partialx}right){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1181.png"
 ALT="\begin{displaymath}
j = \frac{{\rm i} \hbar}{2 m}\left(\psi \frac{\partial\ps...
...partial x} - \psi^\ast \frac{\partial\psi}{\partial x}\right)
\end{displaymath}">|; 

$key = q/displaystyleH_0,psi_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2119.png"
 ALT="$\displaystyle H_0 \psi_1$">|; 

$key = q/psi_{211};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2270.png"
 ALT="$\psi_{211}$">|; 

$key = q/theta=tan^{-1}(yslashx);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img205.png"
 ALT="$\theta=\tan^{-1}(y/x)$">|; 

$key = q/{displaymath}left|int_a^b,f^ast(x),g(x),dxright|^{,2}leqint_a^b|f(x)|^{,2},dx,int_a^b|g(x)|^{,2},dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="399" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img612.png"
 ALT="\begin{displaymath}
\left\vert\int_a^b f^\ast(x) g(x) dx\right\vert^{ 2}\leq...
...\vert f(x)\vert^{ 2} dx \int_a^b \vert g(x)\vert^{ 2} dx,
\end{displaymath}">|; 

$key = q/{displaymath}psi_0({{bf{r})=frac{1}{sqrt{pi},a_0^{,3slash2}},{{rm{e}^{-rslasha_0}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2780.png"
 ALT="\begin{displaymath}
\psi_0({\bf r}) = \frac{1}{\sqrt{\pi} a_0^{ 3/2}} {\rm e}^{-r/a_0}
\end{displaymath}">|; 

$key = q/{displaymath}H(x_1,x_2,ldots,x_N,t)=sum_{i=1,N}frac{p_i^{,2}}{2,m_i}+V(x_1,x_2,ldots,x_N,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="442" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1091.png"
 ALT="\begin{displaymath}
H(x_1,x_2,\ldots, x_N, t) = \sum_{i=1,N}\frac{p_i^{ 2}}{2 m_i}
+ V(x_1,x_2,\ldots, x_N, t).
\end{displaymath}">|; 

$key = q/{displaymath}tau_{2P}=(w_{2Prightarrow1S})^{-1}=1.6,{{rm{ns}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2674.png"
 ALT="\begin{displaymath}
\tau_{2P} = (w_{2P\rightarrow 1S})^{-1} = 1.6 {\rm ns}.
\end{displaymath}">|; 

$key = q/{displaymath}E_F=frac{l_F^{,2},pi^2,hbar^2}{2,m_e,a^2}=frac{pi^2,hbar^2}{2,m,a^2}left(frac{3,N}{pi}right)^{2slash3},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1250.png"
 ALT="\begin{displaymath}
E_F = \frac{l_F^{ 2} \pi^2 \hbar^2}{2 m_e a^2}=\frac{\pi^2 \hbar^2}{2 m a^2}\left(\frac{3 N}{\pi}\right)^{2/3},
\end{displaymath}">|; 

$key = q/n+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1020.png"
 ALT="$n+1$">|; 

$key = q/{{rm{sinc}^2(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="${\rm sinc}^2(x)$">|; 

$key = q/E=-Z^2,E_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1692.png"
 ALT="$E=-Z^2 E_0$">|; 

$key = q/displaystyley_l(k,r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2974.png"
 ALT="$\displaystyle y_l(k r)$">|; 

$key = q/{displaymath}x'=x_1-x_2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1113.png"
 ALT="\begin{displaymath}
x' = x_1-x_2
\end{displaymath}">|; 

$key = q/displaystyleint{{rm{e}^{-beta,x},x,dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2746.png"
 ALT="$\displaystyle \int {\rm e}^{-\beta x} x dx$">|; 

$key = q/l=4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1571.png"
 ALT="$l=4$">|; 

$key = q/n=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1568.png"
 ALT="$n=2$">|; 

$key = q/{displaymath}R=1.5times10^{-15},A^{1slash3},{{rm{m}=2.0times10^{-15},Z_1^{1slash3},{{rm{m}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img912.png"
 ALT="\begin{displaymath}
R = 1.5\times 10^{-15} A^{1/3} {\rm m}=2.0\times 10^{-15} Z_1^{1/3} {\rm m}
\end{displaymath}">|; 

$key = q/(O^nomath_inline{dagnomath_inline{)^nomath_inline{dagnomath_inline{=O;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img547.png"
 ALT="$(O^\dag )^\dag =O$">|; 

$key = q/displaystylefrac{3}{4},hbar^2,chi_pm.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1781.png"
 ALT="$\displaystyle \frac{3}{4} \hbar^2 \chi_\pm.$">|; 

$key = q/E_{200};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2282.png"
 ALT="$E_{200}$">|; 

$key = q/S_p^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2379.png"
 ALT="$S_p^{ 2}$">|; 

$key = q/psirightarrowpsi_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2843.png"
 ALT="$\psi\rightarrow\psi_0$">|; 

$key = q/displaystylefrac{sin(z-l,pislash2)}{z},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2913.png"
 ALT="$\displaystyle \frac{\sin(z - l \pi/2)}{z},$">|; 

$key = q/displaystyleJ_z,psi^{(2)}_{l,s;j,m_j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1918.png"
 ALT="$\displaystyle J_z \psi^{(2)}_{l,s;j,m_j}$">|; 

$key = q/V(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1490.png"
 ALT="$V(r)$">|; 

$key = q/displaystylec_-(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1847.png"
 ALT="$\displaystyle c_-(t)$">|; 

$key = q/m_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1109.png"
 ALT="$m_1$">|; 

$key = q/n=1,2,3,cdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img802.png"
 ALT="$n=1,2,3,\cdots$">|; 

$key = q/{displaymath}psi({{bf{r},t)=psi_0,{{rm{e}^{,{{rm{i},({{bf{k}cdot{{bf{r}-omega,t)},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img226.png"
 ALT="\begin{displaymath}
\psi({\bf r},t) = \psi_0 {\rm e}^{ {\rm i} ({\bf k}\cdot{\bf r}-\omega t)},
\end{displaymath}">|; 

$key = q/-{{rm{i},hbar,partialslashpartialx_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1344.png"
 ALT="$-{\rm i} \hbar \partial/\partial x_i$">|; 

$key = q/Nslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img296.png"
 ALT="$N/2$">|; 

$key = q/E=hbar,omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img638.png"
 ALT="$E=\hbar \omega$">|; 

$key = q/{displaymath}p^2=r^{-2}left[({{bf{r}cdot{{bf{p})^2-{{rm{i},hbar,{{bf{r}cdot{{bf{p}+L^2right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img1526.png"
 ALT="\begin{displaymath}
p^2 = r^{-2}\left[({\bf r}\cdot{\bf p})^2- {\rm i} \hbar {\bf r}\cdot{\bf p}+L^2\right].
\end{displaymath}">|; 

$key = q/1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1857.png"
 ALT="$1/2$">|; 

$key = q/displaystyleX(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1228.png"
 ALT="$\displaystyle X(x)$">|; 

$key = q/{displaymath}tanalpha=frac{Dslash2}{f},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img414.png"
 ALT="\begin{displaymath}
\tan\alpha = \frac{D/2}{f},
\end{displaymath}">|; 

$key = q/{displaymath}chi^{(2)}_{0,0}=frac{1}{sqrt{2}}left(chi^{(1)}_{-1slash2,1slash2}-chi^{(1)}_{1slash2,-1slash2}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="272" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2085.png"
 ALT="\begin{displaymath}
\chi^{(2)}_{0,0} = \frac{1}{\sqrt{2}}\left(\chi^{(1)}_{-1/2,1/2}-\chi^{(1)}_{1/2,-1/2}\right).
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{E}=frac{e,{{bf{r}}{4piepsilon_0,r^3}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img2305.png"
 ALT="\begin{displaymath}
{\bf E} = \frac{e {\bf r}}{4\pi\epsilon_0 r^3}
\end{displaymath}">|; 

$key = q/[k_{min},(k_{min}-1)-l,(l+1)]=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1602.png"
 ALT="$[k_{min} (k_{min}-1)-l (l+1)]=0$">|; 

$key = q/displaystyle{{rm{Re}left[A,{{rm{e}^{,{{rm{i},(k,x-omega,t+varphi)}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img222.png"
 ALT="$\displaystyle {\rm Re}\left[A {\rm e}^{ {\rm i} (k x-\omega t+\varphi)}\right]$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter03slashfig05.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="563" HEIGHT="338" BORDER="0"
 SRC="|."$dir".q|img264.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter03/fig05.eps}}
\end{figure}">|; 

$key = q/Zgg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img913.png"
 ALT="$Z\gg 1$">|; 

$key = q/{displaymath}mu_B=frac{e,hbar}{2,m_e}=5.788times10^{-5},{{rm{eVslashT}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2350.png"
 ALT="\begin{displaymath}
\mu_B = \frac{e \hbar}{2 m_e} = 5.788\times 10^{-5} {\rm eV/T}
\end{displaymath}">|; 

$key = q/d(y^2)=2,y,dy=2,x,X,sintheta,dtheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2792.png"
 ALT="$d(y^2)=2 y dy = 2 x X \sin\theta d\theta$">|; 

$key = q/{{bf{v}=hbar,{{bf{k}slashm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2839.png"
 ALT="${\bf v} = \hbar {\bf k}/m$">|; 

$key = q/l_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1242.png"
 ALT="$l_z$">|; 

$key = q/R=2.5,a_0=1.33times10^{-10},{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2830.png"
 ALT="$R = 2.5 a_0 = 1.33\times 10^{-10} {\rm m}$">|; 

$key = q/beta_{l+};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2951.png"
 ALT="$\beta_{l+}$">|; 

$key = q/sqrt{lambda-y^2}slashy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$\sqrt {\lambda - y^2}/y$">|; 

$key = q/{mitDelta}E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img636.png"
 ALT="${\mit\Delta}E$">|; 

$key = q/psi(x_1,x_2,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1134.png"
 ALT="$\psi(x_1,x_2,t)$">|; 

$key = q/pm1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1920.png"
 ALT="$\pm 1/2$">|; 

$key = q/m_eslashm_psimeq1slash1836;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1584.png"
 ALT="$m_e/m_p\simeq 1/1836$">|; 

$key = q/p_1equivp_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1312.png"
 ALT="$p_1\equiv p_x$">|; 

$key = q/{displaymath}psi(x,t)proptoint_{-infty}^{infty}bar{psi}(k),exp!left[,{{rm{i}left_0)+frac{1}{2},phi_0'',(k-k_0)^{,2}right}right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="514" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img360.png"
 ALT="\begin{displaymath}
\psi(x,t)\propto \int_{-\infty}^{\infty} \bar{\psi}(k) 
\e...
...(k-k_0) + \frac{1}{2} \phi_0'' (k-k_0)^{ 2}\right\}\right],
\end{displaymath}">|; 

$key = q/Vneq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2945.png"
 ALT="$V\neq 0$">|; 

$key = q/dllD;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img268.png"
 ALT="$d\ll D$">|; 

$key = q/J_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1879.png"
 ALT="$J_y$">|; 

$key = q/{displaymath}frac{N_f}{N_i}=frac{exp(-E_fslashk_B,T)}{exp(-E_islashk_B,T)}=exp(,hbar,omega_{if}slashk_B,T).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2610.png"
 ALT="\begin{displaymath}
\frac{N_f}{N_i} = \frac{\exp(-E_f/k_B T)}{\exp(-E_i/k_B T)}
= \exp( \hbar  \omega_{if}/k_B T).
\end{displaymath}">|; 

$key = q/psi=Y_{l,m}(theta,phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1485.png"
 ALT="$\psi=Y_{l,m}(\theta,\phi)$">|; 

$key = q/1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img80.png"
 ALT="$1$">|; 

$key = q/n,l,m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2176.png"
 ALT="$n,l,m$">|; 

$key = q/nu=1slashT=omegaslash2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img161.png"
 ALT="$\nu=1/T=\omega/2\pi$">|; 

$key = q/b_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img770.png"
 ALT="$b_1$">|; 

$key = q/psirightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img963.png"
 ALT="$\psi\rightarrow 0$">|; 

$key = q/0,1,cdots,n-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1666.png"
 ALT="$0,1,\cdots, n-1$">|; 

$key = q/{displaymath}|c_0|^{,2}=1-sum_{n>0}|c_n|^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2699.png"
 ALT="\begin{displaymath}
\vert c_0\vert^{ 2} = 1-\sum_{n&gt;0}\vert c_n\vert^{ 2}.
\end{displaymath}">|; 

$key = q/{displaymath}S^2=S_+,S_-+S_z^{,2}-hbar,S_z.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1762.png"
 ALT="\begin{displaymath}
S^2 = S_+ S_-+S_z^{ 2}-\hbar S_z.
\end{displaymath}">|; 

$key = q/|R|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img819.png"
 ALT="$\vert R\vert^{ 2}$">|; 

$key = q/{displaymath}P(s)=A,s,expleft(-frac{s}{s_0}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img131.png"
 ALT="\begin{displaymath}
P(s) = A s \exp\left(-\frac{s}{s_0}\right),
\end{displaymath}">|; 

$key = q/x<-aslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img962.png"
 ALT="$x&lt;-a/2$">|; 

$key = q/{scriptstylesqrt{(l-m)slash(2,l+1)}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1968.png"
 ALT="${\scriptstyle\sqrt{(l-m)/(2 l+1)}}$">|; 

$key = q/{displaymath}{calR}_0(r)=B,frac{sin(k',r)}{r},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2995.png"
 ALT="\begin{displaymath}
{\cal R}_0(r) = B  \frac{\sin (k' r)}{r},
\end{displaymath}">|; 

$key = q/{displaymath}xequiv{{rm{i},hbar,frac{partial}{partialp}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img599.png"
 ALT="\begin{displaymath}
x\equiv{\rm i} \hbar \frac{\partial}{\partial p}
\end{displaymath}">|; 

$key = q/x,f(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img526.png"
 ALT="$x f(x)$">|; 

$key = q/displaystyle{{rm{i},hbar,L_y.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1324.png"
 ALT="$\displaystyle {\rm i} \hbar L_y.$">|; 

$key = q/{displaymath}frac{d}{dt}int_{-infty}^{infty}|psi(x,t)|^{,2},dx=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img469.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\int_{-\infty}^{\infty}\vert\psi(x,t)\vert^{ 2}  dx = 0,
\end{displaymath}">|; 

$key = q/{displaymath}m_+=+l.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1427.png"
 ALT="\begin{displaymath}
m_+ =+ l.
\end{displaymath}">|; 

$key = q/psi(x,E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1144.png"
 ALT="$\psi(x,E)$">|; 

$key = q/{displaymath}x,delta(x-x')=x',delta(x-x'),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img709.png"
 ALT="\begin{displaymath}
x \delta(x-x') = x' \delta(x-x'),
\end{displaymath}">|; 

$key = q/V_0<E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img848.png"
 ALT="$V_0&lt;E$">|; 

$key = q/displaystylelangle1,0,0|x|2,1,pm1rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2661.png"
 ALT="$\displaystyle \langle 1,0,0\vert x\vert 2,1,\pm 1\rangle$">|; 

$key = q/a_0rightarrowa_0slashZ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2767.png"
 ALT="$a_0\rightarrow a_0/Z$">|; 

$key = q/{displaymath}N=2timesfrac{1}{8}timesfrac{4,pi}{3},l_F^{,3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1247.png"
 ALT="\begin{displaymath}
N = 2\times\frac{1}{8}\times \frac{4 \pi}{3} l_F^{ 3}.
\end{displaymath}">|; 

$key = q/x=L;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img754.png"
 ALT="$x=L$">|; 

$key = q/k',asimeq4.49;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3011.png"
 ALT="$k' a\simeq 4.49$">|; 

$key = q/{displaymath}{{rm{i},hbar,frac{partialpsi}{partialt}=-frac{hbar^2}{2,m},nabla^2psi+V,psi.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1206.png"
 ALT="\begin{displaymath}
{\rm i} \hbar \frac{\partial\psi}{\partial t} = - \frac{\hbar^2}{2 m} 
\nabla^2\psi + V \psi.
\end{displaymath}">|; 

$key = q/|psi|rightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img804.png"
 ALT="$\vert\psi\vert\rightarrow 0$">|; 

$key = q/{displaymath}L_i=epsilon_{ijk},x_j,p_k.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1501.png"
 ALT="\begin{displaymath}
L_i = \epsilon_{ijk} x_j p_k.
\end{displaymath}">|; 

$key = q/displaystylesum_i|c_i|^2,a_i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img689.png"
 ALT="$\displaystyle \sum_i \vert c_i\vert^2 a_i,$">|; 

$key = q/r=|z|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img214.png"
 ALT="$r=\vert z\vert$">|; 

$key = q/v=sqrt{2,Eslashm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img928.png"
 ALT="$v= \sqrt{2 E/m}$">|; 

$key = q/{displaymath}chi_+equivleft(array{{c}10array{right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1796.png"
 ALT="\begin{displaymath}
\chi_+ \equiv \left(\begin{array}{c}1 0\end{array}\right),
\end{displaymath}">|; 

$key = q/phi(k);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img358.png"
 ALT="$\phi(k)$">|; 

$key = q/d^2psislashdx^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img789.png"
 ALT="$d^2\psi/d x^2$">|; 

$key = q/u_0({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2848.png"
 ALT="$u_0({\bf r})$">|; 

$key = q/{mitDelta}xsim10^{-10},{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img395.png"
 ALT="${\mit\Delta} x\sim 10^{-10} {\rm m}$">|; 

$key = q/{displaymath}frac{dx}{dt}=c.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img241.png"
 ALT="\begin{displaymath}
\frac{dx}{dt} = c.
\end{displaymath}">|; 

$key = q/{displaymath}sigma_z,chi_pm=pmchi_pm.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1808.png"
 ALT="\begin{displaymath}
\sigma_z \chi_\pm = \pm \chi_\pm.
\end{displaymath}">|; 

$key = q/H_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2097.png"
 ALT="$H_1$">|; 

$key = q/displaystyle-3,e,a_0,|{{bf{E}|.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2286.png"
 ALT="$\displaystyle -3 e a_0 \vert{\bf E}\vert.$">|; 

$key = q/psi^{(1)}_{l,1slash2;m,pm1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1923.png"
 ALT="$\psi^{(1)}_{l,1/2;m,\pm 1/2}$">|; 

$key = q/PproptoV^{-5slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1267.png"
 ALT="$P\propto V^{-5/3}$">|; 

$key = q/{displaymath}X=frac{m_1,x_1+m_2,x_2}{m_1+m_2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1114.png"
 ALT="\begin{displaymath}
X = \frac{m_1 x_1+m_2 x_2}{m_1+m_2}
\end{displaymath}">|; 

$key = q/{displaymath}int|psi(x_1,x_2,ldots,x_N,t)|^2,dx_1,dx_2ldotsdx_N=1{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="345" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1078.png"
 ALT="\begin{displaymath}
\int \vert\psi(x_1,x_2,\ldots, x_N,t)\vert^2 dx_1 dx_2\ldots dx_N = 1
\end{displaymath}">|; 

$key = q/{displaymath}c_n^{(1)}=-frac{i}{hbar}int_0^tH_{ni}(t'),exp(,{{rm{i},omega_{ni},t'),dt'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2477.png"
 ALT="\begin{displaymath}
c_n^{(1)} = -\frac{i}{\hbar}\int_0^t H_{ni}(t') \exp( {\rm i} \omega_{ni} t') dt'.
\end{displaymath}">|; 

$key = q/x,pneqp,x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img553.png"
 ALT="$x p\neq p x$">|; 

$key = q/{displaymath}frac{d^2psi}{dy^2}-y^2,psisimeq0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1004.png"
 ALT="\begin{displaymath}
\frac{d^2\psi}{dy^2} - y^2 \psi \simeq 0.
\end{displaymath}">|; 

$key = q/{displaymath}DeltaE_{l,1slash2;j,m_j}=frac{e^2,hbar^2}{16pi,epsilon_0,m_e^{,2},cl,(l+1)-3slash4}{l,(l+1slash2),(l+1),n^3}right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="474" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2323.png"
 ALT="\begin{displaymath}
\Delta E_{l,1/2;j,m_j}= \frac{e^2 \hbar^2}{16\pi \epsilon_...
...t[\frac{j (j+1)-l (l+1)-3/4}{l (l+1/2) (l+1) n^3}\right],
\end{displaymath}">|; 

$key = q/{displaymath}L_z,(L_-,Y_{l,m})=(m-1),hbar,(L_-,Y_{l,m}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1376.png"
 ALT="\begin{displaymath}
L_z (L_- Y_{l,m}) = (m-1) \hbar (L_- Y_{l,m}).
\end{displaymath}">|; 

$key = q/displaystyle(u_1-u_2)leftslashsqrt{2},right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img780.png"
 ALT="$\displaystyle (u_1-u_2)\left/\sqrt{2},\right.$">|; 

$key = q/{displaymath}Omega=sqrt{gamma^2+(omega-omega_{21})^2slash4}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img2426.png"
 ALT="\begin{displaymath}
\Omega = \sqrt{\gamma^2 + (\omega-\omega_{21})^2/4}.
\end{displaymath}">|; 

$key = q/{displaymath}P(x,y,z,t)=|psi(x,y,z,t)|^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1178.png"
 ALT="\begin{displaymath}
P(x,y,z,t) = \vert\psi(x,y,z,t)\vert^2.
\end{displaymath}">|; 

$key = q/m=4,m_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img920.png"
 ALT="$m=4 m_p$">|; 

$key = q/{displaymath}P_{l,m}(u)=(-1)^{l},frac{(1-u^2)^{mslash2}}{2^l,l!}left(frac{d}{du}right)^{l+m}(1-u^2)^l,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="382" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1467.png"
 ALT="\begin{displaymath}
P_{l,m}(u) = (-1)^{l} \frac{(1-u^2)^{m/2}}{2^l l!}\left(\frac{d}{du}\right)^{l+m} (1-u^2)^l,
\end{displaymath}">|; 

$key = q/1slash6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img74.png"
 ALT="$1/6$">|; 

$key = q/omega_{nm}=(E_n-E_m)slashhbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2468.png"
 ALT="$\omega_{nm}=(E_n-E_m)/\hbar$">|; 

$key = q/l>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1607.png"
 ALT="$l&gt;0$">|; 

$key = q/E_0(Z)=Z^2,E_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2762.png"
 ALT="$E_0(Z) = Z^2 E_0$">|; 

$key = q/{displaymath}int_0^inftyexp(-mu,r'),sin(q,r'),dr'=frac{q}{q^2+mu^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img2885.png"
 ALT="\begin{displaymath}
\int_0^\infty \exp(-\mu  r')  \sin(q r')   dr' = \frac{q}{q^2+ \mu^2}.
\end{displaymath}">|; 

$key = q/tan(k',a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3004.png"
 ALT="$\tan (k' a)$">|; 

$key = q/displaystyleL^2,psi^{(2)}_{l,s;j,m_j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1912.png"
 ALT="$\displaystyle L^2 \psi^{(2)}_{l,s;j,m_j}$">|; 

$key = q/displaystylelangler^2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1653.png"
 ALT="$\displaystyle \langle r^2\rangle$">|; 

$key = q/';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="9" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1218.png"
 ALT="$'$">|; 

$key = q/|bar{psi}|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img317.png"
 ALT="$\vert\bar{\psi}\vert^{ 2}$">|; 

$key = q/{displaymath}dot{c}_pm=mp{{rm{i},frac{Omega}{2},c_pm.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1840.png"
 ALT="\begin{displaymath}
\dot{c}_\pm = \mp {\rm i} \frac{\Omega}{2} c_\pm.
\end{displaymath}">|; 

$key = q/0leqrleqa;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1538.png"
 ALT="$0\leq r \leq a$">|; 

$key = q/displaystyleint_{-infty}^inftyP(u),du,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img123.png"
 ALT="$\displaystyle \int_{-\infty}^\infty P(u) du,$">|; 

$key = q/displaystyleS_2^{,2},chi_{s_1,s_2;s,m_s}^{(2)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2052.png"
 ALT="$\displaystyle S_2^{ 2}  \chi_{s_1,s_2;s,m_s}^{(2)}$">|; 

$key = q/{displaymath}psi_{nlm}^{(1)}=sum_{k=1,N_n}langlen,k,m|n,l^{(1)},mrangle,psi_{nkm}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2246.png"
 ALT="\begin{displaymath}
\psi_{nlm}^{(1)}= \sum_{k=1,N_n}\langle n,k,m\vert n,l^{(1)},m\rangle \psi_{nkm}.
\end{displaymath}">|; 

$key = q/2pi,a^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2987.png"
 ALT="$2\pi  a^2$">|; 

$key = q/sigma_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3030.png"
 ALT="$\sigma_l$">|; 

$key = q/{displaymath}psi(x_1,x_2,t)=psi_{x'}(x',t),psi_X(X,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1122.png"
 ALT="\begin{displaymath}
\psi(x_1,x_2,t) = \psi_{x'}(x',t) \psi_X(X,t).
\end{displaymath}">|; 

$key = q/displaystylegamma,expleft[-{{rm{i},(omega-omega_{21}),tright],c_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2416.png"
 ALT="$\displaystyle \gamma \exp\left[-{\rm i} (\omega-\omega_{21}) t\right] c_1,$">|; 

$key = q/N(t)gg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img932.png"
 ALT="$N(t)\gg 1$">|; 

$key = q/z=r,costheta+{{rm{i},r,sintheta=r,(costheta+{{rm{i},sintheta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="333" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img213.png"
 ALT="$z=r \cos\theta + {\rm i} r \sin\theta= r (\cos\theta+{\rm i} \sin\theta)$">|; 

$key = q/y+{mitDelta}y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img279.png"
 ALT="$y+{\mit\Delta}y$">|; 

$key = q/y^i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1012.png"
 ALT="$y^i$">|; 

$key = q/{displaymath}alphasimeqfrac{D}{2,f},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img415.png"
 ALT="\begin{displaymath}
\alpha\simeq \frac{D}{2 f},
\end{displaymath}">|; 

$key = q/S_1^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2030.png"
 ALT="$S_1^{ 2}$">|; 

$key = q/{displaymath}frac{d}{dt}int_{-infty}^{infty}psi^{ast},psi,dx=int_{-infty}^{inftyi+psi^ast,frac{partialpsi}{partialt}right),dx=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="396" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img470.png"
 ALT="\begin{displaymath}
\frac{d}{dt}\int_{-\infty}^{\infty}\psi^{\ast} \psi dx=
\i...
...\psi
+\psi^\ast \frac{\partial\psi}{\partial t}\right) dx=0.
\end{displaymath}">|; 

$key = q/displaystylesqrt{frac{2}{3}},psi^{(1)}_{-1,1slash2}-sqrt{frac{1}{3}},psi^{(1)}_{0,-1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1990.png"
 ALT="$\displaystyle \sqrt{\frac{2}{3}} \psi^{(1)}_{-1,1/2}
-\sqrt{ \frac{1}{3}} \psi^{(1)}_{0,-1/2},$">|; 

$key = q/langlex^2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img746.png"
 ALT="$\langle x^2\rangle$">|; 

$key = q/displaystyle-frac{1}{2,m_e,c^2}left[E_n^{,2}+2,E_nleft(frac{e^2}{4piepsilon_0}riepsilon_0}right)^2leftlanglefrac{1}{r^{,2}}rightrangleright].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2300.png"
 ALT="$\displaystyle -\frac{1}{2 m_e c^2}\left[
E_n^{ 2} + 2 E_n\left(\frac{e^2}{4...
...e^2}{4\pi\epsilon_0}\right)^2\left\langle\frac{1}{r^{ 2}}\right\rangle\right].$">|; 

$key = q/{displaymath}sigma_{{rm{total}=frac{4pi}{k^2},sin^2delta_0=4pi,a^2left(frac{1}{k^2,a^2}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img3018.png"
 ALT="\begin{displaymath}
\sigma_{\rm total} = \frac{4\pi}{k^2}  \sin^2\delta_0 = 4\pi  a^2
\left(\frac{1}{k^2  a^2}\right).
\end{displaymath}">|; 

$key = q/{displaymath}c_n^{(0)}(t)=delta_{ni}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2474.png"
 ALT="\begin{displaymath}
c_n^{(0)} (t) = \delta_{ni}.
\end{displaymath}">|; 

$key = q/{displaymath}E_y(x,t)={{rm{Re}[psi(x,t)]=|bar{psi}|,cos(k,x-omega,t+varphi),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="375" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img231.png"
 ALT="\begin{displaymath}
E_y(x,t) = {\rm Re}[\psi(x,t)] = \vert\bar{\psi}\vert \cos(k x-\omega t +\varphi),
\end{displaymath}">|; 

$key = q/l,m=0,0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="$l,m=0,0$">|; 

$key = q/{displaymath}delta_0simeqk,aleft[frac{tan(k',a)}{k',a}-1right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img3006.png"
 ALT="\begin{displaymath}
\delta_0 \simeq k a \left[ \frac{\tan( k' a)}{k' a} -1\right].
\end{displaymath}">|; 

$key = q/l_F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1246.png"
 ALT="$l_F$">|; 

$key = q/{displaymath}epsilon=frac{2,E}{hbar,omega}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1000.png"
 ALT="\begin{displaymath}
\epsilon = \frac{2 E}{\hbar \omega}.
\end{displaymath}">|; 

$key = q/displaystyle-frac{cosz}{z^2}-frac{sinz}{z}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1558.png"
 ALT="$\displaystyle - \frac{\cos z}{z^2} - \frac{\sin z}{z}.$">|; 

$key = q/{displaymath}langlef(u)+g(u)rangle=sum_{i=1}^{M}P(u_i),[f(u_i)+g(u_i)]=sum_{i=1}^{M}P(u_i),f(u_i)+sum_{i=1}^{M}P(u_i),g(u_i),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="580" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img106.png"
 ALT="\begin{displaymath}
\langle f(u)+g(u)\rangle = \sum_{i=1}^{M}P(u_i) [f(u_i)+g(u...
...
= \sum_{i=1}^{M}P(u_i) f(u_i)+ \sum_{i=1}^{M}P(u_i) g(u_i),
\end{displaymath}">|; 

$key = q/f(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img579.png"
 ALT="$f(0)$">|; 

$key = q/-j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2351.png"
 ALT="$-j$">|; 

$key = q/{displaymath}leftlangle(u-langleurangle)^2rightrangle=leftlangle(u^2-2,u,langleue-2,langleurangle,langleurangle+langleurangle^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="520" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img113.png"
 ALT="\begin{displaymath}
\left \langle (u-\langle u\rangle )^2\right\rangle = \left\l...
...ngle-2 \langle u\rangle \langle u\rangle+\langle u\rangle^2,
\end{displaymath}">|; 

$key = q/displaystylepsi^{(2)}_{3slash2,pm3slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1994.png"
 ALT="$\displaystyle \psi^{(2)}_{3/2, \pm 3/2},$">|; 

$key = q/omegaslash|k|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img322.png"
 ALT="$\omega/\vert k\vert$">|; 

$key = q/V_0llE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img853.png"
 ALT="$V_0\ll E$">|; 

$key = q/E<V_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img855.png"
 ALT="$E&lt; V_0$">|; 

$key = q/{displaymath}|psi(x)|^{,2}=|psi_1|^{,2},expleft(-2,int_{x_1}^x|k(x')|,dx'right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="327" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img890.png"
 ALT="\begin{displaymath}
\vert\psi(x)\vert^{ 2} = \vert\psi_1\vert^{ 2} \exp\left(-2 \int_{x_1}^x \vert k(x')\vert dx'\right),
\end{displaymath}">|; 

$key = q/r'slashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2854.png"
 ALT="$r'/r$">|; 

$key = q/E<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img798.png"
 ALT="$E&lt;0$">|; 

$key = q/psi_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2825.png"
 ALT="$\psi_+$">|; 

$key = q/k_{min}=l+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1605.png"
 ALT="$k_{min}=l+1$">|; 

$key = q/O+O^nomath_inline{dagnomath_inline{;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img548.png"
 ALT="$O+O^\dag $">|; 

$key = q/{displaymath}H(t)=H_0+H_1(t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2387.png"
 ALT="\begin{displaymath}
H(t) = H_0 + H_1(t).
\end{displaymath}">|; 

$key = q/|E_0|llm_e,c^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1626.png"
 ALT="$\vert E_0\vert\ll m_e c^2$">|; 

$key = q/v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img188.png"
 ALT="$v$">|; 

$key = q/displaystylepsi^{(2)}_{1slash2,-1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1989.png"
 ALT="$\displaystyle \psi^{(2)}_{1/2,-1/2}$">|; 

$key = q/E>-V_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img972.png"
 ALT="$E&gt; -V_0$">|; 

$key = q/{displaymath}A,psi_i=a_i,psi_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img679.png"
 ALT="\begin{displaymath}
A \psi_i = a_i \psi_i,
\end{displaymath}">|; 

$key = q/k=k_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img357.png"
 ALT="$k=k_0$">|; 

$key = q/phi(p,t)=bar{psi}(k,t)slashsqrt{hbar};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img573.png"
 ALT="$\phi(p,t)= \bar{\psi}(k,t)/\sqrt{\hbar}$">|; 

$key = q/textstylerightarrow;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2453.png"
 ALT="$\textstyle \rightarrow$">|; 

$key = q/{displaymath}langlef(x)rangle=int_{-infty}^{infty}f(x),|psi|^{,2},dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img491.png"
 ALT="\begin{displaymath}
\langle f(x)\rangle = \int_{-\infty}^{\infty} f(x) \vert\psi\vert^{ 2} dx.
\end{displaymath}">|; 

$key = q/I;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1822.png"
 ALT="$I$">|; 

$key = q/displaystyle0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1083.png"
 ALT="$\displaystyle 0,$">|; 

$key = q/2,pm2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="$2,\pm 2$">|; 

$key = q/displaystyler,costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1350.png"
 ALT="$\displaystyle r \cos\theta.$">|; 

$key = q/displaystyley,p_z-z,p_y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1305.png"
 ALT="$\displaystyle y p_z - z p_y,$">|; 

$key = q/displaystylesum_{n',l',m'}langle1,0,0|z|n',l',m'rangle,langlen',l',m'|z|1,0,0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2229.png"
 ALT="$\displaystyle \sum_{n',l',m'}\langle 1,0,0\vert z\vert n',l',m'\rangle \langle n',l',m'\vert z\vert 1,0,0\rangle$">|; 

$key = q/{displaymath}M=m_1+m_2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1118.png"
 ALT="\begin{displaymath}
M = m_1+ m_2
\end{displaymath}">|; 

$key = q/{displaymath}frac{d^2psi}{dx^2}=-k^2,psi,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img814.png"
 ALT="\begin{displaymath}
\frac{d^2 \psi}{d x^2} = - k^2 \psi,
\end{displaymath}">|; 

$key = q/displaystylefrac{e,B}{2,m_e},left(m_j,hbar+langlel,1slash2;j,m_j|S_z|l,1slash2;j,m_jrangleright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2342.png"
 ALT="$\displaystyle \frac{e B}{2 m_e} \left(m_j \hbar + \langle l,1/2;j,m_j\vert S_z\vert l,1/2;j,m_j\rangle\right),$">|; 

$key = q/displaystylealpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img371.png"
 ALT="$\displaystyle \alpha$">|; 

$key = q/{displaymath}H=frac{p^2}{2,m_e}-frac{e,{{bf{A}!cdot!{{bf{p}}{m_e}+frac{e^2A^2}{2,m_e}+e,phi+V_0(r).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2536.png"
 ALT="\begin{displaymath}
H = \frac{p^2}{2 m_e} -\frac{e {\bf A}\!\cdot\!{\bf p}}{m_e}
+\frac{ e^2 A^2}{2 m_e}+ e  \phi + V_0(r).
\end{displaymath}">|; 

$key = q/|R|^{,2}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img851.png"
 ALT="$\vert R\vert^{ 2}=0$">|; 

$key = q/{displaymath}D=frac{4}{X},int_0^X{{rm{e}^{-2,x},x^2,dx+4int_X^infty{{rm{e}^{-2,x},x,dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="327" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2812.png"
 ALT="\begin{displaymath}
D =\frac{4}{X} \int_0^X {\rm e}^{-2 x} x^2 dx
+ 4\int_X^\infty {\rm e}^{-2 x} x dx,
\end{displaymath}">|; 

$key = q/displaystyle{{rm{i},k,(1-R);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img858.png"
 ALT="$\displaystyle {\rm i} k (1-R)$">|; 

$key = q/j=l+1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1953.png"
 ALT="$j=l+1/2$">|; 

$key = q/{displaymath}frac{partial|psi|^2}{partialt}+frac{partialj}{partialx}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1180.png"
 ALT="\begin{displaymath}
\frac{\partial\vert\psi\vert^2}{\partial t} + \frac{\partial j}{\partial x} = 0,
\end{displaymath}">|; 

$key = q/E>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img797.png"
 ALT="$E&gt;0$">|; 

$key = q/displaystylek^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img953.png"
 ALT="$\displaystyle k^2$">|; 

$key = q/psi^{(2)}_{3slash2,1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2007.png"
 ALT="$\psi^{(2)}_{3/2,1/2}$">|; 

$key = q/M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img77.png"
 ALT="$M$">|; 

$key = q/pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1575.png"
 ALT="$\pi$">|; 

$key = q/{displaymath}psi(x,t)=int_{-infty}^{infty}bar{psi}(k),{{rm{e}^{,{{rm{i},(k,x-omega,t)},dk,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img427.png"
 ALT="\begin{displaymath}
\psi(x,t)=\int_{-\infty}^{\infty}\bar{\psi}(k) {\rm e}^{ {\rm i} (k x-\omega t)} dk,
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{x,X}int_{|x-X|}^{x+X}{{rm{e}^{-y},y,dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2794.png"
 ALT="$\displaystyle \frac{1}{x X}\int_{\vert x-X\vert}^{x+X}
{\rm e}^{-y} y dy$">|; 

$key = q/{displaymath}psi_{E,boson}(x_1,x_2)=frac{1}{sqrt{2}}left[psi(x_1,E_a),psi(x_2,E_b)+psi(x_2,E_a),psi(x_1,E_b)right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="523" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1145.png"
 ALT="\begin{displaymath}
\psi_{E boson}(x_1,x_2) = \frac{1}{\sqrt{2}}\left[\psi(x_1,E_a) \psi(x_2,E_b)+\psi(x_2,E_a) \psi(x_1,E_b)\right],
\end{displaymath}">|; 

$key = q/P^{abs}_{irightarrowf}ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2573.png"
 ALT="$P^{abs}_{i\rightarrow f}\ll 1$">|; 

$key = q/0leql<n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2241.png"
 ALT="$0\leq l &lt; n$">|; 

$key = q/{displaymath}langleArangle=sum_{i,j}c_i^ast,c_j,A_{ij}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2110.png"
 ALT="\begin{displaymath}
\langle A\rangle = \sum_{i,j} c_i^\ast c_j A_{ij}.
\end{displaymath}">|; 

$key = q/epsilon_{321}=epsilon_{132}=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1506.png"
 ALT="$\epsilon_{321}=\epsilon_{132}=-1$">|; 

$key = q/{displaymath}V(x)=left{array{{lcl}V_0&mbox{hspace{1cm}}&mbox{for{0leqxleqa{}[0.5ex]0&&mbox{otherwise}array{right.,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="307" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img812.png"
 ALT="\begin{displaymath}
V(x) = \left\{\begin{array}{lcl}
V_0&amp;\mbox{\hspace{1cm}}&amp;\mb...
...eq x\leq a$} [0.5ex]
0&amp;&amp;\mbox{otherwise}
\end{array}\right.,
\end{displaymath}">|; 

$key = q/{displaymath}{{rm{i},hbar,frac{dc_n^{(1)}}{dt}=sum_mH_{nm},exp(,{{rm{i},omega_{n,t),c_m^{(0)}=H_{ni},exp(,{{rm{i},omega_{ni},t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="437" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img2475.png"
 ALT="\begin{displaymath}
{\rm i} \hbar \frac{dc_n^{(1)}}{dt} = \sum_m H_{nm} \exp(...
...{nm} t) c_m^{(0)} = H_{ni} \exp( {\rm i} \omega_{ni} t),
\end{displaymath}">|; 

$key = q/r=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1539.png"
 ALT="$r=0$">|; 

$key = q/u_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img96.png"
 ALT="$u_1$">|; 

$key = q/{displaymath}epsilon=2,n+1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1019.png"
 ALT="\begin{displaymath}
\epsilon = 2 n+1,
\end{displaymath}">|; 

$key = q/q^{,2}=(2mslashhbar^2),(E-W);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1059.png"
 ALT="$q^{ 2}= (2m/\hbar^2) (E-W)$">|; 

$key = q/{displaymath}{{rm{i},hbar,frac{dc_n(t)}{dt}=sum_mH_{nm}(t),expleft(,{{rm{i},omega_{nm},tright)c_m(t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2406.png"
 ALT="\begin{displaymath}
{\rm i} \hbar \frac{dc_n(t)}{dt} = \sum_m H_{nm}(t) \exp\left( {\rm i} 
\omega_{nm} t\right)c_m(t),
\end{displaymath}">|; 

$key = q/{displaymath}int_{-1}^1P_n(mu),P_m(mu),dmu=frac{delta_{nm}}{n+1slash2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2917.png"
 ALT="\begin{displaymath}
\int_{-1}^1 P_n(\mu)  P_m(\mu) d\mu = \frac{\delta_{nm}}{n+1/2},
\end{displaymath}">|; 

$key = q/{displaymath}int_{-infty}^inftypsi_1^{ast},(O,psi_2),dx=int_{-infty}^infty(O,psi_1)^ast,psi_2,dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img602.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty \psi_1^{\ast} (O \psi_2) dx = \int_{-\infty}^\infty
(O \psi_1)^\ast \psi_2 dx,
\end{displaymath}">|; 

$key = q/{displaymath}c_i=psi_i^nomath_inline{dagnomath_inline{,psi_a.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2106.png"
 ALT="\begin{displaymath}
c_i = \psi_i^\dag \psi_a.
\end{displaymath}">|; 

$key = q/{displaymath}DeltaE=-frac{1}{2},alpha,|{{bf{E}|^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img2216.png"
 ALT="\begin{displaymath}
\Delta E = -\frac{1}{2} \alpha \vert{\bf E}\vert^2.
\end{displaymath}">|; 

$key = q/{{bf{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2307.png"
 ALT="${\bf v}$">|; 

$key = q/{displaymath}H=H_0+V({{bf{r}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2835.png"
 ALT="\begin{displaymath}
H = H_0 + V({\bf r}),
\end{displaymath}">|; 

$key = q/f(u);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img102.png"
 ALT="$f(u)$">|; 

$key = q/omega=sqrt{Kslashm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img998.png"
 ALT="$\omega = \sqrt{K/m}$">|; 

$key = q/displaystylep_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1197.png"
 ALT="$\displaystyle p_z$">|; 

$key = q/{displaymath}({{bf{x}_{nl})_k=langlen,k,m|n,l^{(1)},mrangle,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img2259.png"
 ALT="\begin{displaymath}
({\bf x}_{nl})_k = \langle n,k,m\vert n,l^{(1)},m\rangle,
\end{displaymath}">|; 

$key = q/tany=infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img985.png"
 ALT="$\tan y =\infty$">|; 

$key = q/displaystylefrac{(k,r)^l}{(2,l+1)!!},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2973.png"
 ALT="$\displaystyle \frac{(k r)^l}{(2 l+1)!!},$">|; 

$key = q/F_-(X);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="$F_-(X)$">|; 

$key = q/{displaymath}psi_p(x,p')propto{{rm{e}^{+{{rm{i},p',xslashhbar}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img723.png"
 ALT="\begin{displaymath}
\psi_p(x,p') \propto {\rm e}^{+{\rm i} p' x/\hbar}.
\end{displaymath}">|; 

$key = q/{displaymath}leftlangle|mbox{{boldmath{{epsilon{}!cdot!{{bf{d}_{if}|^{,2}rightrangle_{av}=frac{d_{if}^{,2}}{3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2594.png"
 ALT="\begin{displaymath}
\left\langle \vert\mbox{\boldmath$\epsilon$}\!\cdot\!{\bf d}_{if}\vert^{ 2}\right\rangle_{av} = \frac{d_{if}^{ 2}}{3}.
\end{displaymath}">|; 

$key = q/{displaymath}psi_a=sum_ic_i,psi_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2103.png"
 ALT="\begin{displaymath}
\psi_a = \sum_i c_i \psi_i.
\end{displaymath}">|; 

$key = q/m_e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1251.png"
 ALT="$m_e$">|; 

$key = q/{displaymath}psi(x,t)=int_{-infty}^{infty}bar{psi}(k),{{rm{e}^{,{{rm{i},(k,x-omega,t)},dk.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img318.png"
 ALT="\begin{displaymath}
\psi(x,t) = \int_{-\infty}^{\infty} \bar{\psi}(k) {\rm e}^{ {\rm i} (k x-\omega t)} dk.
\end{displaymath}">|; 

$key = q/{{bf{q}={{bf{k}-{{bf{k}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2876.png"
 ALT="${\bf q} = {\bf k}-{\bf k}'$">|; 

$key = q/-V_0<E<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img981.png"
 ALT="$-V_0&lt; E &lt; 0$">|; 

$key = q/{displaymath}A,chi_{a}=a,chi_{a}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1736.png"
 ALT="\begin{displaymath}
A \chi_{a} = a \chi_{a}.
\end{displaymath}">|; 

$key = q/displaystyle1-P_2(t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2430.png"
 ALT="$\displaystyle 1-P_2(t),$">|; 

$key = q/H_2^+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2776.png"
 ALT="$H_2^+$">|; 

$key = q/{displaymath}lambda=21.1,{{rm{cm}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img2386.png"
 ALT="\begin{displaymath}
\lambda = 21.1 {\rm cm}.
\end{displaymath}">|; 

$key = q/hbar,omega_{nm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2473.png"
 ALT="$\hbar \omega_{nm}$">|; 

$key = q/{displaymath}psi=sum_ic_i,psi_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img685.png"
 ALT="\begin{displaymath}
\psi = \sum_i c_i \psi_i,
\end{displaymath}">|; 

$key = q/{displaymath}ointpsi^ast(theta,phi),psi(theta,phi),dOmegageq0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1405.png"
 ALT="\begin{displaymath}
\oint \psi^\ast(\theta,\phi) \psi(\theta,\phi) d\Omega \geq 0,
\end{displaymath}">|; 

$key = q/{displaymath}Y_{l,-l+2}(theta,phi)simL_+,Y_{l,-l+1}simfrac{{{rm{e}^{-{{rm{i},(l-intheta}frac{d}{dtheta}right)^2(sintheta)^{2,l}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="470" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1456.png"
 ALT="\begin{displaymath}
Y_{l,-l+2}(\theta,\phi) \sim L_+ Y_{l,-l+1}\sim \frac{{\rm ...
...{1}{\sin\theta}\frac{d}{d\theta}\right)^2
(\sin\theta)^{2 l}.
\end{displaymath}">|; 

$key = q/{displaymath}psi_{E,dist}(x_1,x_2)=psi(x_1,E_a),psi(x_2,E_b).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1150.png"
 ALT="\begin{displaymath}
\psi_{E dist}(x_1,x_2) = \psi(x_1,E_a) \psi(x_2,E_b).
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{j}'=n,frac{|f({{bf{k},{{bf{k}')|^2}{r^2},{{bf{v}',{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img2865.png"
 ALT="\begin{displaymath}
{\bf j}' = n \frac{\vert f({\bf k},{\bf k}')\vert^2}{r^2} {\bf v}',
\end{displaymath}">|; 

$key = q/|{{bf{k}'|=|{{bf{k}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2871.png"
 ALT="$\vert{\bf k}'\vert=\vert{\bf k}\vert$">|; 

$key = q/E_n-E_0>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2701.png"
 ALT="$E_n-E_0&gt;0$">|; 

$key = q/psi_1=psi_2=psi_a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img655.png"
 ALT="$\psi_1=\psi_2=\psi_a$">|; 

$key = q/{displaymath}psi_0(x)=frac{{{rm{e}^{-x^2slash2,d^2}}{pi^{1slash4},sqrt{d}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1026.png"
 ALT="\begin{displaymath}
\psi_0(x) = \frac{{\rm e}^{-x^2/2 d^2}}{\pi^{1/4} \sqrt{d}},
\end{displaymath}">|; 

$key = q/{displaymath}langleHrangle=8,E_0-frac{5}{2},E_0=frac{11}{2},E_0=-74.8,{{rm{eV}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="325" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img2753.png"
 ALT="\begin{displaymath}
\langle H\rangle = 8 E_0 - \frac{5}{2} E_0 = \frac{11}{2} E_0 = -74.8 {\rm eV}.
\end{displaymath}">|; 

$key = q/3times10^{-10},{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img450.png"
 ALT="$3\times 10^{-10} {\rm m}$">|; 

$key = q/psi_1(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img603.png"
 ALT="$\psi_1(x)$">|; 

$key = q/{displaymath}frac{k}{|k'|}ggk^{-1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img881.png"
 ALT="\begin{displaymath}
\frac{k}{\vert k'\vert}\gg k^{-1}.
\end{displaymath}">|; 

$key = q/{displaymath}x=j,(j+1)-l,(l+1)-3slash4.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1948.png"
 ALT="\begin{displaymath}
x = j (j+1) - l (l+1) - 3/4.
\end{displaymath}">|; 

$key = q/displaystyleS_z,psi^{(1)}_{l,s;m,m_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1905.png"
 ALT="$\displaystyle S_z \psi^{(1)}_{l,s;m,m_s}$">|; 

$key = q/|c_2|^2=1-x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img694.png"
 ALT="$\vert c_2\vert^2=1-x$">|; 

$key = q/displaystyle[l,(l+1)-m,(m+1)],hbar^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1385.png"
 ALT="$\displaystyle [l (l+1)-m (m+1)] \hbar^2,$">|; 

$key = q/{displaymath}nusimeq2times10^{28},,{{rm{yr}^{-1}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img930.png"
 ALT="\begin{displaymath}
\nu\simeq 2\times 10^{28}  {\rm yr}^{-1}
\end{displaymath}">|; 

$key = q/{displaymath}f({{bf{k},{{bf{k}')simeqfrac{m}{2pi,hbar^2}int{{rm{e}^{,{{rm{i},({{-{{bf{k'})cdot{{bf{r}'},V({{bf{r'}),d^3{{bf{r}'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img2875.png"
 ALT="\begin{displaymath}
f({\bf k},{\bf k}') \simeq \frac{m}{2\pi \hbar^2}
\int {\rm...
...} ({\bf k}-{\bf k'})\cdot{\bf r}'} V({\bf r'}) d^3{\bf r}'.
\end{displaymath}">|; 

$key = q/displaystylelangleArangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img688.png"
 ALT="$\displaystyle \langle A\rangle$">|; 

$key = q/m=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1574.png"
 ALT="$m=2$">|; 

$key = q/V_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="$V_0$">|; 

$key = q/s=1slash2,m_s=pm1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1937.png"
 ALT="$s=1/2, m_s=\pm 1/2$">|; 

$key = q/displaystyleomega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img367.png"
 ALT="$\displaystyle \omega_0$">|; 

$key = q/{displaymath}Y_{l,l-1}sim{{rm{e}^{,{{rm{i},(l-1),phi}left(frac{d}{dtheta}+l,cotthetaright)(sintheta)^l.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1447.png"
 ALT="\begin{displaymath}
Y_{l,l-1}\sim {\rm e}^{ {\rm i} (l-1) \phi}\left(\frac{d}{d\theta} +l \cot\theta\right)(\sin\theta)^l.
\end{displaymath}">|; 

$key = q/displaystyles,(s+1),hbar^{,2},chi_{s,m_s}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1745.png"
 ALT="$\displaystyle s (s+1) \hbar^{ 2} \chi_{s,m_s}.$">|; 

$key = q/psi-psi_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2864.png"
 ALT="$\psi-\psi_0$">|; 

$key = q/x=pmaslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img973.png"
 ALT="$x=\pm a/2$">|; 

$key = q/{displaymath}sigma_lsimeqfrac{4pi}{k^2},(2,l+1),frac{Gamma^2slash4}{(E-E_0)^2+Gamma^2slash4}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img3029.png"
 ALT="\begin{displaymath}
\sigma_l \simeq \frac{4\pi}{k^2}  (2 l+1) 
\frac{\Gamma^2/4}{(E-E_0)^2 + \Gamma^2/4}.
\end{displaymath}">|; 

$key = q/displaystylem_s,hbar,chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2048.png"
 ALT="$\displaystyle m_s \hbar
 \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}.$">|; 

$key = q/k_{min}=-l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1604.png"
 ALT="$k_{min}=-l$">|; 

$key = q/displaystylelanglen,l,m|H_1|n,l,mrangle=-frac{1}{8,m_e^{,3},c^2},langlen,l,m|p^4|n,l,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="401" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2293.png"
 ALT="$\displaystyle \langle n,l,m\vert H_1\vert n,l,m\rangle = - \frac{1}{8 m_e^{ 3} c^2} 
\langle n,l,m\vert p^4\vert n,l,m\rangle$">|; 

$key = q/d_{if};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2614.png"
 ALT="$d_{if}$">|; 

$key = q/{displaymath}{mitDeltax},{mitDelta}psimh,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img426.png"
 ALT="\begin{displaymath}
{\mit\Delta x} {\mit\Delta} p\sim h,
\end{displaymath}">|; 

$key = q/mu=costheta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2739.png"
 ALT="$\mu=\cos\theta_1$">|; 

$key = q/{displaymath}{mitDelta}y=frac{D,lambda}{d}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img277.png"
 ALT="\begin{displaymath}
{\mit\Delta}y = \frac{D \lambda}{d}.
\end{displaymath}">|; 

1;

