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


$key = q/displaystyleP_{irightarrown}(t)=frac{|H_{in}|^{,2},t^2}{hbar^2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1908.png"
 ALT="$\displaystyle P_{i\rightarrow n} (t) = \frac{\vert H_{in}\vert^{ 2} t^2}{\hbar^2},$">|; 

$key = q/displaystylefrac{dp_i}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img244.png"
 ALT="$\displaystyle \frac{d p_i}{dt}$">|; 

$key = q/psi_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1247.png"
 ALT="$ \psi_-$">|; 

$key = q/|s,jrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1270.png"
 ALT="$ \vert s, j\rangle$">|; 

$key = q/displaystylelanglep_x'|rangle=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img405.png"
 ALT="$\displaystyle \langle p_x'\vert \rangle = 1.$">|; 

$key = q/displaystylefrac{dP_{l,m}}{dxi}=-frac{1}{sqrt{1-xi^{,2}}},P_{l,,m+1}-frac{m,xi}{1)}{sqrt{1-xi^{,2}}},P_{l,,m-1}+frac{m,xi}{1-xi^{,2}},P_{l,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="616" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img917.png"
 ALT="$\displaystyle \frac{d P_{l m}}{d\xi} = -\frac{1}{\sqrt{1-\xi^{ 2}}} P_{l  ...
...+1)}{\sqrt{1-\xi^{ 2}}} P_{l  m-1} + \frac{m \xi} {1-\xi^{ 2}}  P_{l m}$">|; 

$key = q/displaystyle{rmi},hbar,frac{d|At_0rangle}{dt_0}={rmi},hbarlim_{trightarrowt_0}fre}{t-t_0}={rmi},hbar,tau(t_0),|At_0rangle=H(t_0),|At_0rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="459" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img552.png"
 ALT="$\displaystyle {\rm i} \hbar  \frac{d\vert At_0\rangle}{dt_0} = {\rm i} \hbar...
...0} = {\rm i} \hbar \tau(t_0) \vert At_0\rangle = H(t_0)  \vert At_0\rangle.$">|; 

$key = q/displaystyle{mbox{boldmathmu}}=-frac{e,{bfS}}{m_e}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1614.png"
 ALT="$\displaystyle {\mbox{\boldmath$\mu$}} = - \frac{e  {\bf S}}{m_e}.$">|; 

$key = q/displaystyle[u,v]_{qm}=frac{[u,v]}{{rmi},hbar}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img292.png"
 ALT="$\displaystyle [u, v]_{qm} = \frac{[u, v]}{{\rm i}  \hbar}.$">|; 

$key = q/displaystyleS^2,|s,s_zrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1022.png"
 ALT="$\displaystyle S^2  \vert s, s_z\rangle$">|; 

$key = q/m=n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1487.png"
 ALT="$ m=n$">|; 

$key = q/displaystylep_x={rmi},hbar,lim_{xrightarrowx_0}frac{D(x,x_0)-1}{x-x_0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img556.png"
 ALT="$\displaystyle p_x = {\rm i} \hbar \lim_{x\rightarrow x_0} \frac{D(x,x_0)-1}{x-x_0},$">|; 

$key = q/displaystylelangle2|,H_1,|2rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1450.png"
 ALT="$\displaystyle \langle 2 \vert H_1  \vert 2\rangle,$">|; 

$key = q/displaystyle{bfS}={bfS}_e+{bfS}_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1746.png"
 ALT="$\displaystyle {\bf S} = {\bf S}_e + {\bf S}_p$">|; 

$key = q/{mitDelta}E=|E_n-E_i|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1906.png"
 ALT="$ {\mit\Delta} E = \vert E_n - E_i\vert$">|; 

$key = q/displaystyleL^2(L^+|l,mrangle)=hbar^2,f(l,m),(L^+|l,mrangle),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img793.png"
 ALT="$\displaystyle L^2 (L^+ \vert l, m\rangle) = \hbar^2  f(l,m)  (L^+ \vert l,m\rangle),$">|; 

$key = q/{bfS}cdot{bfL}=(hbarslash2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1199.png"
 ALT="$ {\bf S}\cdot {\bf L} = (\hbar/2) $">|; 

$key = q/displaystylecdot{bfL});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2801.png"
 ALT="$\displaystyle \cdot{\bf L})$">|; 

$key = q/displaystylelangleq_1'cdotsq_N'|,frac{partial}{partialq_i}=frac{partial}{partialq_i'},langleq_1'cdotsq_N'|,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img397.png"
 ALT="$\displaystyle \langle q_1'\cdots q_N'\vert  \frac{\partial}{\partial q_i} = \frac{\partial}{\partial q_i'}  \langle q_1'\cdots q_N'\vert,$">|; 

$key = q/S_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1173.png"
 ALT="$ S_k$">|; 

$key = q/m_1+m_2=m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1407.png"
 ALT="$ m_1+m_2=m$">|; 

$key = q/H_0,L^2,S^2,L_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1619.png"
 ALT="$ H_0, L^2, S^2, L_z$">|; 

$key = q/{bfx}equiv(x_1,x_2,x_3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img615.png"
 ALT="$ {\bf x}\equiv
(x_1, x_2, x_3)$">|; 

$key = q/displaystylexi=xi^{dag}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img102.png"
 ALT="$\displaystyle \xi= \xi^{\dag }.$">|; 

$key = q/displaystylelanglen'|,x,|nrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img697.png"
 ALT="$\displaystyle \langle n'\vert x \vert n\rangle$">|; 

$key = q/displaystyleS^2=S_x^{,2}+S_y^{,2}+S_z^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1015.png"
 ALT="$\displaystyle S^2 = S_x^{ 2}+S_y^{ 2} + S_z^{ 2}.$">|; 

$key = q/displaystylelangle-|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1148.png"
 ALT="$\displaystyle \langle - \vert$">|; 

$key = q/displaystyle=+hbar,L^-,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img777.png"
 ALT="$\displaystyle =+\hbar L^-,$">|; 

$key = q/displaystyle=sum_{xi'}langleA|xi'ranglelanglexi'|,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img135.png"
 ALT="$\displaystyle = \sum_{\xi'} \langle A\vert\xi'\rangle \langle\xi'\vert,$">|; 

$key = q/g_{mu,nu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2555.png"
 ALT="$ g_{\mu \nu}$">|; 

$key = q/{mitDelta}E_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1984.png"
 ALT="$ {\mit\Delta} E_i$">|; 

$key = q/m_1=1slash2,m_2=1slash2,j=1,m=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1404.png"
 ALT="$ m_1=1/2, m_2=1/2, j=1, m=1$">|; 

$key = q/displaystyleP_2(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1819.png"
 ALT="$\displaystyle P_2(t)$">|; 

$key = q/displaystyle|x'rangle,|x''rangle,|x'''rangle,~ldots,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img112.png"
 ALT="$\displaystyle \vert x'\rangle, \vert x''\rangle, \vert x'''\rangle,&nbsp;\ldots,$">|; 

$key = q/m_1,hbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1315.png"
 ALT="$ m_1 \hbar$">|; 

$key = q/displaystylecdot{bfB}=H_0+H_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1836.png"
 ALT="$\displaystyle \cdot {\bf B} = H_0 + H_1,$">|; 

$key = q/displaystylem=m_1+m_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1367.png"
 ALT="$\displaystyle m = m_1 + m_2.$">|; 

$key = q/|n,lrangle';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1547.png"
 ALT="$ \vert n,l\rangle'$">|; 

$key = q/n'neqn;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1570.png"
 ALT="$ n'\neq n$">|; 

$key = q/displaystyleU_I(t_0,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1869.png"
 ALT="$\displaystyle U_I(t_0, t)$">|; 

$key = q/displaystylex_i(t)=x_i(0)+left[frac{p_i(0)}{m}right]t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img602.png"
 ALT="$\displaystyle x_i(t) = x_i(0) + \left[\frac{p_i(0)}{m}\right] t,$">|; 

$key = q/displaystyleJ^{,2},|j_1,j_2;j,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1356.png"
 ALT="$\displaystyle J^{ 2}  \vert j_1,j_2; j,m\rangle$">|; 

$key = q/displaystyle={bfa}cdot{bfb}+{rmi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1211.png"
 ALT="$\displaystyle = {\bf a} \cdot {\bf b} +{\rm i} $">|; 

$key = q/displaystyle{rmi},hbar,frac{dc_2}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1805.png"
 ALT="$\displaystyle {\rm i} \hbar  \frac{d c_2}{dt}$">|; 

$key = q/x'rightarrowx'-deltax;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img478.png"
 ALT="$ x'\rightarrow x'- \delta x$">|; 

$key = q/displaystyleointd{mitOmega},({calY}_{l,m}^-)^dagger,{bfL}cdot{bfS},{calY}_{l,m}^-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1670.png"
 ALT="$\displaystyle \oint d{\mit\Omega} ({\cal Y}_{l  m}^-)^\dagger  {\bf L} \cdot{\bf S}  {\cal Y}_{l m}^-$">|; 

$key = q/displaystylem,frac{d^2{bfx}}{dt^2}=frac{d{bfp}}{dt}=-nablaV({bfx}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img613.png"
 ALT="$\displaystyle m \frac{d^2 {\bf x}}{d t^2} = \frac{d{\bf p}}{dt} = - \nabla V({\bf x}).$">|; 

$key = q/displaystylefrac{dq_i}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img242.png"
 ALT="$\displaystyle \frac{d q_i}{d t}$">|; 

$key = q/displaystyle(m-m'),langlen,l,m|,z,|n',l',m'rangle=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1502.png"
 ALT="$\displaystyle (m - m')  \langle n,l, m\vert z \vert n',l',m'\rangle = 0,$">|; 

$key = q/g_nu^{~mu}=g^{,mu}_{~nu}=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2565.png"
 ALT="$ g_\nu^{&nbsp;\mu}=g^{ \mu}_{&nbsp;\nu} =1$">|; 

$key = q/displaystylesqrt{j,(j+1)-m,(mpm1)},|j_1,j_2;j,mpm1rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1386.png"
 ALT="$\displaystyle \sqrt{j (j+1)- m (m\pm 1)}  \vert j_1,j_2;j,m\pm 1\rangle$">|; 

$key = q/displaystyle|{bfa}|^{,2},|,{bfb}|^{,2}geq|{bfa}cdot{bfb}|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img169.png"
 ALT="$\displaystyle \vert{\bf a}\vert^{ 2}  \vert {\bf b}\vert^{ 2} \geq \vert{\bf a}\cdot {\bf b}\vert^{ 2}$">|; 

$key = q/p^i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2629.png"
 ALT="$ p^i$">|; 

$key = q/displaystyle[u,v_1+v_2];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img497.png"
 ALT="$\displaystyle [u, v_1 + v_2]$">|; 

$key = q/displaystyle|l+1slash2,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1633.png"
 ALT="$\displaystyle \vert l+1/2, m\rangle$">|; 

$key = q/displaystylelanglephi,frac{partial}{partialq_i}=-langlefrac{partialphi}{partialq_i}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img388.png"
 ALT="$\displaystyle \langle \phi  \frac{\partial }{\partial q_i} = -\langle \frac{\partial \phi} {\partial q_i}.$">|; 

$key = q/displaystyle{mitDelta}E=IpmJ,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2528.png"
 ALT="$\displaystyle {\mit\Delta}E = I\pm J,$">|; 

$key = q/displaystyle=P_{31},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2412.png"
 ALT="$\displaystyle = P_{31},$">|; 

$key = q/displaystylepsi(q_1'cdotsq_N')=langleq_1'cdotsq_N'|psirangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img377.png"
 ALT="$\displaystyle \psi(q_1'\cdots q_N') = \langle q_1'\cdots q_N'\vert\psi\rangle.$">|; 

$key = q/lpm1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1661.png"
 ALT="$ l\pm 1/2$">|; 

$key = q/displaystyle|A_Rrangle=T_z({mitDelta}varphi),|Arangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1078.png"
 ALT="$\displaystyle \vert A_R\rangle = T_z({\mit\Delta}\varphi)  \vert A\rangle.$">|; 

$key = q/|xi'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img118.png"
 ALT="$ \vert\xi'\rangle$">|; 

$key = q/displaystyleW_{1rightarrow2}=N_1,w^{rmabs}_{1rightarrow2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2105.png"
 ALT="$\displaystyle W_{1\rightarrow 2} = N_1 w^{\rm abs}_{1\rightarrow 2},$">|; 

$key = q/displaystylep_i=-{rmi},hbarfrac{partial}{partialq_i}+frac{partialF(q_1cdotsq_N)}{partialq_i}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img393.png"
 ALT="$\displaystyle p_i = -{\rm i}  \hbar \frac{\partial}{\partial q_i} + \frac{\partial F(q_1\cdots q_N) }{\partial q_i}.$">|; 

$key = q/Nrightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="$ N\rightarrow\infty$">|; 

$key = q/displaystylepsi(r,theta,varphi)=R(r),Y_{l,m}(theta,varphi).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img933.png"
 ALT="$\displaystyle \psi(r, \theta, \varphi) = R(r)  Y_{l m}(\theta, \varphi).$">|; 

$key = q/displaystylebeta_{l+}=k,aleft[frac{cosdelta_l,j_l'(k,a)-sindelta_l,eta_l'(k,a)}{cosdelta_l,j_l(k,a)-sindelta_l,eta_l(k,a)}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2270.png"
 ALT="$\displaystyle \beta_{l+} = k a \left[\frac{ \cos\delta_l j_l'(k a) - \sin\de...
... \eta_l'(k a)}{\cos\delta_l   j_l(k a) - \sin\delta_l \eta_l(k a)}\right],$">|; 

$key = q/B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="$ B$">|; 

$key = q/l,(l+1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img905.png"
 ALT="$ l (l+1)$">|; 

$key = q/displaystylenablacdot{bfE}=frac{rho}{epsilon_0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2931.png"
 ALT="$\displaystyle \nabla\cdot{\bf E} = \frac{\rho}{\epsilon_0}.$">|; 

$key = q/displaystyleT_z({mitDelta}varphi)=exp(-{rmi},S_z,{mitDelta}varphislashhbar).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1077.png"
 ALT="$\displaystyle T_z({\mit\Delta}\varphi) = \exp(-{\rm i}  S_z  {\mit\Delta}\varphi/\hbar).$">|; 

$key = q/t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="10" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img239.png"
 ALT="$ t$">|; 

$key = q/[cdots,cdots];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img503.png"
 ALT="$ [\cdots,\cdots]$">|; 

$key = q/k'=k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2169.png"
 ALT="$ k'=k$">|; 

$key = q/lambda_{23}=pm1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2439.png"
 ALT="$ \lambda_{23}=\pm 1$">|; 

$key = q/J_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1325.png"
 ALT="$ J_z$">|; 

$key = q/displaystylepsi(q_1cdotsq_N)rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img376.png"
 ALT="$\displaystyle \psi(q_1\cdots q_N)\rangle.$">|; 

$key = q/displaystyleH=frac{{p}^2}{2,M}+V(r).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img924.png"
 ALT="$\displaystyle H = \frac{{p}^2}{2 M} + V(r).$">|; 

$key = q/displaystyle{mitDelta}E_{nlmpm}=frac{e,B}{2,m_e},left(m,hbar+langlelpm1slash2,m|,S_z,|lpm1slash2,mrangle,right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="390" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1698.png"
 ALT="$\displaystyle {\mit\Delta} E_{nlm\pm} = \frac{e  B}{2  m_e}  \left(m \hbar + \langle l\pm 1/2, m\vert  S_z \vert l\pm 1/2, m\rangle  \right).$">|; 

$key = q/x^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2536.png"
 ALT="$ x^2$">|; 

$key = q/displaystylea_0=frac{4pi,epsilon_0,hbar^2}{mu,e^2}=5.3times10^{-11},,,{rmmeters};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img979.png"
 ALT="$\displaystyle a_0 =\frac{4\pi  \epsilon_0 \hbar^2}{\mu  e^2} = 5.3\times 10^{-11}   {\rm meters}$">|; 

$key = q/displaystyle|j_1-j_2|leqjleqj_1+j_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1372.png"
 ALT="$\displaystyle \vert j_1-j_2\vert \leq j \leq j_1+j_2.$">|; 

$key = q/displaystyle{bfL}times{bfx}+{bfx}times{bfL}=2,{rmi},hbar,{bfx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2818.png"
 ALT="$\displaystyle {\bf L}\times {\bf x} + {\bf x}\times{\bf L} = 2 {\rm i} \hbar {\bf x},$">|; 

$key = q/displaystyleT_z({mitDelta}varphi),|Arangle={rme}^{-{rmi},{mitDelta}varphislash2}rme}^{,{rmi},{mitDelta}varphislash2},langle-|Arangle|-rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1101.png"
 ALT="$\displaystyle T_z({\mit\Delta}\varphi) \vert A\rangle = {\rm e}^{-{\rm i} {\m...
... e}^{ {\rm i} {\mit\Delta}\varphi/2}  \langle -\vert A\rangle \vert-\rangle.$">|; 

$key = q/p^3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2572.png"
 ALT="$ p^3$">|; 

$key = q/langle1slash2,1slash2;1slash2,1slash2|1slash2,1slash2;1,1rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1405.png"
 ALT="$ \langle 1/2, 1/2; 1/2, 1/2
\vert 1/2, 1/2; 1, 1\rangle$">|; 

$key = q/{mitPhi}^{,mu}=(phi,c,{bfA});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2623.png"
 ALT="$ {\mit\Phi}^{ \mu} = (\phi,c {\bf A})$">|; 

$key = q/displaystyleeta_l(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2230.png"
 ALT="$\displaystyle \eta_l(y)$">|; 

$key = q/displaystyle[J_{1,i},J_{2,j}]=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1305.png"
 ALT="$\displaystyle [J_{1 i}, J_{2 j}] = 0,$">|; 

$key = q/(1,1slash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1723.png"
 ALT="$ (1,1/2)$">|; 

$key = q/m_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img941.png"
 ALT="$ m_p$">|; 

$key = q/displaystyle=left({array}{rr}0&sigma_i[0.5ex]-sigma_i&0{array}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2606.png"
 ALT="$\displaystyle = \left(\begin{array}{rr} 0&amp; \sigma_i\ [0.5ex]-\sigma_i&amp;0\end{array}\right),$">|; 

$key = q/displaystyle=-{rmi},hbar,frac{partialG}{partialx_i},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img594.png"
 ALT="$\displaystyle = - {\rm i} \hbar \frac{\partial G}{\partial x_i},$">|; 

$key = q/displaystyleE_2,|2rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1441.png"
 ALT="$\displaystyle E_2  \vert 2\rangle.$">|; 

$key = q/displaystylex;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img888.png"
 ALT="$\displaystyle x$">|; 

$key = q/displaystyleU=2,epsilon_0,c,omega^2,|A_0|^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2059.png"
 ALT="$\displaystyle U = 2 \epsilon_0 c \omega^2 \vert A_0\vert^{ 2},$">|; 

$key = q/5.59;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1733.png"
 ALT="$ 5.59$">|; 

$key = q/L_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1042.png"
 ALT="$ L_j$">|; 

$key = q/L^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img760.png"
 ALT="$ L^2$">|; 

$key = q/s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1024.png"
 ALT="$ s$">|; 

$key = q/displaystylefrac{d}{dx},psirangle=frac{dpsi}{dx}rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img336.png"
 ALT="$\displaystyle \frac{d}{dx}  \psi\rangle = \frac{d\psi}{dx}\rangle.$">|; 

$key = q/displaystyle|+ranglelangle+|+|-ranglelangle-|=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1035.png"
 ALT="$\displaystyle \vert+\rangle \langle +\vert + \vert-\rangle \langle -\vert = 1.$">|; 

$key = q/displaystylefrac{partialrho}{partialt}+nabla'cdot{bfj}=2,frac{{rmIm}(V)}{hbar},rho,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img660.png"
 ALT="$\displaystyle \frac{\partial \rho}{\partial t} + \nabla'\cdot {\bf j} = 2 \frac{{\rm Im}(V)}{\hbar} \rho,
$">|; 

$key = q/displaystyle|k',k''rangle_+=frac{1}{sqrt{2}},left(|k'rangle,|k''rangle+|k''rangle,|k'rangleright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2394.png"
 ALT="$\displaystyle \vert k' k''\rangle_+ = \frac{1}{\sqrt{2}} \left(\vert k'\rangle \vert k''\rangle + \vert k''\rangle \vert k'\rangle\right),$">|; 

$key = q/-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2393.png"
 ALT="$ -1$">|; 

$key = q/2,|epsilon|<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1465.png"
 ALT="$ 2 \vert\epsilon\vert&lt;1$">|; 

$key = q/H(q_i,p_i,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img246.png"
 ALT="$ H(q_i, p_i, t)$">|; 

$key = q/displaystyleX,(Y,|Arangle);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img88.png"
 ALT="$\displaystyle X (Y \vert A\rangle)$">|; 

$key = q/displaystyleexp(,{rmi},G,lambda),A,exp(-{rmi},G,lambda);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1086.png"
 ALT="$\displaystyle \exp( {\rm i}  G \lambda) A  \exp(-{\rm i}  G  \lambda)$">|; 

$key = q/displaystylecdot{bfp}times{bfL}={rmi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2803.png"
 ALT="$\displaystyle \cdot{\bf p}\times {\bf L}= {\rm i} $">|; 

$key = q/displaystylefrac{E}{c}=(p^2+m_e^2,c^2)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2581.png"
 ALT="$\displaystyle \frac{E}{c}=(p^2+ m_e^2 c^2)^{1/2},$">|; 

$key = q/displaystylelangleAd|Adrangle=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img449.png"
 ALT="$\displaystyle \langle Ad\vert Ad\rangle = 1.$">|; 

$key = q/x={rmi},hbar,dslashdp_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img514.png"
 ALT="$ x = {\rm i} \hbar d/dp_x$">|; 

$key = q/p^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2571.png"
 ALT="$ p^2$">|; 

$key = q/(A+B)^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img219.png"
 ALT="$ (A+B)^n$">|; 

$key = q/displaystyle{bfj}({bfx}',t)=-left(frac{{rmi},hbar}{2,m}right)left[psi^ast,nabla',psiright]=left(frac{hbar}{m}right){rmIm}(psi^ast,nabla'psi).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="436" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img649.png"
 ALT="$\displaystyle {\bf j}({\bf x}', t) = - \left(\frac{{\rm i}  \hbar}{2 m}\right...
...\psi\right] = \left(\frac{\hbar}{m} \right){\rm Im} (\psi^\ast  \nabla' \psi).$">|; 

$key = q/|j_1,j_2;m_1,m_2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1341.png"
 ALT="$ \vert j_1,j_2; m_1,m_2\rangle$">|; 

$key = q/n>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1530.png"
 ALT="$ n&gt;1$">|; 

$key = q/displaystylex^{,mu'}=a^{,mu}_{~nu},x^nu,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2655.png"
 ALT="$\displaystyle x^{ \mu'} = a^{ \mu}_{&nbsp;\nu} x^\nu,$">|; 

$key = q/displaystylefrac{du}{dt}=sum_{i=1,N}left(frac{partialu}{partialq_i}frac{dq_i}{dtialu}{partialp_i}frac{partialH}{partialq_i}right)=[u,H]_{cl},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="522" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img255.png"
 ALT="$\displaystyle \frac{du}{dt} = \sum_{i=1,N} \left(\frac{\partial u}{\partial q_i...
...c{\partial u}{\partial p_i}\frac{\partial H}{\partial q_i}\right) =[u, H]_{cl},$">|; 

$key = q/displaystylelangle-|,H_1,|+rangle=langle+|,H_1,|-rangle^ast=frac{e,hbar,B_1}{2,m_e}exp(,{rmi},omega,t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="333" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1844.png"
 ALT="$\displaystyle \langle - \vert H_1 \vert + \rangle = \langle + \vert H_1 \vert - \rangle^\ast = \frac{e  \hbar  B_1}{2  m_e} \exp( {\rm i} \omega  t).$">|; 

$key = q/displaystyle=gammaexp[-{rmi},(omega-omega_{21}),t,],c_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1806.png"
 ALT="$\displaystyle = \gamma \exp[-{\rm i}  (\omega-\omega_{21}) t ] c_1,$">|; 

$key = q/s+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2907.png"
 ALT="$ s+1$">|; 

$key = q/displaystylembox{boldmath{{Sigma{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2736.png"
 ALT="$\displaystyle \mbox{\boldmath$\Sigma$}$">|; 

$key = q/displaystylelanglexi'|xi''rangle=delta_{xi'xi''}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img128.png"
 ALT="$\displaystyle \langle \xi'\vert\xi''\rangle = \delta_{\xi' \xi''}.$">|; 

$key = q/0leqmleql;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img907.png"
 ALT="$ 0\leq m\leq l$">|; 

$key = q/s_z=-1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1071.png"
 ALT="$ s_z = -1/2$">|; 

$key = q/psi_{nlm}({bfx});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2498.png"
 ALT="$ \psi_{nlm}({\bf x})$">|; 

$key = q/3,e,a_0,|{bfE}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1607.png"
 ALT="$ 3 e a_0  \vert{\bf E}\vert$">|; 

$key = q/displaystylelangle({mitDeltax})^2rangle=sigma^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img509.png"
 ALT="$\displaystyle \langle ({\mit\Delta x})^2\rangle = \sigma^2,$">|; 

$key = q/displaystyle=-frac{rmi}{hbar},H_{ii},int_{-infty}^texp(eta,t'),dt'=-frac{rmi}{hbar},H_{ii},frac{exp(eta,t)}{eta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="346" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1960.png"
 ALT="$\displaystyle = -\frac{\rm i}{\hbar} H_{ii}  \int_{-\infty}^t \exp(\eta  t') dt'= -\frac{\rm i}{\hbar} H_{ii}  \frac{\exp( \eta  t)}{\eta},$">|; 

$key = q/displaystylesqrt{2},langle1slash2,-1slash2|1,0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1414.png"
 ALT="$\displaystyle \sqrt{2}  \langle 1/2, -1/2\vert 1, 0\rangle$">|; 

$key = q/P_{23};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2428.png"
 ALT="$ P_{23}$">|; 

$key = q/k'a=pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2337.png"
 ALT="$ k'a = \pi/2$">|; 

$key = q/cosalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$ \cos\alpha$">|; 

$key = q/W_{2rightarrow1}=W_{1rightarrow2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2108.png"
 ALT="$ W_{2\rightarrow 1}=W_{1\rightarrow 2}$">|; 

$key = q/displaystyle={rmi},omega,A^dag,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img717.png"
 ALT="$\displaystyle = {\rm i} \omega A^\dag ,$">|; 

$key = q/-lleqmleql;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img915.png"
 ALT="$ -l\leq m \leq l$">|; 

$key = q/|k'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img519.png"
 ALT="$ \vert k'\rangle$">|; 

$key = q/displaystyley;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img890.png"
 ALT="$\displaystyle y$">|; 

$key = q/|{bfx}'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img619.png"
 ALT="$ \vert{\bf x}'\rangle$">|; 

$key = q/displaystyle{mitDelta}E_{n,l,m_l,m_s}=langlen,l,m_l,m_s|,H_B,|n,l,m_l,m_srangle=frac{e,hbar,B}{2,m_e},(m_l+2,m_s).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="456" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1709.png"
 ALT="$\displaystyle {\mit\Delta} E_{n l m_l  m_s}= \langle n,l,m_l, m_s\vert  H_B \vert n,l,m_l, m_s\rangle = \frac{e  \hbar  B} {2  m_e}  (m_l + 2  m_s).$">|; 

$key = q/y,psi_brightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2886.png"
 ALT="$ y \psi_b\rightarrow 0$">|; 

$key = q/r=|{bfx}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1425.png"
 ALT="$ r=\vert{\bf x}\vert$">|; 

$key = q/dot{z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2712.png"
 ALT="$ \dot{z}$">|; 

$key = q/displaystyleA_0(r)=exp(-{rmi},k,a)left[frac{cos(k,a),sin(k,r)-sin(k,a),cos(k,r)}{k,r}right]=exp(-{rmi},k,a),frac{sin[k,(r-a)]}{k,r}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="644" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2288.png"
 ALT="$\displaystyle A_0(r) = \exp(-{\rm i}  k a) \left[\frac{\cos (k a)  \sin (k\...
...cos( k r)}{k r}\right] =\exp(-{\rm i}  k a)  \frac{ \sin[k (r-a)]}{k r}.$">|; 

$key = q/a_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2880.png"
 ALT="$ a_2$">|; 

$key = q/displaystyle=frac{|2,0,0rangle+|2,1,0rangle}{sqrt{2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1593.png"
 ALT="$\displaystyle = \frac{\vert 2,0,0\rangle + \vert 2,1,0\rangle}{\sqrt{2}},$">|; 

$key = q/displaystyle{rmsinc}(x)equivfrac{sinx}{x}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1896.png"
 ALT="$\displaystyle {\rm sinc}(x)\equiv \frac{\sin x}{x}.$">|; 

$key = q/{mitPhi}_mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2673.png"
 ALT="$ {\mit\Phi}_\mu$">|; 

$key = q/displaystylec_i(t)simeq1+left(frac{-{rmi}}{hbar}right)H_{ii},frac{exp(eta,t)}{etc{|H_{mi}|^{,2}exp(2,eta,t)}{2,eta,(E_i-E_m+{rmi},hbar,eta)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="643" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1964.png"
 ALT="$\displaystyle c_i(t) \simeq 1 + \left(\frac{-{\rm i}}{\hbar}\right) H_{ii}  \f...
...i}\vert^{ 2} \exp(2 \eta  t)} {2 \eta (E_i -E_m + {\rm i} \hbar  \eta)}.$">|; 

$key = q/displaystyle=[s,(s+1)-j,(j-1)]^{1slash2},hbar,|s,j-1rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1277.png"
 ALT="$\displaystyle = [s (s+1) - j  (j-1)]^{1/2}  \hbar  \vert s, j-1\rangle.$">|; 

$key = q/displaystyleL_x,x+L_y,y+L_z,z=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1517.png"
 ALT="$\displaystyle L_x  x + L_y  y + L_z  z = 0.$">|; 

$key = q/displaystyle=sqrt{2},frac{2^7}{3^5},a_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2134.png"
 ALT="$\displaystyle = \sqrt{2} \frac{2^7}{3^5} a_0,$">|; 

$key = q/a^{,mu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2543.png"
 ALT="$ a^{ \mu}$">|; 

$key = q/displaystylefrac{c_s'}{a_2}+frac{c_s}{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2909.png"
 ALT="$\displaystyle \frac{c_s'}{a_2} + \frac{c_s}{a}$">|; 

$key = q/c,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2539.png"
 ALT="$ c t$">|; 

$key = q/(3p)_{3slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1685.png"
 ALT="$ (3p)_{3/2}$">|; 

$key = q/rrightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img937.png"
 ALT="$ r\rightarrow
\infty$">|; 

$key = q/displaystyle=frac{gamma^2slashhbar^2}{gamma^2slashhbar^2+(omega-omega_{21})^{,2}slashhbar^2+(omega-omega_{21})^{,2}slash4]^{1slash2},tright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="454" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1820.png"
 ALT="$\displaystyle = \frac{\gamma^2/\hbar^2}{ \gamma^2/\hbar^2 + (\omega-\omega_{21}...
..., \sin^2\left([\gamma^2/\hbar^2+ (\omega-\omega_{21})^{ 2}/4]^{1/2} t\right),$">|; 

$key = q/R_k({mitDelta}varphi)=exp(-{rmi},J_k,{mitDelta}varphislashhbar);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1100.png"
 ALT="$ R_k ({\mit\Delta}\varphi) = \exp(-{\rm i} J_k  {\mit\Delta}\varphi/\hbar)$">|; 

$key = q/(3p)_{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1684.png"
 ALT="$ (3p)_{1/2}$">|; 

$key = q/dsigmaslashd{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2175.png"
 ALT="$ d\sigma/d{\mit\Omega}$">|; 

$key = q/displaystylepsi(x')={rme}^{,{rmi},k',x'},u(x'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img711.png"
 ALT="$\displaystyle \psi(x') = {\rm e}^{ {\rm i} k' x'} u(x'),
$">|; 

$key = q/{bfS}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2453.png"
 ALT="$ {\bf S}_1$">|; 

$key = q/displaystyle=E,|k''',k',k''rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2420.png"
 ALT="$\displaystyle = E \vert k''' k' k''\rangle,$">|; 

$key = q/displaystyle{mitDelta}E_{000}=frac{8}{3},g_p,frac{m_e}{m_p},alpha^2,|E_0|=5.88times10^{-6},{rmeV},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1755.png"
 ALT="$\displaystyle {\mit\Delta} E_{000} = \frac{8}{3} g_p \frac{m_e}{m_p} \alpha^2 \vert E_0\vert = 5.88\times 10^{-6} {\rm eV},$">|; 

$key = q/omega,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1122.png"
 ALT="$ \omega  t$">|; 

$key = q/phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2034.png"
 ALT="$ \phi$">|; 

$key = q/displaystyle[A,A^dag];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img666.png"
 ALT="$\displaystyle [A,A^\dag ]$">|; 

$key = q/langleB|Branglegeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img791.png"
 ALT="$ \langle B\vert B\rangle\geq 0$">|; 

$key = q/displaystyle{mitDelta}B=frac{4pi,hbar}{e,g_n,lambdabar,l},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1139.png"
 ALT="$\displaystyle {\mit\Delta} B = \frac{4\pi  \hbar}{e  g_n  \lambdabar  l},$">|; 

$key = q/displaystyleH,T,v_t+{rmi},hbar,T,frac{dv_t}{dt}=v,H,T,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img570.png"
 ALT="$\displaystyle H T  v_t +{\rm i} \hbar  T \frac{d v_t}{dt} = v  H T,$">|; 

$key = q/-{rmi},hbar,partialslashpartialq_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img392.png"
 ALT="$ -{\rm i} \hbar  \partial/\partial q_i$">|; 

$key = q/displaystyle|1slash2,-1slash2rangle=frac{1}{sqrt{2}}left(|1,0rangle+|0,0rangleright).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1421.png"
 ALT="$\displaystyle \vert 1/2,-1/2\rangle = \frac{1}{\sqrt{2}} \left(\vert 1,0\rangle + \vert,0\rangle\right).$">|; 

$key = q/displaystylefrac{dsigma}{d{mitOmega}}simeqleft(frac{Z,Z',e^2}{16pi,epsilon_0,E}right)^2frac{1}{sin^4(thetaslash2)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2208.png"
 ALT="$\displaystyle \frac{d\sigma}{d{\mit\Omega}} \simeq\left(\frac{Z  Z'  e^2}{16\pi \epsilon_0 E}\right)^2 \frac{1}{\sin^4(\theta/2)},$">|; 

$key = q/displaystylefrac{2,m}{hbar^2}frac{|V_0|}{mu^2}geq2.7,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2215.png"
 ALT="$\displaystyle \frac{2 m}{\hbar^2} \frac{\vert V_0\vert} {\mu^2} \geq 2.7,$">|; 

$key = q/displaystyletau(t_0)=lim_{trightarrowt_0}frac{T(t,t_0)-1}{t-t_0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img547.png"
 ALT="$\displaystyle \tau(t_0) = \lim_{t\rightarrow t_0}\frac{T(t, t_0)-1}{t-t_0}.$">|; 

$key = q/displaystyle=nablatimes{bfA}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2031.png"
 ALT="$\displaystyle = \nabla\times {\bf A}.$">|; 

$key = q/displaystyle=|2,1,-1rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1599.png"
 ALT="$\displaystyle = \vert 2,1,-1\rangle.$">|; 

$key = q/f(theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2253.png"
 ALT="$ f(\theta)$">|; 

$key = q/psi({bfx}');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1251.png"
 ALT="$ \psi({\bf x}')$">|; 

$key = q/pi,r^2,I;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1105.png"
 ALT="$ \pi  r^2  I$">|; 

$key = q/|Adrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img459.png"
 ALT="$ \vert Ad\rangle$">|; 

$key = q/|l,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img781.png"
 ALT="$ \vert l, m\rangle$">|; 

$key = q/E+dE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1913.png"
 ALT="$ E+dE$">|; 

$key = q/gammaslashdeltax;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img469.png"
 ALT="$ \gamma/\delta x$">|; 

$key = q/mu=costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2257.png"
 ALT="$ \mu = \cos\theta$">|; 

$key = q/displaystylelangle1|,H_1,|2rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1452.png"
 ALT="$\displaystyle \langle 1\vert H_1 \vert 2\rangle.$">|; 

$key = q/cos^2(thetaslash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1291.png"
 ALT="$ \cos^2(\theta/2)$">|; 

$key = q/psi_A(x)rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img327.png"
 ALT="$ \psi_A(x)\rangle$">|; 

$key = q/4pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1103.png"
 ALT="$ 4\pi$">|; 

$key = q/gamma^0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2609.png"
 ALT="$ \gamma^0$">|; 

$key = q/displaystylealpha_i=gamma^5,Sigma_i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2730.png"
 ALT="$\displaystyle \alpha_i = \gamma^5 \Sigma_i,$">|; 

$key = q/E_0=-e^2slash(8pi,epsilon_0,a_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2509.png"
 ALT="$ E_0=-e^2/(8\pi \epsilon_0 a_0)$">|; 

$key = q/L_z,S_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1625.png"
 ALT="$ L_z, S_z,$">|; 

$key = q/{mitDelta}varphi_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img882.png"
 ALT="$ {\mit\Delta}\varphi_y$">|; 

$key = q/psi_{nlm}({bfx}')=langle{bfx}'|n,l,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1526.png"
 ALT="$ \psi_{nlm}({\bf x}') = \langle {\bf x}'\vert n,l,m\rangle$">|; 

$key = q/displaystyleP_{12},|k'rangle,|k''rangle=|k''rangle,|k'rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2380.png"
 ALT="$\displaystyle P_{12} \vert k'\rangle \vert k''\rangle = \vert k''\rangle \vert k'\rangle.$">|; 

$key = q/E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img675.png"
 ALT="$ E$">|; 

$key = q/{mitDelta}E,{mitDelta}tsimhbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1998.png"
 ALT="$ {\mit\Delta} E  {\mit\Delta} t \sim \hbar$">|; 

$key = q/{mitDelta}y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img489.png"
 ALT="$ {\mit\Delta} y$">|; 

$key = q/cdot{bfx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2843.png"
 ALT="$ \cdot{\bf x}$">|; 

$key = q/displaystyle=-frac{e,hbar}{2,m_e},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2752.png"
 ALT="$\displaystyle = - \frac{e \hbar}{2 m_e} $">|; 

$key = q/L^+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img768.png"
 ALT="$ L^+$">|; 

$key = q/displaystyle=left({array}{c}0010{array}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="111" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1589.png"
 ALT="$\displaystyle = \left( \begin{array}{c} 0 \ 0 \ 1 \ 0 \end{array} \right),$">|; 

$key = q/displaystyleR_{2,0}(r)=frac{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="323" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1007.png"
 ALT="$\displaystyle R_{2 0}(r)= \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/n=0,1,2,cdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img682.png"
 ALT="$ n=0,1,2,\cdots$">|; 

$key = q/displaystyle{alpha_i,alpha_j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2590.png"
 ALT="$\displaystyle \{\alpha_i,\alpha_j\}$">|; 

$key = q/displaystylechi(1,1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2466.png"
 ALT="$\displaystyle \chi(1,1)$">|; 

$key = q/k'ggk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2326.png"
 ALT="$ k' \gg k$">|; 

$key = q/psi({bfx}')equivlangle{bfx'}|Erangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1239.png"
 ALT="$ \psi({\bf x}') \equiv \langle {\bf x'}\vert E\rangle$">|; 

$key = q/displaystyle[x_i,x_j];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img738.png"
 ALT="$\displaystyle [x_i, x_j]$">|; 

$key = q/displaystylelangleA|=sum_{i=1,N}alpha_i^ast,langlei|,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="$\displaystyle \langle A\vert = \sum_{i=1,N}\alpha_i^\ast  \langle i\vert,$">|; 

$key = q/(2,j_2+1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1376.png"
 ALT="$ (2 j_2+1)$">|; 

$key = q/displaystyleR_x({mitDelta}varphi_x),|0,0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img877.png"
 ALT="$\displaystyle R_x({\mit\Delta} \varphi_x)  \vert,0\rangle$">|; 

$key = q/displaystylef^{,2}_{21}=|langle2|,x,|1rangle|^{,2}+|langle2|,y,|1rangle|^{,2}+|langle2|,z,|1rangle|^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="313" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2113.png"
 ALT="$\displaystyle f^{ 2}_{21} = \vert\langle 2\vert x \vert 1\rangle\vert^{ 2}+...
... \vert 1\rangle\vert^{ 2}+ \vert\langle 2\vert z \vert 1\rangle\vert^{ 2}.$">|; 

$key = q/displaystylerightarrowleft(!{array}{c}10{array}!right)equivchi_+,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1143.png"
 ALT="$\displaystyle \rightarrow \left(\!\begin{array}{c}1\ 0\end{array}\!\right) \equiv \chi_+,$">|; 

$key = q/N_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2104.png"
 ALT="$ N_2$">|; 

$key = q/E_i-E_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1941.png"
 ALT="$ E_i-E_m$">|; 

$key = q/displaystyleH_I(t_0,t)=exp[+{rmi},H_0,(t-t_0)slashhbar],H_1,exp[-{rmi},H_0,(t-t_0)slashhbar],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="432" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1861.png"
 ALT="$\displaystyle H_I(t_0,t) = \exp[ +{\rm i}   H_0 (t-t_0)/\hbar]   H_1  \exp[ -{\rm i}   H_0 (t-t_0)/\hbar],$">|; 

$key = q/displaystylelanglexi''|,xi=xi''^{,ast},langlexi''|.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img121.png"
 ALT="$\displaystyle \langle \xi''\vert \xi = \xi''^{ \ast}  \langle \xi''\vert.$">|; 

$key = q/displaystylelanglel,1slash2|l+1slash2,l+1slash2rangle=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1651.png"
 ALT="$\displaystyle \langle l, 1/2\vert l+1/2, l+1/2\rangle = 1.$">|; 

$key = q/={bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2083.png"
 ALT="$ = {\bf e}_z$">|; 

$key = q/A_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2049.png"
 ALT="$ A_0$">|; 

$key = q/displaystylesimeqleft[left(frac{-{rmi}}{hbar}right)H_{ii}+left(frac{-{rmi}}{hbar]leftslashleft(1-frac{rmi}{hbar}frac{H_{ii}}{eta}right)right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="508" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1968.png"
 ALT="$\displaystyle \simeq \left[\left( \frac{-{\rm i}}{\hbar}\right) H_{ii} + \left(...
...a}\right]\left/\left( 1- \frac{\rm i}{\hbar} \frac{H_{ii}}{\eta} \right)\right.$">|; 

$key = q/{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1045.png"
 ALT="$ {\bf S}$">|; 

$key = q/displaystylesigma_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1168.png"
 ALT="$\displaystyle \sigma_2$">|; 

$key = q/displaystyleleft[{mitDelta}xi,{mitDelta}etaright];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img182.png"
 ALT="$\displaystyle \left[ {\mit\Delta} \xi, {\mit\Delta} \eta\right]$">|; 

$key = q/displaystylep_i=-{rmi},hbarfrac{partial}{partialq_i}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img394.png"
 ALT="$\displaystyle p_i = -{\rm i} \hbar \frac{\partial}{\partial q_i}.$">|; 

$key = q/displaystylecdot{bfx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2848.png"
 ALT="$\displaystyle \cdot{\bf x},$">|; 

$key = q/displaystylew_{2prightarrow1s}=left(frac{2}{3}right)^8alpha^5,frac{m_e,c^2}{hbar}=6.26times10^8,{rms}^{-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2126.png"
 ALT="$\displaystyle w_{2p\rightarrow 1s} = \left(\frac{2}{3}\right)^8\alpha^5 \frac{m_e c^2}{\hbar}=6.26\times 10^8 {\rm s}^{-1},
$">|; 

$key = q/{bfS}_e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1741.png"
 ALT="$ {\bf S}_e$">|; 

$key = q/-e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2928.png"
 ALT="$ -e$">|; 

$key = q/displaystylev_t=T^{dag},v,T.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img567.png"
 ALT="$\displaystyle v_t = T^{\dag }  v  T.$">|; 

$key = q/displaystyle|i,t_0,trangle=exp[-{rmi},H_0,(t-t_0)slashhbar],U_I(t_0,t),|irangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="318" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1864.png"
 ALT="$\displaystyle \vert i, t_0, t\rangle = \exp[-{\rm i}   H_0 (t-t_0)/\hbar]  U_I(t_0, t)  \vert i\rangle.$">|; 

$key = q/-{rmi},hbar,dslashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img350.png"
 ALT="$ -{\rm i} \hbar  d/dx$">|; 

$key = q/displaystyle=frac{3,hbar^2}{4},|pmrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1030.png"
 ALT="$\displaystyle = \frac{3  \hbar^2}{4} \vert\pm \rangle.$">|; 

$key = q/displaystyle(langleA|+c^ast,langleB|),(|Arangle+c,|Brangle)geq0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img170.png"
 ALT="$\displaystyle (\langle A\vert + c^\ast  \langle B\vert)  (\vert A\rangle + c  \vert B\rangle) \geq 0,$">|; 

$key = q/n,l,m=1,0,0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2526.png"
 ALT="$ n,l,m=1,0,0$">|; 

$key = q/displaystylesum_{n'}|langlen|,x,|n'rangle|^{,2},(E_{n'}-E_{n})=frac{hbar^2}{2,m},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img708.png"
 ALT="$\displaystyle \sum_{n'} \vert\langle n\vert x \vert n'\rangle\vert^{ 2} (E_{n'}-E_{n})= \frac{\hbar^2}{2 m},
$">|; 

$key = q/displaystylecdot{bfB}=left(frac{e}{m_e}right){bfS}cdot{bfB}=omega,S_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1119.png"
 ALT="$\displaystyle \cdot {\bf B} = \left(\frac{e}{m_e}\right) {\bf S} \cdot {\bf B} = \omega  S_z,$">|; 

$key = q/p_x'+dp_x';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img408.png"
 ALT="$ p_x'+dp_x'$">|; 

$key = q/displaystyle=hbar,omegaleft(frac{1}{2}+A^dag,Aright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img669.png"
 ALT="$\displaystyle =\hbar \omega\left(\frac{1}{2} + A^\dag A\right),$">|; 

$key = q/displaystyle=langlep_x'|Arangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img432.png"
 ALT="$\displaystyle = \langle p_x'\vert A\rangle.$">|; 

$key = q/displaystyle|3rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1596.png"
 ALT="$\displaystyle \vert 3\rangle$">|; 

$key = q/displaystylealpha_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2599.png"
 ALT="$\displaystyle \alpha_i$">|; 

$key = q/displaystyle=-{rmi},hbar,p_y.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2763.png"
 ALT="$\displaystyle = -{\rm i} \hbar p_y.$">|; 

$key = q/dot{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2708.png"
 ALT="$ \dot{x}$">|; 

$key = q/displaystyle[L^2,x];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1511.png"
 ALT="$\displaystyle [L^2, x]$">|; 

$key = q/c_{s-1}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2882.png"
 ALT="$ c_{s-1}'$">|; 

$key = q/displaystylepsi({bfx}',t)=psi({bfx'},t_0)exp[-{rmi},H',(t-t_0)slashhbar].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img633.png"
 ALT="$\displaystyle \psi({\bf x}', t) = \psi({\bf x'}, t_0) \exp[ -{\rm i} H' (t-t_0)/\hbar].$">|; 

$key = q/displaystyletau_i=frac{hbar}{{mitGamma}_i},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1989.png"
 ALT="$\displaystyle \tau_i =\frac{\hbar}{{\mit\Gamma}_i},$">|; 

$key = q/displaystyle=[s,(s+1)-j,(j+1)]^{1slash2},hbar,|s,j+1rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1275.png"
 ALT="$\displaystyle = [s (s+1) - j  (j+1)]^{1/2}  \hbar  \vert s, j+1\rangle,$">|; 

$key = q/l=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img987.png"
 ALT="$ l=1$">|; 

$key = q/|Crangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$ \vert C\rangle$">|; 

$key = q/displaystyle=left(!{array}{rrr}1&0&00&0&00&0&-1{array}!right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="88" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1287.png"
 ALT="$\displaystyle = \left(\!\begin{array}{rrr} 1 &amp;0&amp;0\ 0&amp;0&amp;0\ 0&amp;0&amp;-1\end{array}\!\right).$">|; 

$key = q/displaystyleH_1,|n,l^{(1)}rangle=lambda_{n,l},|n,l^{(1)}rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1557.png"
 ALT="$\displaystyle H_1  \vert n, l^{(1)}\rangle = \lambda_{n l}  \vert n, l^{(1)}\rangle.$">|; 

$key = q/displaystyleP_{12},H,P_{12},|k''rangle,|k'rangle=E,|k''rangle,|k'rangle=H,|k''rangle,|k'rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="337" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2389.png"
 ALT="$\displaystyle P_{12} H P_{12} \vert k''\rangle \vert k'\rangle = E \vert k''\rangle \vert k'\rangle = H \vert k''\rangle \vert k'\rangle,$">|; 

$key = q/|{bfx'}rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img617.png"
 ALT="$ \vert{\bf x'}\rangle$">|; 

$key = q/displaystyle={mitDelta}eta,|~rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img175.png"
 ALT="$\displaystyle = {\mit\Delta} \eta  \vert&nbsp;\rangle,$">|; 

$key = q/psirangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img334.png"
 ALT="$ \psi\rangle$">|; 

$key = q/displaystylefrac{partial}{partialt}intd^{,3}x',rho({bfx'},t)=frac{2}{hbar},{rmIm}!intd^{,3}x',V({bfx}'),rho({bfx'},t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img657.png"
 ALT="$\displaystyle \frac{\partial}{\partial t} \int d^{ 3} x' \rho({\bf x'}, t) = \frac{2}{\hbar}  {\rm Im}\!\int d^{ 3} x' V({\bf x}')  \rho({\bf x'}, t).$">|; 

$key = q/displaystyle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="9" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1219.png"
 ALT="$\displaystyle ,$">|; 

$key = q/f({bfk}',{bfk});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2187.png"
 ALT="$ f({\bf k}', {\bf k})$">|; 

$key = q/E=p^2slash2,m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2209.png"
 ALT="$ E= p^2/2 m$">|; 

$key = q/displaystyle=left(frac{-{rmi}}{hbar}right)^2sum_m|H_{mi}|^{,2}int_{-infty}^tdt'iexp[(eta+{rmi},omega_{im}),t']exp[(eta+{rmi},omega_{mi}),t''];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="543" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1962.png"
 ALT="$\displaystyle = \left(\frac{-{\rm i}}{\hbar} \right)^2 \sum_m \vert H_{mi}\vert...
...xp[(\eta+ {\rm i} \omega_{im})  t'] \exp[ (\eta+ {\rm i} \omega_{mi})  t'']$">|; 

$key = q/displaystyleH=frac{{bfp}_1^{,2}}{2,m}+frac{{bfp}_2^{,2}}{2,m}+V_{rmpair}(|{bfx}_1-{bfx}_2|)+V_{rmext}({bfx}_1)+V_{rmext}({bfx}_2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="412" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2385.png"
 ALT="$\displaystyle H = \frac{{\bf p}_1^{ 2}}{2 m} + \frac{{\bf p}_2^{ 2}}{2 m} +...
...vert{\bf x}_1-{\bf x}_2\vert) + V_{\rm ext}({\bf x}_1)+ V_{\rm ext}({\bf x}_2).$">|; 

$key = q/displaystylec_n(t)=c_n^{(1)}+c_n^{(2)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1933.png"
 ALT="$\displaystyle c_n(t) = c_n^{(1)}+ c_n^{(2)}$">|; 

$key = q/displaystylecotdelta_l(E)=cotdelta_l(E_0)+left(frac{dcotdelta_l}{dE}right)_{E=E_}{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="679" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2345.png"
 ALT="$\displaystyle \cot \delta_l(E) = \cot \delta_l(E_0) +\left( \frac{ d \cot\delta...
...ft(\frac{1}{\sin^2\delta_l}\frac{d\delta_l}{d E}\right)_{E=E_0} (E-E_0)+\cdots.$">|; 

$key = q/V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img609.png"
 ALT="$ V$">|; 

$key = q/displaystylelanglex'|_{rmnew}={rme}^{,{rmi},gamma'}langlex'|_{rmold},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img352.png"
 ALT="$\displaystyle \langle x'\vert _{\rm new} = {\rm e}^{ {\rm i} \gamma'}\langle x'\vert _{\rm old},$">|; 

$key = q/R_y({mitDelta}varphi_y),R_x({mitDelta}varphi_x)neqR_x({mitDelta}varphi_x),R_y({mitDelta}varphi_y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img883.png"
 ALT="$ R_y({\mit\Delta}\varphi_y) R_x({\mit\Delta}\varphi_x)
\neq R_x({\mit\Delta}\varphi_x) R_y({\mit\Delta}\varphi_y)$">|; 

$key = q/pmc;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2709.png"
 ALT="$ \pm c$">|; 

$key = q/displaystyle={rmi},hbar,{bfJ}_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1304.png"
 ALT="$\displaystyle = {\rm i} \hbar {\bf J}_2.$">|; 

$key = q/displaystyleH;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img668.png"
 ALT="$\displaystyle H$">|; 

$key = q/y=rslasha_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2865.png"
 ALT="$ y=r/a_0$">|; 

$key = q/displaystyle2,{rmi},(g^nu_{~alpha},gamma_beta-g^nu_{~beta},gamma_alpha)=[gamma^nu,sigma_{alpha,beta}].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2692.png"
 ALT="$\displaystyle 2 {\rm i} (g^\nu_{&nbsp;\alpha} \gamma_\beta -g^\nu_{&nbsp;\beta} \gamma_\alpha) = [\gamma^\nu, \sigma_{\alpha \beta}].$">|; 

$key = q/displaystyle=exp(0),|0,0rangle=|0,0rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img881.png"
 ALT="$\displaystyle =\exp(0) \vert,0\rangle = \vert,0\rangle.$">|; 

$key = q/displaystylepsi({bfx})simeqfrac{1}{(2pi)^{3slash2}}sum_lC_l,frac{exp[,{rmi},(k,r-{rmi},(k,r-l,pislash2+delta_l)]}{2,{rmi},k,r},P_l(costheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="624" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2248.png"
 ALT="$\displaystyle \psi({\bf x}) \simeq \frac{1}{(2\pi)^{3/2}} \sum_l C_l  \frac{\e...
...p[-{\rm i} (k r - l \pi/2+ \delta_l)] }{2 {\rm i} k r}  P_l(\cos\theta),$">|; 

$key = q/displaystylej^{,i};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2641.png"
 ALT="$\displaystyle j^{ i}$">|; 

$key = q/displaystyle(X,Y)^{dag}=Y^{dag},X^{dag}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img100.png"
 ALT="$\displaystyle (X  Y)^{\dag } = Y^{\dag }  X^{\dag }.$">|; 

$key = q/H(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img554.png"
 ALT="$ H(t)$">|; 

$key = q/displaystylehbar,gamma(x)=int^xdx',f(x');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img365.png"
 ALT="$\displaystyle \hbar \gamma(x) = \int^x dx'  f(x')$">|; 

$key = q/displaystyleL^-,Y_{l,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img920.png"
 ALT="$\displaystyle L^-  Y_{l m}$">|; 

$key = q/langle{bfx}'|psirangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2143.png"
 ALT="$ \langle {\bf x}'\vert\psi\rangle$">|; 

$key = q/m_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1622.png"
 ALT="$ m_l$">|; 

$key = q/E_0=-13.6,{rmeV};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2505.png"
 ALT="$ E_0=-13.6 {\rm eV}$">|; 

$key = q/exp(-{rmi},k',a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img525.png"
 ALT="$ \exp(-{\rm i} k' a)$">|; 

$key = q/l=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img876.png"
 ALT="$ l=0$">|; 

$key = q/displaystyleint_{-infty}^{+infty}dx,exp(,{rmi},a,x)=2pi,delta(a),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img424.png"
 ALT="$\displaystyle \int_{-\infty}^{+\infty}dx  \exp( {\rm i}  a x) = 2\pi \delta (a),$">|; 

$key = q/displaystylelangleS_krangle=langleA|,S_k,|Arangle=sum_pmlangleA|pmranglelanglepmglepm|Arangle=left(frac{hbar}{2}right),chi^{dag},sigma_k,chi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="451" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1174.png"
 ALT="$\displaystyle \langle S_k \rangle = \langle A\vert S_k  \vert A\rangle = \sum...
...\vert A\rangle = \left(\frac{\hbar}{2}\right)  \chi^{\dag }  \sigma_k  \chi.$">|; 

$key = q/displaystyleH_0,|nrangle=E_n,|nrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1467.png"
 ALT="$\displaystyle H_0  \vert n\rangle = E_n  \vert n\rangle,$">|; 

$key = q/displaystyleleft.-|k'''rangle,|k''rangle,|k'rangle-|k'rangle,|k'''rangle,|k''rangle-|k''rangle,|k'rangle,|k'''rangleright).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2450.png"
 ALT="$\displaystyle \left.-\vert k'''\rangle \vert k''\rangle \vert k'\rangle - \ve...
...,\vert k''\rangle- \vert k''\rangle \vert k'\rangle \vert k'''\rangle\right).$">|; 

$key = q/S_e^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1748.png"
 ALT="$ S_e^{ 2}$">|; 

$key = q/displaystyleL^+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img770.png"
 ALT="$\displaystyle L^+$">|; 

$key = q/r=a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2280.png"
 ALT="$ r=a$">|; 

$key = q/displaystylelanglen'|,p_x^{,2},|nrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img703.png"
 ALT="$\displaystyle \langle n'\vert p_x^{ 2} \vert n\rangle$">|; 

$key = q/displaystyle=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1889.png"
 ALT="$\displaystyle = 0$">|; 

$key = q/displaystylelangle({mitDelta}x_i)^2rangle_tlangle({mitDelta}x_i)^2rangle_{t=0}geqfrac{hbar^2,t^2}{4,m^2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img605.png"
 ALT="$\displaystyle \langle ({\mit\Delta} x_i)^2\rangle_t \langle ({\mit\Delta} x_i)^2\rangle_{t=0} \geq \frac{\hbar^2  t^2}{4  m^2}.$">|; 

$key = q/j_1geqj_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1373.png"
 ALT="$ j_1\geq j_2$">|; 

$key = q/displaystylepsi({bfx})simeqfrac{1}{(2pi)^{3slash2}}sum_{l=0,infty}left[A_l,frac{}{k,r}-B_l,frac{cos(k,r-l,pislash2)}{k,r}right]P_l(costheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="538" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2245.png"
 ALT="$\displaystyle \psi ({\bf x} ) \simeq \frac{1}{(2\pi)^{3/2}} \sum_{l=0,\infty}\l...
...\pi/2)}{k r} - B_l \frac{\cos(k r -l \pi/2)}{k r} \right] P_l(\cos\theta),$">|; 

$key = q/V({bfx})=V(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2217.png"
 ALT="$ V({\bf x}) = V(r)$">|; 

$key = q/displaystylepsi=left({array}{c}psi_0[0.5ex]psi_1[0.5ex]psi_2[0.5ex]psi_3{array}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="124" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2626.png"
 ALT="$\displaystyle \psi= \left(\begin{array}{c}\psi_0\ [0.5ex]\psi_1\ [0.5ex]\psi_2\ [0.5ex]\psi_3\end{array}\right)$">|; 

$key = q/k,all1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2292.png"
 ALT="$ k a\ll 1$">|; 

$key = q/displaystyleu_2,v_2-v_2,u_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img284.png"
 ALT="$\displaystyle u_2  v_2 - v_2  u_2$">|; 

$key = q/displaystyle={bfx}cdot{bfp}+{rmi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2833.png"
 ALT="$\displaystyle = {\bf x}\cdot{\bf p} + {\rm i} $">|; 

$key = q/f(x)=x^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img315.png"
 ALT="$ f(x)=x^2$">|; 

$key = q/displaystyle[H,P_{12}]=[H,P_{23}]=[H,P_{31}]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="269" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2427.png"
 ALT="$\displaystyle [H,P_{12}] = [H,P_{23}] = [H,P_{31}] = 0.$">|; 

$key = q/displaystyleH=frac{left|{bfp}+e,{bfA}right|^{,2}}{2,m_e}-e,phi+V_0(r),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2032.png"
 ALT="$\displaystyle H = \frac{\left\vert{\bf p} + e  {\bf A}\right\vert^{ 2} }{2 m_e}- e  \phi + V_0(r),$">|; 

$key = q/J_{2,z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1314.png"
 ALT="$ J_{2 z}$">|; 

$key = q/psi(x)rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img329.png"
 ALT="$ \psi(x)\rangle$">|; 

$key = q/p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img237.png"
 ALT="$ p$">|; 

$key = q/displaystyle|1,0rangle=frac{1}{sqrt{2}}left(|1slash2,-1slash2rangle+|-1slash2,1slash2rangleright).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1418.png"
 ALT="$\displaystyle \vert 1,0\rangle = \frac{1}{\sqrt{2}} \left(\vert 1/2,-1/2\rangle + \vert-1/2,1/2\rangle\right).$">|; 

$key = q/s_z=pm1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1134.png"
 ALT="$ s_z = \pm 1/2$">|; 

$key = q/pi,hbarslash2,gamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1826.png"
 ALT="$ \pi  \hbar/2 \gamma$">|; 

$key = q/displaystyleleft[mbox{boldmathSigma}cdot({bfp}+e,{bfA})right]^{,2}=({bfp}+e,{bfA},mbox{boldmathSigma}cdot({bfp}+e,{bfA})times({bfp}+e,{bfA}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="440" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2741.png"
 ALT="$\displaystyle \left[\mbox{\boldmath$\Sigma$}\cdot({\bf p}+e {\bf A})\right]^{\...
...,\mbox{\boldmath$\Sigma$}\cdot ({\bf p}+e {\bf A})\times ({\bf p}+e {\bf A}).$">|; 

$key = q/|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="$ \vert A\rangle$">|; 

$key = q/langlex',y',z'|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1186.png"
 ALT="$ \langle x', y', z'\vert$">|; 

$key = q/displaystyleH_0,|1rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1437.png"
 ALT="$\displaystyle H_0  \vert 1\rangle$">|; 

$key = q/k';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img520.png"
 ALT="$ k'$">|; 

$key = q/{bfp}=-({rmi}slashhbar)nabla;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img925.png"
 ALT="$ {\bf p} = -({\rm i}/\hbar)
\nabla$">|; 

$key = q/displaystyle|2rangle';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1463.png"
 ALT="$\displaystyle \vert 2\rangle'$">|; 

$key = q/displaystylex_{mu'}=a_{mu}^{~nu},x_nu.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2657.png"
 ALT="$\displaystyle x_{\mu'} = a_{\mu}^{&nbsp;\nu} x_\nu.$">|; 

$key = q/displaystyle=sum_{i=1,N}{bfL}_itimessum_{j=1,N}{bfL}_j=sum_{i=1,N}{bfL}_itimes{bum_{i,j=1,N}^{ineqj}({bfL}_itimes{bfL}_j+{bfL}_jtimes{bfL}_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="78" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img757.png"
 ALT="$\displaystyle = \sum_{i=1,N} {\bf L}_i\times \sum_{j=1,N} {\bf L}_j = \sum_{i=1...
...m_{i,j = 1,N}^{i\neq j}( {\bf L}_i\times {\bf L}_j + {\bf L}_j\times {\bf L}_i)$">|; 

$key = q/displaystylefrac{dsigma}{d{mitOmega}}=left(frac{2,m_e,e^2}{4pi,epsilon_0,hbar^{,2},q^2}right)^2left(1-frac{16}{[4+(q,a_0)^2]^{,2}}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2370.png"
 ALT="$\displaystyle \frac{d\sigma}{d{\mit\Omega}} = \left(\frac{2 m_e e^2}{4\pi \e...
...n_0 \hbar^{ 2} q^2}\right)^2\left(1-\frac{16}{[4+(q a_0)^2]^{ 2}}\right),
$">|; 

$key = q/|Brangle=f(x),|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img316.png"
 ALT="$ \vert B\rangle = f(x) \vert A\rangle$">|; 

$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)-2,hbar^2,(L^2,z-z,L^2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="495" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1515.png"
 ALT="$\displaystyle = - 4  \hbar^2  L_y (y  L_z - L_y  z) + 4 \hbar^2  L_x (L_x  z - x  L_z)- 2  \hbar^2 (L^2  z - z  L^2).$">|; 

$key = q/nneq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1495.png"
 ALT="$ n\neq 1$">|; 

$key = q/displaystylecdot{bfx}times{bfp}=r,p_r+{rmi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2834.png"
 ALT="$\displaystyle \cdot{\bf x}\times {\bf p} = r p_r + {\rm i} $">|; 

$key = q/displaystyle=L_x-{rmi},L_y.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img773.png"
 ALT="$\displaystyle = L_x -{\rm i}  L_y.$">|; 

$key = q/displaystyleint_{-infty}^inftydx,f(x),delta(x-x')=f(x');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img194.png"
 ALT="$\displaystyle \int_{-\infty}^\infty dx f(x) \delta(x-x') =f(x')$">|; 

$key = q/displaystyleA_{rmnew}={rme}^{-{rmi},gamma}A_{rmold},{rme}^{,{rmi},gamma}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img362.png"
 ALT="$\displaystyle A_{\rm new} = {\rm e}^{-{\rm i} \gamma} A_{\rm old} {\rm e}^{ {\rm i} \gamma}.$">|; 

$key = q/displaystyle=frac{1}{sqrt{2,l+1}}left({array}{c}pmsqrt{lpmm+1slash2},,Y_{l,,m-1st{lmpm+1slash2},,Y_{l,,m+1slash2}(theta,varphi){array}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="351" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1658.png"
 ALT="$\displaystyle = \frac{1}{\sqrt{2 l+1}}\left( \begin{array}{c} \pm \sqrt{l\pm m...
....5ex] \sqrt{l\mp m+1/2}  Y_{l  m+1/2}(\theta, \varphi) \end{array} \right).$">|; 

$key = q/displaystylec_i^{(2)}(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1961.png"
 ALT="$\displaystyle c_i^{(2)} (t)$">|; 

$key = q/q_2';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img382.png"
 ALT="$ q_2'$">|; 

$key = q/displaystyleE_n=(n+1slash2),hbar,omega,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img681.png"
 ALT="$\displaystyle E_n = (n+1/2) \hbar \omega,
$">|; 

$key = q/|Atrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img528.png"
 ALT="$ \vert At\rangle$">|; 

$key = q/({mitDelta}xi)^2equiv{mitDelta}xi,{mitDelta}xi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img165.png"
 ALT="$ ({\mit\Delta} \xi)^2 \equiv {\mit\Delta} \xi {\mit\Delta} \xi$">|; 

$key = q/displaystyleS_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1040.png"
 ALT="$\displaystyle S_z$">|; 

$key = q/langleA|,v,|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img566.png"
 ALT="$ \langle A\vert v \vert A\rangle$">|; 

$key = q/k,hbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2830.png"
 ALT="$ k \hbar$">|; 

$key = q/m_{rmmax}(m_{rmmax}+1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img812.png"
 ALT="$ m_{\rm max}(m_{\rm max} + 1)$">|; 

$key = q/j_1,j_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1422.png"
 ALT="$ j_1, j_2$">|; 

$key = q/-|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="$ -\vert A\rangle$">|; 

$key = q/displaystyle=2,{rmi},Sigma_3,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2770.png"
 ALT="$\displaystyle =2 {\rm i} \Sigma_3,$">|; 

$key = q/displaystyleleft.+sqrt{j_2,(j_2+1)-m_2',(m_2'pm1)},|j_1,j_2;m_1',m_2'pm1rangleright);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="371" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1388.png"
 ALT="$\displaystyle \left.+\sqrt{ j_2 (j_2+1)- m_2' (m_2'\pm 1)}  \vert j_1, j_2; m_1', m_2'\pm 1\rangle\right)$">|; 

$key = q/1slashk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2309.png"
 ALT="$ 1/k$">|; 

$key = q/displaystyleH,|k'',k',k'''rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2421.png"
 ALT="$\displaystyle H \vert k'' k' k'''\rangle$">|; 

$key = q/displaystylej^{,mu}=c,psi^dag,gamma^0,gamma^{,mu},psi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2637.png"
 ALT="$\displaystyle j^{ \mu} = c \psi^\dag \gamma^0 \gamma^{ \mu} \psi.$">|; 

$key = q/displaystylex(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img720.png"
 ALT="$\displaystyle x(t)$">|; 

$key = q/displaystylerightarrow(0,1)equivchi_-^{dag}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1149.png"
 ALT="$\displaystyle \rightarrow (0, 1) \equiv \chi_-^{\dag }.$">|; 

$key = q/displaystyle=delta_{i,n},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1875.png"
 ALT="$\displaystyle = \delta_{i n},$">|; 

$key = q/displaystylelangle1,0,0|,z,|2,1,0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2133.png"
 ALT="$\displaystyle \langle 1,0,0\vert z \vert 2,1,0\rangle$">|; 

$key = q/displaystylerho({bfx'},t)=|psi({bfx'},t)|^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img642.png"
 ALT="$\displaystyle \rho({\bf x'}, t) = \vert\psi({\bf x'}, t)\vert^{ 2},$">|; 

$key = q/displaystyle=(;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1220.png"
 ALT="$\displaystyle =($">|; 

$key = q/H_1rightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2145.png"
 ALT="$ H_1\rightarrow 0$">|; 

$key = q/displaystyleP_{irightarrowi}(t)=|c_i|^{,2}=exp(-{mitGamma}_i,tslashhbar),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1985.png"
 ALT="$\displaystyle P_{i\rightarrow i} (t) = \vert c_i\vert^{ 2} = \exp(-{\mit\Gamma}_i t/ \hbar),$">|; 

$key = q/gammarightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2369.png"
 ALT="$ \gamma\rightarrow \infty$">|; 

$key = q/A_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2243.png"
 ALT="$ A_l$">|; 

$key = q/H_1=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1782.png"
 ALT="$ H_1= 0$">|; 

$key = q/|n,l,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1496.png"
 ALT="$ \vert n,l,m\rangle$">|; 

$key = q/displaystyle|Arangle=|Brangle+|Crangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$\displaystyle \vert A\rangle = \vert B\rangle + \vert C\rangle,$">|; 

$key = q/displaystyle[Sigma_1,gamma^5];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2767.png"
 ALT="$\displaystyle [\Sigma_1,\gamma^5]$">|; 

$key = q/displaystylelanglephi(q_1cdotsq_N),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img378.png"
 ALT="$\displaystyle \langle \phi(q_1\cdots q_N),$">|; 

$key = q/displaystylepsi({bfx}_1',{bfx}_2';s,m)=phi({bfx}_1',{bfx}_2'),chi(s,m),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2465.png"
 ALT="$\displaystyle \psi({\bf x}_1',{\bf x}_2'; s,m) = \phi({\bf x}_1', {\bf x}_2') \chi(s,m),$">|; 

$key = q/displaystyle=sqrt{n},|E_{n-1}rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img687.png"
 ALT="$\displaystyle = \sqrt{n} \vert E_{n-1}\rangle,$">|; 

$key = q/displaystylecdot{bfx})=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2846.png"
 ALT="$\displaystyle \cdot{\bf x}) =$">|; 

$key = q/partialslashpartialx^{,mu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2575.png"
 ALT="$ \partial/\partial x^{ \mu}$">|; 

$key = q/displaystyle[L_x,p_x];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2759.png"
 ALT="$\displaystyle [L_x,p_x]$">|; 

$key = q/P_{12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2379.png"
 ALT="$ P_{12}$">|; 

$key = q/displaystylee_{n'l'nl}=langlen',l'|,H_1,|n,lrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1551.png"
 ALT="$\displaystyle e_{n'l'nl} = \langle n',l'\vert H_1 \vert n,l\rangle.$">|; 

$key = q/displaystyleV({bfx})=V_1(r)+V_2(r)left[3,frac{(mbox{boldmathsigma}_ncdot{bfx}),(right]+V_3(r),mbox{boldmathsigma}_ncdotmbox{boldmathsigma}_p,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="511" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1424.png"
 ALT="$\displaystyle V({\bf x}) = V_1(r) + V_2(r)\left[3 \frac{(\mbox{\boldmath $\sig...
...\right] + V_3(r) \mbox{\boldmath $\sigma$}_n\cdot\mbox{\boldmath $\sigma$}_p,
$">|; 

$key = q/displaystyleP_{irightarrown}(t)simeq|c_n^{(1)}|^{,2}=frac{4,|H_{ni}|^{,2}}{|E_n-E_i|^{,2}},sin^2left[frac{(E_n-E_i),t}{2,hbar}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="366" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1894.png"
 ALT="$\displaystyle P_{i\rightarrow n}(t) \simeq \vert c_n^{(1)}\vert^{ 2} = \frac{4...
...vert E_n - E_i\vert^{ 2}}  \sin^2\left[ \frac{(E_n-E_i) t}{2 \hbar}\right],$">|; 

$key = q/{mitDelta}l=0,pm2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2096.png"
 ALT="$ {\mit\Delta} l =0, \pm 2$">|; 

$key = q/displaystyle{rmi},hbar,frac{partial}{partialt},|A,t_0,trangle=H,|A,t_0,trangle=(H_0+H_1),|A,t_0,trangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="367" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1788.png"
 ALT="$\displaystyle {\rm i} \hbar  \frac{\partial}{\partial t} \vert A, t_0, t\rangle = H \vert A,t_0,t\rangle= (H_0+H_1)  \vert A,t_0,t\rangle.$">|; 

$key = q/{rmi},(A-A^dag);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img222.png"
 ALT="$ {\rm i} (A-A^\dag )$">|; 

$key = q/displaystylechi=left({array}{c}cosalphasinalpha,exp(,{rmi},beta){array}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1294.png"
 ALT="$\displaystyle \chi =\left(\begin{array}{c}\cos\alpha\ \sin\alpha \exp( {\rm i} \beta)\end{array}\right)
$">|; 

$key = q/psi(r,theta,varphi)=R(r),Y_{l,m}(theta,phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img916.png"
 ALT="$ \psi(r, \theta, \varphi) = R(r)  Y_{l,m}(\theta, \phi)$">|; 

$key = q/displaystylecdot{bfL})=L^{,2}+{rmi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2789.png"
 ALT="$\displaystyle \cdot{\bf L}) = L^{ 2} + {\rm i} $">|; 

$key = q/displaystylep^{,mu}rightarrowp^{,mu}+frac{e}{c},{mitPhi}^{,mu},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2622.png"
 ALT="$\displaystyle p^{ \mu} \rightarrow p^{ \mu} + \frac{e}{c} {\mit\Phi}^{ \mu},$">|; 

$key = q/|n,m_{rmmax}-1rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img810.png"
 ALT="$ \vert n, m_{\rm max}-1\rangle$">|; 

$key = q/displaystylefrac{partialrho}{partialt}+nabla'cdot{bfj}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img647.png"
 ALT="$\displaystyle \frac{\partial \rho}{\partial t} + \nabla'\cdot {\bf j} = 0.$">|; 

$key = q/displaystylelangleA|AranglelangleB|Branglegeq|langleA|Brangle|^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img168.png"
 ALT="$\displaystyle \langle A\vert A\rangle \langle B\vert B\rangle \geq \vert\langle A\vert B\rangle\vert^{ 2},$">|; 

$key = q/langlexirangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img167.png"
 ALT="$ \langle\xi\rangle$">|; 

$key = q/2,hbar^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1751.png"
 ALT="$ 2 \hbar^2$">|; 

$key = q/L^+|l,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img794.png"
 ALT="$ L^+ \vert l,m\rangle$">|; 

$key = q/partial_muequivpartialslashpartialx^{,mu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2614.png"
 ALT="$ \partial_\mu\equiv \partial/\partial x^{ \mu}$">|; 

$key = q/displaystyletandelta_l=frac{j_l(k,a)}{eta_l(k,a)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2285.png"
 ALT="$\displaystyle \tan \delta_l = \frac{j_l(k a)}{\eta_l(k a)}.$">|; 

$key = q/^{(1)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1873.png"
 ALT="$ ^{(1)}$">|; 

$key = q/displaystylesigma_{rmtotal}=sum_{l=0,k,a}frac{2pi}{k^2},(2,l+1)simeq2pi,a^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2307.png"
 ALT="$\displaystyle \sigma_{\rm total} = \sum_{l=0,k a} \frac{2\pi}{k^2}  (2 l+1) \simeq 2\pi  a^2.$">|; 

$key = q/displaystyle=gammaexp[+{rmi},(omega-omega_{21}),t,],c_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1804.png"
 ALT="$\displaystyle = \gamma \exp[+{\rm i}  (\omega-\omega_{21}) t ] c_2,$">|; 

$key = q/displaystyleJ_{1z},|j_1,j_2;m_1,m_2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1346.png"
 ALT="$\displaystyle J_{1z}  \vert j_1,j_2; m_1,m_2\rangle$">|; 

$key = q/chi^dagger,sigma_k,chi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1232.png"
 ALT="$ \chi^\dagger  \sigma_k  \chi$">|; 

$key = q/displaystylef(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2874.png"
 ALT="$\displaystyle f(y)$">|; 

$key = q/displaystylea_l={rmi}^l,(2,l+1),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2240.png"
 ALT="$\displaystyle a_l = {\rm i}^l  (2 l+1),$">|; 

$key = q/q,p-p,q=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img238.png"
 ALT="$ q p-p q = 0$">|; 

$key = q/{bfk}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2168.png"
 ALT="$ {\bf k}'$">|; 

$key = q/i=1,2,3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2574.png"
 ALT="$ i=1,2,3$">|; 

$key = q/displaystyleU=frac{1}{2}left(frac{epsilon_0,E_0^{,2}}{2}+frac{B_0^{,2}}{2,mu_0}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2056.png"
 ALT="$\displaystyle U = \frac{1}{2}\left(\frac{\epsilon_0 E_0^{ 2}}{2}+ \frac{B_0^{ 2}}{2 \mu_0} \right),$">|; 

$key = q/psi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1009.png"
 ALT="$ \psi$">|; 

$key = q/delta_{i,j}=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="$ \delta_{i j}=1$">|; 

$key = q/displaystyledot{alpha}_1(t)=dot{alpha}_1(0),expleft(frac{-2,{rmi},H,t}{hbar}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2719.png"
 ALT="$\displaystyle \dot{\alpha}_1(t) = \dot{\alpha}_1(0) \exp\left(\frac{-2 {\rm i} H t}{\hbar}\right).$">|; 

$key = q/displaystylebeta_{l-}=left.frac{1}{(u_lslashr)}frac{d(u_lslashr)}{dr}right|_{r=a}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2281.png"
 ALT="$\displaystyle \beta_{l-} = \left.\frac{1}{(u_l/r)} \frac{d(u_l/r)}{dr}\right\vert _{r=a}.$">|; 

$key = q/displaystylecosalpha=langlem-1slash2,1slash2|l+1slash2,mrangle=sqrt{frac{l+m+1slash2}{2,l+1}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="398" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1652.png"
 ALT="$\displaystyle \cos\alpha=\langle m-1/2, 1/2\vert l+1/2, m\rangle = \sqrt{\frac{l+m+1/2}{2 l+1}}.$">|; 

$key = q/displaystyle=langlex',y',z'|langle+||Aranglerangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1064.png"
 ALT="$\displaystyle = \langle x', y', z' \vert\langle +\vert\vert A\rangle\rangle,$">|; 

$key = q/displaystyleX,|Arangle=Y,|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img82.png"
 ALT="$\displaystyle X  \vert A\rangle = Y \vert A\rangle$">|; 

$key = q/displaystylefrac{d}{dx},psirangle=frac{dpsi}{dx}ranglestackrel{rmDC}{longleftrightarrow}langlefrac{dpsi^ast}{dx}=-langlepsi^ast,frac{d}{dx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="281" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img344.png"
 ALT="$\displaystyle \frac{d}{dx}  \psi\rangle = \frac{d\psi}{dx}\rangle \stackrel{\r...
...trightarrow} \langle \frac{d\psi^\ast}{dx} = -\langle \psi^\ast  \frac{d}{dx},$">|; 

$key = q/|V_{in}^dagger|^{,2}=|V_{ni}|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2020.png"
 ALT="$ \vert V_{in}^\dagger\vert^{ 2} = \vert V_{ni}\vert^{ 2}$">|; 

$key = q/displaystylepsi_{nlmpm}=R_{n,l}(r),{calY}_{l,m}^pm.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1659.png"
 ALT="$\displaystyle \psi_{nlm\pm} = R_{n l}(r)  {\cal Y}_{l m}^\pm.$">|; 

$key = q/displaystyle|-rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1144.png"
 ALT="$\displaystyle \vert-\rangle$">|; 

$key = q/displaystyleequiv{mitDelta}xi,{mitDelta}eta-{mitDelta}eta,{mitDelta}xi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img183.png"
 ALT="$\displaystyle \equiv {\mit\Delta}\xi  {\mit\Delta} \eta -{\mit\Delta} \eta  {\mit\Delta} \xi,$">|; 

$key = q/displaystylepsi^pm({bfx})simeqfrac{exp(,{rmi},{bfk}cdot{bfx})}{(2pi)^{3slash2}}-d^3x',exp(mp{rmi},{bfk}'cdot{bfx}'),V({bfx}'),psi^pm({bfx}').;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="574" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2171.png"
 ALT="$\displaystyle \psi^\pm({\bf x}) \simeq \frac{\exp( {\rm i} {\bf k}\cdot{\bf x...
...exp(\mp {\rm i}  {\bf k}' \cdot {\bf x}')  V({\bf x}')  \psi^\pm ({\bf x}').$">|; 

$key = q/H_1neq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1783.png"
 ALT="$ H_1\neq 0$">|; 

$key = q/E_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img685.png"
 ALT="$ E_n$">|; 

$key = q/displaystyleH_0,|phirangle=E,|phirangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2138.png"
 ALT="$\displaystyle H_0  \vert\phi\rangle = E  \vert\phi\rangle,$">|; 

$key = q/displaystyle=2,A_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2041.png"
 ALT="$\displaystyle = 2  A_0  $">|; 

$key = q/displaystylelangle1,0,0|,x,|2,1,pm1rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2129.png"
 ALT="$\displaystyle \langle 1,0,0\vert x \vert 2,1,\pm 1\rangle$">|; 

$key = q/displaystyleR_x({mitDelta}varphi_x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img868.png"
 ALT="$\displaystyle R_x({\mit\Delta}\varphi_x)$">|; 

$key = q/|Rt_0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img535.png"
 ALT="$ \vert Rt_0\rangle$">|; 

$key = q/displaystyler^2,epsilon^2=(;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2840.png"
 ALT="$\displaystyle r^2 \epsilon^2 = ($">|; 

$key = q/|p_x'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img401.png"
 ALT="$ \vert p_x'\rangle$">|; 

$key = q/3,hbarslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1371.png"
 ALT="$ 3 \hbar/2$">|; 

$key = q/displaystylepsi^dag,A^dag,gamma^0,gamma^{,mu},A,psi=a^{,mu}_{~nu},psi^dag,gamma^0,gamma^nu,psi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2701.png"
 ALT="$\displaystyle \psi^\dag A^\dag \gamma^0 \gamma^{ \mu} A \psi= a^{ \mu}_{&nbsp;\nu} \psi^\dag \gamma^0 \gamma^\nu \psi,$">|; 

$key = q/displaystyle=frac{-{rmi}}{hbar}int_0^tdt'left[V_{ni},exp({rmi},omega,t')+V_{ni}^dagger,exp(-{rmi},omega,t')right]exp(,{rmi},omega_{ni},t');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2001.png"
 ALT="$\displaystyle = \frac{-{\rm i}}{\hbar} \int_0^t dt'\left[V_{ni}  \exp({\rm i}\...
...^\dagger  \exp(-{\rm i} \omega  t')\right]\exp( {\rm i}  \omega_{ni}  t')$">|; 

$key = q/displaystylef(theta)=-frac{2,m,V_0}{hbar^2,mu}frac{1}{q^2+mu^2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2200.png"
 ALT="$\displaystyle f(\theta) = - \frac{2 m  V_0}{\hbar^2 \mu} \frac{1}{q^2 + \mu^2},$">|; 

$key = q/displaystyleX,|Aranglestackrel{rmDC}{longleftrightarrow}langleA|,X^{dag}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img98.png"
 ALT="$\displaystyle X  \vert A\rangle \stackrel{\rm DC}{\longleftrightarrow} \langle A\vert X^{\dag }.$">|; 

$key = q/displaystyleq_irangle=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img416.png"
 ALT="$\displaystyle q_i \rangle = 0,$">|; 

$key = q/displaystylelangleA|,xi^{,2},|Aranglegeq0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img788.png"
 ALT="$\displaystyle \langle A\vert \xi^{ 2} \vert A\rangle\geq 0,$">|; 

$key = q/alpha_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2922.png"
 ALT="$ \alpha_3$">|; 

$key = q/displaystyle|tilde{A}rangle=left(frac{1}{sqrt{langleA|Arangle}}right)|Arangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img74.png"
 ALT="$\displaystyle \vert\tilde{A}\rangle = \left(\frac{1}{\sqrt{\langle A\vert A \rangle }}\right)\vert A\rangle,$">|; 

$key = q/displaystyle,cosleft(frac{omega}{c},{bfn}cdot{bfx}-omega,tright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2043.png"
 ALT="$\displaystyle  \cos\left (\frac{\omega}{c}  {\bf n}\cdot{\bf x} - \omega  t\right),$">|; 

$key = q/grightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2888.png"
 ALT="$ g\rightarrow 0$">|; 

$key = q/delta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1138.png"
 ALT="$ \delta$">|; 

$key = q/displaystyle|Rrangle=|Arangle+|Brangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img443.png"
 ALT="$\displaystyle \vert R\rangle = \vert A\rangle + \vert B\rangle$">|; 

$key = q/displaystylew_{irightarrow[n]}=left.frac{2pi}{hbar},overline{left|H_{ni}+sum_mfrnm},H_{mi}}{E_m-E_i}right|^{,2}}rho(E_n)right|_{E_nsimeqE_i},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="346" HEIGHT="92" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1936.png"
 ALT="$\displaystyle w_{i\rightarrow [n]} =\left. \frac{2\pi}{\hbar}  \overline{ \lef...
... H_{mi}}{E_m - E_i}\right\vert^{ 2}} \rho(E_n) \right\vert _{E_n \simeq E_i},$">|; 

$key = q/{mitDelta}eta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img178.png"
 ALT="$ {\mit\Delta} \eta$">|; 

$key = q/displaystyleH_0=frac{e,B_0}{m_e},S_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1837.png"
 ALT="$\displaystyle H_0 = \frac{e B_0}{m_e}  S_z,$">|; 

$key = q/c,p^0+e,{mitPhi}^{,0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2919.png"
 ALT="$ c p^0 + e {\mit\Phi}^{ 0}$">|; 

$key = q/langleA|,xi,|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img542.png"
 ALT="$ \langle A\vert \xi \vert A\rangle$">|; 

$key = q/displaystyle=P_{12},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2382.png"
 ALT="$\displaystyle = P_{12},$">|; 

$key = q/|Brangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$ \vert B\rangle$">|; 

$key = q/displaystyleT,v_t=v,T.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img568.png"
 ALT="$\displaystyle T  v_t = v  T.$">|; 

$key = q/displaystyleL,|n,lrangle=L_{n,l},|n,lrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1543.png"
 ALT="$\displaystyle L \vert n,l\rangle = L_{n l}  \vert n, l\rangle,$">|; 

$key = q/displaystyleH=-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1118.png"
 ALT="$\displaystyle H = -$">|; 

$key = q/displaystylefrac{d}{dx},x,psirangle=frac{d(x,psi)}{dx}rangle=x,frac{d}{dx},psirangle+psirangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="268" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img346.png"
 ALT="$\displaystyle \frac{d}{dx}  x  \psi\rangle = \frac{d (x \psi)}{dx}\rangle = x \frac{d}{dx} \psi\rangle + \psi\rangle.$">|; 

$key = q/displaystylex^{,mu'},x_{mu'}=x^{,mu},x_mu,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2658.png"
 ALT="$\displaystyle x^{ \mu'} x_{\mu'} = x^{ \mu} x_\mu,$">|; 

$key = q/n!!=n,(n-2),(n-4)cdots1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2297.png"
 ALT="$ n!! = n (n-2) (n-4)\cdots 1$">|; 

$key = q/2,s+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1256.png"
 ALT="$ 2 s+1$">|; 

$key = q/g_nu^{~mu}=g^{,mu}_{~nu}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2567.png"
 ALT="$ g_\nu^{&nbsp;\mu}=g^{ \mu}_{&nbsp;\nu}=0$">|; 

$key = q/R_{k,l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1096.png"
 ALT="$ R_{k l}$">|; 

$key = q/displaystyle{mitDelta}_i=H_{ii}+{calP}sum_{mneqi}frac{|H_{mi}|^{,2}}{E_i-E_m}-{rmi},pisum_{mneqi}|H_{mi}|^{,2},delta(E_i-E_m).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="400" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1975.png"
 ALT="$\displaystyle {\mit\Delta}_i = H_{ii} + {\cal P}\sum_{m\neq i} \frac{\vert H_{m...
...E_m} - {\rm i} \pi \sum_{m\neq i} \vert H_{mi}\vert^{ 2}  \delta(E_i - E_m).$">|; 

$key = q/displaystylesum_{j}sum_{m}|j_1,j_2;j,mranglelanglej_1,j_2;j,m|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1361.png"
 ALT="$\displaystyle \sum_{j}\sum_{m} \vert j_1,j_2; j, m\rangle \langle j_1,j_2; j, m\vert$">|; 

$key = q/lambda_{31};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2436.png"
 ALT="$ \lambda_{31}$">|; 

$key = q/p_x';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img400.png"
 ALT="$ p_x'$">|; 

$key = q/displaystylec_i(t)=exp!left(frac{-{rmi},{mitDelta}_i,t}{hbar}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1977.png"
 ALT="$\displaystyle c_i(t) = \exp\!\left(\frac{-{\rm i}  {\mit\Delta}_i  t}{\hbar}\right).$">|; 

$key = q/displaystylec_1,|Arangle+c_2,|Arangle=(c_1+c_2),|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$\displaystyle c_1  \vert A\rangle + c_2  \vert A\rangle = (c_1 + c_2)  \vert A\rangle$">|; 

$key = q/displaystyle=langlex'|Arangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img430.png"
 ALT="$\displaystyle = \langle x'\vert A\rangle,$">|; 

$key = q/T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img537.png"
 ALT="$ T$">|; 

$key = q/{bfk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2155.png"
 ALT="$ {\bf k}$">|; 

$key = q/displaystyle[u_1+u_2,v]_{cl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img259.png"
 ALT="$\displaystyle [u_1+ u_2, v]_{cl}$">|; 

$key = q/sigma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img507.png"
 ALT="$ \sigma$">|; 

$key = q/displaystylesigma_{rmtotal}=frac{4pi}{k^2}sum_{l=0,infty}(2,l+1),sin^2delta_l,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2258.png"
 ALT="$\displaystyle \sigma_{\rm total} = \frac{4\pi}{k^2} \sum_{l=0,\infty} (2 l+1) \sin^2\delta_l,$">|; 

$key = q/displaystyle[x_i(t),x_i(0)]=left[frac{p_i(0),t}{m},x_i(0)right]=frac{-{rmi},hbar,t}{m}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img604.png"
 ALT="$\displaystyle [x_i(t), x_i(0)] = \left[ \frac{p_i(0) t}{m}, x_i(0)\right] = \frac{-{\rm i} \hbar  t} {m}.$">|; 

$key = q/s=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2479.png"
 ALT="$ s=0$">|; 

$key = q/displaystylerightarrowfrac{e,B_0}{m_e},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1848.png"
 ALT="$\displaystyle \rightarrow \frac{e  B_0}{m_e},$">|; 

$key = q/displaystyle{mitGamma}=-frac{2}{[d(cotdelta_0)slashdE]_{E=E_r}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2368.png"
 ALT="$\displaystyle {\mit\Gamma} = - \frac{2}{[d(\cot\delta_0)/dE]_{E=E_r}}.
$">|; 

$key = q/langlej_1,j_2;m_1,m_2|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1369.png"
 ALT="$ \langle j_1, j_2; m_1, m_2\vert$">|; 

$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="162" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1458.png"
 ALT="$\displaystyle E_1 + \frac{\vert e_{12}\vert^{ 2}}{E_1-E_2} + \cdots,$">|; 

$key = q/displaystyleint!int!intdx'dy'dz'left(|psi_+|^{,2}+|psi_-|^{,2}right)=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1073.png"
 ALT="$\displaystyle \int\!\int\!\int dx'dy'dz' \left(\vert\psi_+\vert^{ 2} + \vert\psi_-\vert^{ 2}\right)= 1.$">|; 

$key = q/displaystyleL^2,psi=lambda,hbar^2,psi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img900.png"
 ALT="$\displaystyle L^2  \psi = \lambda  \hbar^2  \psi,$">|; 

$key = q/exp(eta,t)rightarrow1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1955.png"
 ALT="$ \exp(\eta  t)\rightarrow 1$">|; 

$key = q/displaystyleintd^3x,,rho({bfx},0)=1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2649.png"
 ALT="$\displaystyle \int d^3 x  \rho({\bf x},0) = 1,$">|; 

$key = q/displaystyle[u_1,v_1]_{qm},(u_2,v_2-v_2,u_2)=(u_1,v_1-v_1,u_1),[u_2,v_2]_{qm}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img277.png"
 ALT="$\displaystyle [u_1, v_1]_{qm}  (u_2  v_2 - v_2  u_2) = (u_1  v_1-v_1  u_1) [u_2, v_2]_{qm}.$">|; 

$key = q/displaystyle=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1217.png"
 ALT="$\displaystyle = 1$">|; 

$key = q/eta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img152.png"
 ALT="$ \eta$">|; 

$key = q/displaystylelangle1|,H_1,|1rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1448.png"
 ALT="$\displaystyle \langle 1\vert H_1  \vert 1\rangle,$">|; 

$key = q/displaystyle=x,p_y-y,p_x.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img733.png"
 ALT="$\displaystyle = x p_y - y  p_x.$">|; 

$key = q/displaystyle=E_2,|2rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1800.png"
 ALT="$\displaystyle = E_2  \vert 2\rangle.$">|; 

$key = q/displaystyle[p_i,p_j]_{cl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img252.png"
 ALT="$\displaystyle [p_i, p_j]_{cl}$">|; 

$key = q/alphasimeq1slash137;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2120.png"
 ALT="$ \alpha \simeq 1/137$">|; 

$key = q/|langlej_1,j_2;m_1,m_2|j_1,j_2;j,mrangle|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1366.png"
 ALT="$ \vert\langle j_1,j_2;m_1,m_2\vert j_1,j_2;j,m\rangle\vert^{ 2}$">|; 

$key = q/B_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1297.png"
 ALT="$ B_0$">|; 

$key = q/displaystyle-{rmi},hbar,frac{partialpsi^dag}{partialt}=psi^dagleft[-e,phi+c,mbox{boldmathalpha}cdot({bfp}+{bfe},{bfA})+beta,m_e,c^2right]=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="400" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2627.png"
 ALT="$\displaystyle -{\rm i} \hbar \frac{\partial\psi^\dag }{\partial t} =\psi^\dag...
...\boldmath$\alpha$}\cdot({\bf p} +{\bf e} {\bf A})+ \beta m_e c^2\right] = 0,$">|; 

$key = q/displaystylelangleA|Brangle=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.png"
 ALT="$\displaystyle \langle A\vert B \rangle = 0,$">|; 

$key = q/_e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1740.png"
 ALT="$ _e$">|; 

$key = q/c_i(trightarrow-infty)=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1946.png"
 ALT="$ c_i(t\rightarrow-\infty) =1$">|; 

$key = q/{bfJ}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1299.png"
 ALT="$ {\bf J}_1$">|; 

$key = q/displaystyleH=H_0+H_1(t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1776.png"
 ALT="$\displaystyle H = H_0 + H_1(t),$">|; 

$key = q/displaystyle|e_{12}|<frac{|E_1-E_2|}{2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1466.png"
 ALT="$\displaystyle \vert e_{12}\vert &lt; \frac{\vert E_1-E_2\vert}{2}.$">|; 

$key = q/|pmrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1031.png"
 ALT="$ \vert\pm \rangle$">|; 

$key = q/displaystyleP_{irightarrown}(t_0,t)=|c_n^{(0)}+c_n^{(1)}+c_n^{(2)}+cdots|^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="281" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1885.png"
 ALT="$\displaystyle P_{i\rightarrow n} (t_0, t) = \vert c_n^{(0)} + c_n^{(1)} + c_n^{(2)} +\cdots\vert^{ 2}.$">|; 

$key = q/displaystyle[L^2,y];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1509.png"
 ALT="$\displaystyle [L^2, y]$">|; 

$key = q/m_l+2,m_s=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1714.png"
 ALT="$ m_l+2 m_s = -1$">|; 

$key = q/sin^2delta_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2306.png"
 ALT="$ \sin^2\delta_l$">|; 

$key = q/displaystyleH,|Erangle=E,|Erangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1238.png"
 ALT="$\displaystyle H \vert E\rangle = E \vert E\rangle,$">|; 

$key = q/displaystyleint_{-infty}^{+infty}cdotsint_{-infty}^{+infty}dq_1'cdotsdq_N',|psi(q_1'cdotsq_N')|^{,2}=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="345" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img385.png"
 ALT="$\displaystyle \int_{-\infty}^{+\infty} \cdots \int_{-\infty}^{+\infty}dq_1'\cdots dq_N'   \vert\psi(q_1'\cdots q_N')\vert^{ 2}= 1.$">|; 

$key = q/c_s';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2891.png"
 ALT="$ c_s'$">|; 

$key = q/pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2264.png"
 ALT="$ \pi/2$">|; 

$key = q/displaystyleT(t,0)=exp(-{rmi},H,tslashhbar)=exp(-{rmi},S_z,omega,tslashhbar).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1121.png"
 ALT="$\displaystyle T(t,0) = \exp(-{\rm i}  H  t/\hbar) = \exp(-{\rm i}  S_z  \omega  t/\hbar).$">|; 

$key = q/displaystylep_i,|q_1'cdotsq_N'rangle={rmi},hbar,frac{partial}{partialq_i'},|q_1'cdotsq_N'rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img399.png"
 ALT="$\displaystyle p_i \vert q_1'\cdots q_N'\rangle = {\rm i} \hbar \frac{\partial}{\partial q_i'}  \vert q_1'\cdots q_N'\rangle.$">|; 

$key = q/displaystyleleft[left(p^0-frac{e}{c},{mitPhi}^{,0}right)-alpha_1left(p^1-frac{e}p^3-frac{e}{c},{mitPhi}^{,3}right)-beta,m_e,cright]psi^ast=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="635" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2925.png"
 ALT="$\displaystyle \left[\left(p^0-\frac{e}{c} {\mit\Phi}^{ 0}\right)-\alpha_1\lef...
...left(p^3-\frac{e}{c} {\mit\Phi}^{ 3}\right) -\beta m_e c\right]\psi^\ast=0.$">|; 

$key = q/displaystylelanglex'|x''rangle_{rmnew};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img354.png"
 ALT="$\displaystyle \langle x'\vert x''\rangle_{\rm new}$">|; 

$key = q/displaystyleE=Z^{,2},E_{100}+Z^{,2},E_{nlm}+{mitDeltaE},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2524.png"
 ALT="$\displaystyle E = Z^{ 2} E_{100} + Z^{ 2} E_{nlm} + {\mit\Delta E},$">|; 

$key = q/cdot{bff}_{ni};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2091.png"
 ALT="$ \cdot{\bf f}_{ni}$">|; 

$key = q/m_eslashm_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1734.png"
 ALT="$ m_e/m_p$">|; 

$key = q/|l,m+1rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img796.png"
 ALT="$ \vert l, m+1\rangle$">|; 

$key = q/|Brangle+|Crangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="$ \vert B\rangle+\vert C\rangle$">|; 

$key = q/displaystylegamma^0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2603.png"
 ALT="$\displaystyle \gamma^0$">|; 

$key = q/xi';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img119.png"
 ALT="$ \xi'$">|; 

$key = q/displaystylepsi_b(y)=frac{{rme}^{-yslasha}}{y},g(y),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2870.png"
 ALT="$\displaystyle \psi_b(y) = \frac{{\rm e}^{-y/a}}{y} g(y),$">|; 

$key = q/displaystylec^-_{l,,m+1}=(c_{l,m}^+)^ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img824.png"
 ALT="$\displaystyle c^-_{l   m+1} = (c_{l m}^+)^\ast$">|; 

$key = q/E_m=E_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1939.png"
 ALT="$ E_m = E_i$">|; 

$key = q/J_1^{,2},J_2^{,2},J^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1350.png"
 ALT="$ J_1^{ 2}, J_2^{ 2}, J^{ 2}$">|; 

$key = q/langleArangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img361.png"
 ALT="$ \langle A\rangle$">|; 

$key = q/displaystyleL^+|l,m_{rmmax}rangle=|0rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img803.png"
 ALT="$\displaystyle L^+ \vert l, m_{\rm max}\rangle = \vert\rangle.$">|; 

$key = q/p_1cdotsp_N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img414.png"
 ALT="$ p_1\cdots p_N$">|; 

$key = q/langleL_yrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img996.png"
 ALT="$ \langle L_y\rangle$">|; 

$key = q/l=l'=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1523.png"
 ALT="$ l=l'=0$">|; 

$key = q/a_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1771.png"
 ALT="$ a_0$">|; 

$key = q/displaystyle|Arangle=sum_nc_n,|nrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1779.png"
 ALT="$\displaystyle \vert A\rangle = \sum_n c_n  \vert n\rangle,$">|; 

$key = q/t_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img530.png"
 ALT="$ t_0$">|; 

$key = q/displaystyle{mitDelta}E_{nlm-};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1678.png"
 ALT="$\displaystyle {\mit\Delta} E_{nlm-}$">|; 

$key = q/displaystylecdot{bfL}+hbar)^2=J^{,2}+frac{1}{4},hbar^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2795.png"
 ALT="$\displaystyle \cdot {\bf L} + \hbar)^2 = J^{ 2}+\frac{1}{4} \hbar^2.$">|; 

$key = q/displaystylep_x=-{rmi},hbar,left(frac{d}{dx}right)_{rmnew}-hbar,frac{dgamma}{dx}+f(x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img364.png"
 ALT="$\displaystyle p_x = -{\rm i}  \hbar \left(\frac{d}{dx}\right)_{\rm new} - \hbar \frac{d\gamma}{dx} + f(x).$">|; 

$key = q/r_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2513.png"
 ALT="$ r_1$">|; 

$key = q/displaystyle={rmi},hbarleft(-L_x,y-y,L_x+L_y,x+x,L_yright)=2,{rmi},hbar,(L_y,x-L_x,y+{rmi},hbar,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="454" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1507.png"
 ALT="$\displaystyle = {\rm i} \hbar\left( -L_x  y - y  L_x + L_y  x + x  L_y\right)= 2  {\rm i}   \hbar  ( L_y  x - L_x  y + {\rm i} \hbar z)$">|; 

$key = q/displaystylesum_{xi'}|xi'ranglelanglexi'|=1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img138.png"
 ALT="$\displaystyle \sum_{\xi'} \vert\xi'\rangle \langle \xi'\vert = 1,$">|; 

$key = q/displaystyleL^-,|l,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img845.png"
 ALT="$\displaystyle L^-  \vert l,m \rangle$">|; 

$key = q/+hbarslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1243.png"
 ALT="$ +\hbar/2$">|; 

$key = q/displaystylelangle1,0,0|,y,|2,1,pm1rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2131.png"
 ALT="$\displaystyle \langle 1,0,0\vert y \vert 2,1,\pm 1\rangle$">|; 

$key = q/displaystyle|A'rangle=S_k,|Arangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1155.png"
 ALT="$\displaystyle \vert A'\rangle = S_k  \vert A\rangle.$">|; 

$key = q/displaystylee_{11};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1447.png"
 ALT="$\displaystyle e_{11}$">|; 

$key = q/k'a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2333.png"
 ALT="$ k'a$">|; 

$key = q/displaystyle|Arangle=D^{dag},|Adrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img454.png"
 ALT="$\displaystyle \vert A\rangle = D^{\dag }  \vert A d\rangle.$">|; 

$key = q/displaystyle={rme}^{,{rmi},(gamma'-gamma'')},delta(x'-x'')=delta(x'-x'').;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img356.png"
 ALT="$\displaystyle ={ \rm e}^{ {\rm i} (\gamma'-\gamma'')} \delta(x'-x'')= \delta(x'-x'').$">|; 

$key = q/m'=m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1504.png"
 ALT="$ m'=m$">|; 

$key = q/a_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2235.png"
 ALT="$ a_l$">|; 

$key = q/A,A^dag;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img223.png"
 ALT="$ A A^\dag$">|; 

$key = q/displaystyleE_{nl}'=E_n+e_{nlnl}+sum_{n',l'neqn,l}frac{|e_{n'l'nl}|^{,2}}{E_n-E_{n'}}+O(epsilon^3),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="322" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1549.png"
 ALT="$\displaystyle E_{nl}' = E_n + e_{nlnl} + \sum_{n', l' \neq n,l} \frac{\vert e_{n'l'nl}\vert^{ 2}}{E_n - E_{n'}} + O(\epsilon^3),$">|; 

$key = q/displaystyle|Rtrangle=|Atrangle+|Btrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img533.png"
 ALT="$\displaystyle \vert Rt\rangle = \vert At\rangle + \vert B t\rangle.$">|; 

$key = q/displaystyleT^{dag},T=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img541.png"
 ALT="$\displaystyle T^{\dag } T = 1.$">|; 

$key = q/rhorightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img650.png"
 ALT="$ \rho\rightarrow 0$">|; 

$key = q/displaystyle-frac{{rmi},(omega-omega_{21})slash2}{[gamma^2slashhbar^2+(omega-omeslashhbar^2+(omega-omega_{21})^{,2}slash4]^{1slash2},tright).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="624" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1816.png"
 ALT="$\displaystyle - \frac{{\rm i} (\omega-\omega_{21})/2 }{[\gamma^2/\hbar^2 + (\o...
...2] \sin\left( [\gamma^2/\hbar^2+(\omega-\omega_{21})^{ 2}/4]^{1/2} t\right).$">|; 

$key = q/displaystylep^{,mu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2667.png"
 ALT="$\displaystyle p^{ \mu}$">|; 

$key = q/-lleqm<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img911.png"
 ALT="$ -l\leq m&lt; 0$">|; 

$key = q/displaystyle=|k''',k'',k'rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2408.png"
 ALT="$\displaystyle = \vert k''' k'' k'\rangle.$">|; 

$key = q/displaystyleV(r)=frac{V_0,exp(-mu,r)}{mu,r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2197.png"
 ALT="$\displaystyle V(r) = \frac{V_0 \exp(-\mu  r)}{\mu  r},$">|; 

$key = q/langleL_xrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img995.png"
 ALT="$ \langle L_x\rangle$">|; 

$key = q/langlex'|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img304.png"
 ALT="$ \langle x'\vert A\rangle$">|; 

$key = q/displaystyle=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1438.png"
 ALT="$\displaystyle =$">|; 

$key = q/displaystyleR_l(r)=frac{u_l(r)}{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2277.png"
 ALT="$\displaystyle R_l (r) = \frac{u_l(r)}{r},$">|; 

$key = q/a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img202.png"
 ALT="$ a$">|; 

$key = q/displaystylechi=A,left({array}{c}1-2,{rmi}2{array}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1293.png"
 ALT="$\displaystyle \chi = A \left(\begin{array}{c}1-2 {\rm i}\ 2\end{array}\right)
$">|; 

$key = q/displaystylelangleB|Arangle=int_{-infty}^{+infty}dx',langleB|x'ranglelanglex'|Arangle=int_{-infty}^{+infty}dx',psi_B^ast(x'),psi_A'(x'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="429" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img314.png"
 ALT="$\displaystyle \langle B\vert A\rangle = \int_{-\infty}^{+\infty} dx' \langle B...
...\vert A \rangle = \int_{-\infty}^{+\infty} dx' \psi_B^\ast (x')  \psi_A'(x'),$">|; 

$key = q/displaystyle|Erangle=sum_klanglek|Erangle|krangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1468.png"
 ALT="$\displaystyle \vert E\rangle = \sum_k \langle k \vert E\rangle \vert k\rangle,$">|; 

$key = q/delta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2299.png"
 ALT="$ \delta_0$">|; 

$key = q/exp(,{rmi},k,r');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2213.png"
 ALT="$ \exp( {\rm i} k  r')$">|; 

$key = q/displaystyle[gamma^5,Sigma_i]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2734.png"
 ALT="$\displaystyle [\gamma^5, \Sigma_i]=0.$">|; 

$key = q/(2p)_{3slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1772.png"
 ALT="$ (2p)_{3/2}$">|; 

$key = q/displaystylesum_{n>1}|langle1,0,0|,z,|n,1,0rangle|^{,2}=sum_{n',l',m'}langle1,0,langlen',m',l'|,z,|1,0,0rangle=langle1,0,0|,z^2,|1,0,0rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="672" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1537.png"
 ALT="$\displaystyle \sum_{n&gt;1} \vert\langle 1,0,0\vert z \vert n,1,0\rangle\vert^{\...
...m',l'\vert z \vert 1,0,0\rangle= \langle 1,0,0\vert z^2 \vert 1,0,0\rangle,$">|; 

$key = q/displaystyle=2,delta_{ij},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2591.png"
 ALT="$\displaystyle =2 \delta_{ij},$">|; 

$key = q/displaystyleD({mitDelta}x)=expleft({-rmi},p_x,{mitDelta}xslashhbarright).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img488.png"
 ALT="$\displaystyle D({\mit\Delta} x) = \exp\left({-\rm i}  p_x {\mit\Delta} x /\hbar\right).$">|; 

$key = q/displaystylesigma_{rmtotal}simeqfrac{4pi}{k^2}sin^2delta_0=4pi,a^2left[frac{tan(k',a)}{(k',a)}-1right]^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2330.png"
 ALT="$\displaystyle \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}.$">|; 

$key = q/lambda_1=3,e,a_0,|{bfE}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1580.png"
 ALT="$ \lambda_1= 3 e a_0 \vert{\bf E}\vert$">|; 

$key = q/|langleA|xi'rangle|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img131.png"
 ALT="$ \vert\langle A\vert \xi'\rangle\vert^{ 2}$">|; 

$key = q/displaystyleL_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img728.png"
 ALT="$\displaystyle L_x$">|; 

$key = q/displaystylec_n^{(1)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2000.png"
 ALT="$\displaystyle c_n^{(1)}$">|; 

$key = q/{bfe}_r={bfx}slashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1736.png"
 ALT="$ {\bf e}_r = {\bf x}/r$">|; 

$key = q/displaystylerightarrowfrac{sin(y-l,pislash2)}{y},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2232.png"
 ALT="$\displaystyle \rightarrow \frac{\sin(y - l \pi/2)}{y},$">|; 

$key = q/{mitDelta}p_xsimhbarslash{mitDelta}x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img439.png"
 ALT="$ {\mit\Delta} p_x \sim \hbar /{\mit\Delta} x$">|; 

$key = q/1,-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2463.png"
 ALT="$ 1,-1$">|; 

$key = q/lambda_4=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1583.png"
 ALT="$ \lambda_4 =0$">|; 

$key = q/displaystyle=sqrt{n+1},|E_{n+1}rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img689.png"
 ALT="$\displaystyle = \sqrt{n+1} \vert E_{n+1}\rangle.$">|; 

$key = q/displaystylelangleB|Arangle=langleA|Brangle^ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img214.png"
 ALT="$\displaystyle \langle B\vert A \rangle = \langle A\vert B \rangle^\ast
$">|; 

$key = q/displaystylelanglephi,frac{d}{dx}=-langlefrac{dphi}{dx}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img343.png"
 ALT="$\displaystyle \langle \phi  \frac{d}{dx} = - \langle \frac{d\phi}{dx}.$">|; 

$key = q/{bfS}=(hbarslash2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2777.png"
 ALT="$ {\bf S} = (\hbar/2) $">|; 

$key = q/P_{irightarrow[n]};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1921.png"
 ALT="$ P_{i\rightarrow [n]}$">|; 

$key = q/dpsi(x')slashdx';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img332.png"
 ALT="$ d\psi(x')/dx'$">|; 

$key = q/displaystyletau=frac{a_0}{4,alpha^4,c}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img211.png"
 ALT="$\displaystyle \tau=\frac{a_0}{4 \alpha^4 c}.
$">|; 

$key = q/{bfL}=sum_{i=1,N}{bfL}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img755.png"
 ALT="$ {\bf L} = \sum_{i=1,N} {\bf L}_i$">|; 

$key = q/hbar,{bfk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2185.png"
 ALT="$ \hbar { \bf k}$">|; 

$key = q/displaystyle(chi^dagger,sigma_k,chi)'equiv(chi^dagger)'sigma_k,chi'=sum_lR_{k,l},(chi^daggersigma_l,chi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="319" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1233.png"
 ALT="$\displaystyle (\chi^\dagger  \sigma_k  \chi)' \equiv (\chi^\dagger)' \sigma_k  \chi' = \sum_l R_{k l}  (\chi^\dagger \sigma_l  \chi),$">|; 

$key = q/displaystyle(1s)^2(2s)^2(2p)^6(3s).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1681.png"
 ALT="$\displaystyle (1s)^2 (2s)^2(2p)^6(3s).$">|; 

$key = q/gamma_mu=g_{mu,nu},gamma^nu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2690.png"
 ALT="$ \gamma_\mu = g_{\mu \nu} \gamma^\nu$">|; 

$key = q/displaystylelanglexirangle=sum_{xi'}xi'P(xi')=sum_{xi'}xi',|langleA|xi'rangle|^{xi'|Arangle=sum_{xi'}langleA|,xi,|xi'ranglelanglexi'|Arangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="572" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img143.png"
 ALT="$\displaystyle \langle\xi\rangle = \sum_{\xi'} \xi' P(\xi') = \sum_{\xi'} \xi' ...
...= \sum_{\xi'} \langle A\vert \xi \vert\xi'\rangle \langle \xi'\vert A\rangle,$">|; 

$key = q/m_l+2,m_s=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1713.png"
 ALT="$ m_l+2 m_s = 0$">|; 

$key = q/displaystyleJ_1^{,2},|j_1,j_2;m_1,m_2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1342.png"
 ALT="$\displaystyle J_1^{ 2}  \vert j_1,j_2; m_1,m_2\rangle$">|; 

$key = q/|{bfx}|rightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2647.png"
 ALT="$ \vert{\bf x}\vert\rightarrow
\infty$">|; 

$key = q/displaystyle(1+deltax,d_x^{,dag}),(1+deltax,d_x)=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img471.png"
 ALT="$\displaystyle (1+ \delta x d_x^{ \dag })  (1+ \delta x d_x) = 1.$">|; 

$key = q/displaystyle=[L_x^{,2},z]+[L_y^{,2},z]=L_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="494" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1506.png"
 ALT="$\displaystyle = [L_x^{ 2}, z] + [L_y^{ 2}, z] = L_x [L_x, z] + [L_x, z]  L_x + L_y [L_y, z] + [L_y, z]  L_y$">|; 

$key = q/2p(m=1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1609.png"
 ALT="$ 2p(m=1)$">|; 

$key = q/displaystyleL_z=-{rmi},hbar,frac{partial}{partialvarphi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img851.png"
 ALT="$\displaystyle L_z = -{\rm i} \hbar  \frac{\partial}{\partial \varphi}.$">|; 

$key = q/displaystyleleft(p^0-alpha_1,p^1-alpha_2,p^2-alpha_3,p^3-beta,m_e,cright)psi=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2587.png"
 ALT="$\displaystyle \left(p^0 - \alpha_1 p^1-\alpha_2 p^2-\alpha_3 p^3-\beta m_e c\right)\psi = 0,$">|; 

$key = q/delta_{i,j}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="$ \delta_{i j}=0$">|; 

$key = q/E_i+{rmRe}({mitDelta}_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1997.png"
 ALT="$ E_i +{\rm Re}({\mit\Delta}_i)$">|; 

$key = q/beta^{,2}=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2827.png"
 ALT="$ \beta^{ 2}=1$">|; 

$key = q/n=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1761.png"
 ALT="$ n=3$">|; 

$key = q/displaystyleleft|frac{m}{2pi,hbar^2}intd^3x',frac{exp(,{rmi},k,r')}{r'},V({bfx}')right|ll1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2211.png"
 ALT="$\displaystyle \left\vert \frac{m}{2\pi  \hbar^2} \int d^3 x' \frac{ \exp( {\rm i}  k  r')}{r'}  V({\bf x}') \right\vert \ll 1.$">|; 

$key = q/displaystyle{mitDelta}E_{nlm}=frac{alpha^2,E_n}{n^2}left(frac{n}{j+1slash2}-frac{3}{4}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1764.png"
 ALT="$\displaystyle {\mit\Delta} E_{nlm} = \frac{\alpha^2 E_n}{n^2}\left(\frac{n}{j+1/2}-\frac{3}{4}\right).
$">|; 

$key = q/displaystylelangleS_kranglerightarrowsum_lR_{k,l},langleS_lrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1095.png"
 ALT="$\displaystyle \langle S_k \rangle \rightarrow \sum_l R_{k l}  \langle S_l\rangle,$">|; 

$key = q/displaystyleE-V_0=frac{hbar^2,k'^{,2}}{2,m}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2317.png"
 ALT="$\displaystyle E - V_0 = \frac{\hbar^2  k'^{ 2}}{2 m}.$">|; 

$key = q/lambda_{12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2434.png"
 ALT="$ \lambda_{12}$">|; 

$key = q/S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2652.png"
 ALT="$ S$">|; 

$key = q/displaystyleE_n'=E_n+e_{nn}+sum_{kneqn}frac{|e_{nk}|^{,2}}{E_n-E_k}+O(epsilon^3),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1490.png"
 ALT="$\displaystyle E_n' = E_n + e_{nn} + \sum_{k\neq n} \frac{\vert e_{nk}\vert^{ 2}} {E_n-E_k} + O(\epsilon^3),$">|; 

$key = q/3p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1683.png"
 ALT="$ 3p$">|; 

$key = q/d^3x_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2483.png"
 ALT="$ d^3x_2$">|; 

$key = q/displaystylefrac{d}{dxi}!left[(1-xi^{,2}),frac{dP_{l,m}}{dxi}right]-frac{m^2}{1-xi^{,2}},P_{l,m}+l,(l+1),P_{l,m}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img909.png"
 ALT="$\displaystyle \frac{d}{d\xi}\! \left[ (1-\xi^{ 2}) \frac{dP_{l m}}{d\xi}\right] - \frac{m^2}{1-\xi^{ 2}}  P_{l m} + l (l+1) P_{l m} = 0.$">|; 

$key = q/displaystyleexp[,{rmi},(omegaslashc),{bfn}cdot{bfx}]=1+{rmi},frac{omega}{c},{bfn}cdot{bfx}+cdots,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2070.png"
 ALT="$\displaystyle \exp[ {\rm i} (\omega/c) {\bf n}\cdot{\bf x}] = 1 + {\rm i} \frac{\omega}{c}  {\bf n}\cdot{\bf x} + \cdots,$">|; 

$key = q/displaystyleH(t_0)={rmi},hbar,lim_{trightarrowt_0}frac{T(t,t_0)-1}{t-t_0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img557.png"
 ALT="$\displaystyle H(t_0) = {\rm i} \hbar \lim_{t\rightarrow t_0} \frac{T(t, t_0)-1}{t-t_0}.$">|; 

$key = q/partialslashpartialt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2586.png"
 ALT="$ \partial/\partial t$">|; 

$key = q/langle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img325.png"
 ALT="$ \langle$">|; 

$key = q/j_1,(j_1+1),hbar^2,j_2,(j_2+1),hbar^2,j,(j+1),hbar^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1364.png"
 ALT="$ j_1 (j_1+1) \hbar^2, j_2 (j_2+1) \hbar^2, j (j+1) \hbar^2$">|; 

$key = q/displaystylexi,|xi'rangle=xi',|xi'rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img191.png"
 ALT="$\displaystyle \xi  \vert\xi'\rangle =\xi'  \vert\xi'\rangle.$">|; 

$key = q/displaystyle{bfL}times{bfp}+{bfp}times{bfL}=2,{rmi},hbar,{bfp}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2805.png"
 ALT="$\displaystyle {\bf L}\times {\bf p} + {\bf p}\times{\bf L} = 2 {\rm i} \hbar {\bf p}.$">|; 

$key = q/displaystylex(t)=x(0)-frac{hbar^2,c}{4},dot{alpha}_1(0),expleft(frac{-2,{rmi},H,t}{hbar}right)H^{,-2}+c^2,p_x,H^{,-1},t.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="439" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2721.png"
 ALT="$\displaystyle x(t) = x(0)- \frac{\hbar^2 c}{4} \dot{\alpha}_1(0) \exp\left(\frac{-2 {\rm i} H t}{\hbar}\right)H^{ -2} + c^2 p_x H^{ -1} t.$">|; 

$key = q/displaystyle[L^+,L^-];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img778.png"
 ALT="$\displaystyle [L^+, L^-]$">|; 

$key = q/H_B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1693.png"
 ALT="$ H_B$">|; 

$key = q/H_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1434.png"
 ALT="$ H_0$">|; 

$key = q/rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img323.png"
 ALT="$ \rangle$">|; 

$key = q/displaystyleH=-e,phi(r)+c,epsilon,(p_r-{rmi},hbarslashr)+{rmi},c,epsilon,beta,zetaslashr+beta,m_e,c^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="394" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2855.png"
 ALT="$\displaystyle H= - e \phi(r) + c \epsilon (p_r-{\rm i} \hbar/r) + {\rm i} c \epsilon \beta \zeta/r + \beta m_e c^2.$">|; 

$key = q/displaystyle_e=-frac{e}{m_e},{bfS}_e.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1739.png"
 ALT="$\displaystyle _e = - \frac{e}{m_e} {\bf S}_e.$">|; 

$key = q/langleA|Arangle=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img311.png"
 ALT="$ \langle A\vert A\rangle = 1$">|; 

$key = q/displaystylecdot{bfL})=-2,hbar,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2807.png"
 ALT="$\displaystyle \cdot{\bf L}) =-2 \hbar $">|; 

$key = q/displaystyle_p=frac{g_p,e}{2,m_p},{bfS}_p,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1730.png"
 ALT="$\displaystyle _p = \frac{g_p e}{2 m_p} {\bf S}_p,$">|; 

$key = q/displaystyle{rmi},hbar,dot{alpha}_1=alpha_1,H-H,alpha_1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2713.png"
 ALT="$\displaystyle {\rm i} \hbar \dot{\alpha}_1 =\alpha_1 H-H \alpha_1.$">|; 

$key = q/displaystyleV(r)=V_0,exp(-r^2slasha^2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2353.png"
 ALT="$\displaystyle V(r)=V_0 \exp(-r^2/a^2).
$">|; 

$key = q/displaystyle=sum_sc_s',y^s,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2877.png"
 ALT="$\displaystyle = \sum_s c_s' y^s,$">|; 

$key = q/u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img247.png"
 ALT="$ u$">|; 

$key = q/displaystyle[zeta,r]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2820.png"
 ALT="$\displaystyle [\zeta,r] = 0.$">|; 

$key = q/displaystyle=E,|k',k''',k''rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2424.png"
 ALT="$\displaystyle = E \vert k' k''' k''\rangle,$">|; 

$key = q/l_{rmmax}=k,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2304.png"
 ALT="$ l_{\rm max} = k a$">|; 

$key = q/a^{,mu}_{~nu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2656.png"
 ALT="$ a^{ \mu}_{&nbsp;\nu}$">|; 

$key = q/displaystylek'a=sqrt{k^2,a^2+frac{2,m,|V_0|,a^2}{hbar^2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2331.png"
 ALT="$\displaystyle k'a = \sqrt{ k^2  a^2 + \frac{2  m  \vert V_0\vert  a^2}{\hbar^2}},$">|; 

$key = q/displaystyleleft(frac{d^2}{dy^2}-2,frac{d}{dy}-frac{l,(l+1)}{y^2}+frac{2,mu,e^2,a}{4pi,epsilon_0,hbar^2,y}right)f=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img950.png"
 ALT="$\displaystyle \left(\frac{d^2}{dy^2} -2 \frac{d}{dy} -\frac{l (l+1)}{y^2} + \frac{2 \mu  e^2  a}{4\pi  \epsilon_0  \hbar^2 y}\right) f= 0.$">|; 

$key = q/displaystyleexp[,{rmi},(omega-omega_{21}),tslash2],cosleft([gamma^2slashhbar^2+(omega-omega_{21})^{,2}slash4]^{1slash2},tright);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="409" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1815.png"
 ALT="$\displaystyle \exp[ {\rm i} (\omega-\omega_{21}) t/2] \cos\left( [\gamma^2/\hbar^2+(\omega-\omega_{21})^{ 2}/4]^{1/2} t\right)$">|; 

$key = q/q_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img240.png"
 ALT="$ q_i$">|; 

$key = q/displaystylelangle({mitDelta}x)^2rangle,langle({mitDelta}p_x)^2rangle=(n+1slash2)^2,hbar^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img705.png"
 ALT="$\displaystyle \langle({\mit\Delta} x)^2\rangle \langle ({\mit\Delta}p_x)^2\rangle = (n+1/2)^2 \hbar^2
$">|; 

$key = q/A+A^dag;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img221.png"
 ALT="$ A+A^\dag$">|; 

$key = q/displaystylew_{2rightarrow1}^{rmspn}=frac{omega_{21}^{,3},e^2,f_{21}^{,2}}{3pi,epsilon_0,hbar,c^3}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2116.png"
 ALT="$\displaystyle w_{2\rightarrow 1}^{\rm spn} = \frac{\omega_{21}^{ 3} e^2 f_{21}^{ 2}}{3\pi \epsilon_0 \hbar c^3}.$">|; 

$key = q/w_{2rightarrow1}^{rmspn}llomega_{21};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2121.png"
 ALT="$ w_{2\rightarrow 1}^{\rm spn} \ll \omega_{21}$">|; 

$key = q/displaystyleg_{11};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2557.png"
 ALT="$\displaystyle g_{11}$">|; 

$key = q/k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img437.png"
 ALT="$ k$">|; 

$key = q/displaystyleint_{-infty}^{+infty}dx',|x'ranglelanglex'|=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img302.png"
 ALT="$\displaystyle \int_{-\infty}^{+\infty} d x'   \vert x'\rangle\langle x'\vert= 1.$">|; 

$key = q/displaystyleX+Y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img84.png"
 ALT="$\displaystyle X + Y$">|; 

$key = q/R_{kl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1234.png"
 ALT="$ R_{kl}$">|; 

$key = q/H_{LS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1617.png"
 ALT="$ H_{LS}$">|; 

$key = q/displaystylec_1,|Arangle+c_2,|Branglestackrel{rmDC}{longleftrightarrow}c_1^ast,langleA|+c_2^ast,langleB|.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="$\displaystyle c_1  \vert A\rangle + c_2  \vert B\rangle \stackrel{\rm DC}{\longleftrightarrow} c_1^\ast  \langle A\vert + c_2^\ast  \langle B\vert.$">|; 

$key = q/displaystyleleft(frac{H}{c}+frac{e}{c},phiright)^2=left[mbox{boldmathSigma}cdot({bfp}+e,{bfA})right]^{,2}+m_e^{,2},c^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2735.png"
 ALT="$\displaystyle \left(\frac{H}{c}+\frac{e}{c} \phi\right)^2 = \left[\mbox{\boldmath$\Sigma$}\cdot({\bf p}+e {\bf A})\right]^{ 2} + m_e^{ 2} c^2.$">|; 

$key = q/H';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img629.png"
 ALT="$ H'$">|; 

$key = q/{bfn};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1214.png"
 ALT="$ {\bf n}$">|; 

$key = q/n=i+j+1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2918.png"
 ALT="$ n= i + j+1/2$">|; 

$key = q/1slash|x|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1897.png"
 ALT="$ 1/\vert x\vert$">|; 

$key = q/displaystylecdot{bfx}]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2819.png"
 ALT="$\displaystyle \cdot {\bf x}] = 0.$">|; 

$key = q/displaystyle|j_1,j_2;m_1,m_2rangle=sum_{j}sum_mlanglej_1,j_2;j,m|j_1,j_2;m_1,m_2rangle|j_1,j_2;j,mrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="471" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1397.png"
 ALT="$\displaystyle \vert j_1,j_2; m_1,m_2\rangle = \sum_{j}\sum_m \langle j_1, j_2; j, m\vert j_1, j_2; m_1, m_2\rangle \vert j_1,j_2; j,m\rangle.$">|; 

$key = q/displaystyle=sqrt{l,(l+1)-m,(m-1)},hbar,|l,m-1rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img846.png"
 ALT="$\displaystyle = \sqrt{l (l+1)-m (m-1)} \hbar \vert l, m-1\rangle,$">|; 

$key = q/p^0={rmi},hbar,partialslashpartialx_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2579.png"
 ALT="$ p^0={\rm i} \hbar \partial/\partial x_0$">|; 

$key = q/displaystyle=hbar,omega,A^dag.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img673.png"
 ALT="$\displaystyle = \hbar \omega A^\dag .$">|; 

$key = q/displaystyleleft(frac{1}{sintheta}frac{partial}{partialtheta},sintheta,frac{partin^2theta}frac{partial^2}{partialvarphi^2}right)Y+lambda,Y=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img904.png"
 ALT="$\displaystyle \left( \frac{1}{\sin\theta}\frac{\partial}{\partial\theta}  \sin...
...}{\sin^2\theta}\frac{\partial^2} {\partial\varphi^2}\right)Y + \lambda  Y = 0,$">|; 

$key = q/displaystyle=A+{rmi},lambda,[G,A]+left(frac{{rmi}^2lambda^2}{2!}right)[G,[G,A]]+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1087.png"
 ALT="$\displaystyle = A + {\rm i}  \lambda  [G,A] + \left(\frac{{\rm i}^2 \lambda^2}{2!}\right) [G, [G,A]]+$">|; 

$key = q/displaystyleU=intdomega,u(omega),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2062.png"
 ALT="$\displaystyle U = \int d\omega u(\omega),$">|; 

$key = q/displaystyle|Arangle=sum_{n=-infty,infty}c_n,|k'+n,k_arangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img521.png"
 ALT="$\displaystyle \vert A\rangle = \sum_{n=-\infty,\infty}c_n \vert k'+n k_a\rangle,
$">|; 

$key = q/displaystylexi,|xi'eta'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img156.png"
 ALT="$\displaystyle \xi \vert\xi' \eta'\rangle$">|; 

$key = q/displaystyle|Brangle,langleA|Crangle=langleA|Crangle,|Brangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img106.png"
 ALT="$\displaystyle \vert B \rangle \langle A \vert C\rangle = \langle A \vert C\rangle  \vert B \rangle,$">|; 

$key = q/displaystylelanglem|Eranglesimeq-frac{e_{mn}}{E_m-E_n}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1486.png"
 ALT="$\displaystyle \langle m\vert E\rangle \simeq -\frac{e_{mn}}{E_m - E_n}.$">|; 

$key = q/p_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1197.png"
 ALT="$ p_k$">|; 

$key = q/H_{ni}=langlen|,H_1,|irangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1951.png"
 ALT="$ H_{ni} = \langle n\vert H_1 \vert i\rangle$">|; 

$key = q/m_l+2,m_s=-2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1715.png"
 ALT="$ m_l+2 m_s = -2$">|; 

$key = q/c_{nneqi}(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1949.png"
 ALT="$ c_{n \neq i}(t)$">|; 

$key = q/displaystyle{mitDelta}x,{mitDelta}p_xsimhbar.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img441.png"
 ALT="$\displaystyle {\mit\Delta} x  {\mit\Delta} p_x \sim \hbar.$">|; 

$key = q/|Brangle=xi,|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img790.png"
 ALT="$ \vert B\rangle = \xi  \vert A\rangle$">|; 

$key = q/displaystylefrac{dsigma}{d{mitOmega}}=A+B,costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2359.png"
 ALT="$\displaystyle \frac{d\sigma}{d{\mit\Omega} }= A + B \cos\theta.
$">|; 

$key = q/P(xi');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img142.png"
 ALT="$ P(\xi')$">|; 

$key = q/displaystylecdot{bfp}+beta,m_e,c^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2781.png"
 ALT="$\displaystyle \cdot{\bf p} + \beta m_e c^2,$">|; 

$key = q/displaystylerightarrow{bfp}-q,{bfA},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2024.png"
 ALT="$\displaystyle \rightarrow {\bf p} - q { \bf A},$">|; 

$key = q/[u_1,u_2,v_1,v_2]_{qm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img270.png"
 ALT="$ [u_1  u_2, v_1  v_2]_{qm}$">|; 

$key = q/displaystyle{bfL}cdot{bfS},|l+1slash2,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1664.png"
 ALT="$\displaystyle {\bf L} \cdot{\bf S}  \vert l+ 1/2, m\rangle$">|; 

$key = q/displaystyleS^-,|s,jrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1276.png"
 ALT="$\displaystyle S^-  \vert s, j\rangle$">|; 

$key = q/D,exp(,{rmi},gamma);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img466.png"
 ALT="$ D  \exp( {\rm i} \gamma)$">|; 

$key = q/displaystylelangle_{rmnew}=langle_{rmold},{rme}^{,{rmi},gamma}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img360.png"
 ALT="$\displaystyle \langle_{\rm new} = \langle_{\rm old}  {\rm e}^{ {\rm i}  \gamma}.$">|; 

$key = q/U(t_0,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1852.png"
 ALT="$ U(t_0, t)$">|; 

$key = q/displaystylelangle{bfx}'|,H_1,|{bfx}rangle=V({bfx}),delta({bfx}-{bfx}').;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2152.png"
 ALT="$\displaystyle \langle {\bf x}'\vert H_1 \vert{\bf x}\rangle = V({\bf x})  \delta({\bf x} -{\bf x}').$">|; 

$key = q/displaystyle{rmi},hbar,dot{L}_x=[L_x,H].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2757.png"
 ALT="$\displaystyle {\rm i} \hbar \dot{L}_x = [L_x,H].$">|; 

$key = q/displaystyle(E_m+e_{mm}-E),langlem|Erangle+sum_{kneqm}e_{mk},langlek|Erangle=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1471.png"
 ALT="$\displaystyle (E_m + e_{mm} - E)  \langle m\vert E\rangle + \sum_{k\neq m} e_{mk}  \langle k\vert E\rangle = 0,$">|; 

$key = q/displaystyleleftlanglefrac{p_x^{,2}}{2,m}rightrangle=frac{1}{2}leftlanglex,frac{dV}{dx}rightrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img724.png"
 ALT="$\displaystyle \left\langle \frac{p_x^{ 2}}{2 m}\right\rangle =\frac{1}{2}\left\langle x \frac{dV}{dx}\right\rangle.
$">|; 

$key = q/displaystylea_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2548.png"
 ALT="$\displaystyle a_2$">|; 

$key = q/displaystylepsi(x');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img429.png"
 ALT="$\displaystyle \psi(x')$">|; 

$key = q/p_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img348.png"
 ALT="$ p_x$">|; 

$key = q/d^3x_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2482.png"
 ALT="$ d^3 x_1$">|; 

$key = q/S_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1018.png"
 ALT="$ S_z$">|; 

$key = q/psi(x^0,x^1,x^2,x^3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2540.png"
 ALT="$ \psi(x^0,x^1,x^2,x^3)$">|; 

$key = q/displaystylecdot{bfL};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2835.png"
 ALT="$\displaystyle \cdot{\bf L}$">|; 

$key = q/R({mitDelta}varphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img852.png"
 ALT="$ R({\mit\Delta}\varphi)$">|; 

$key = q/displaystyle=left(frac{m,hbar,omega}{2}right)left(-sqrt{n,(n-1)},delta_{n',n-2}-sqrt{(n+1),(n+2)},delta_{n',n+2}+(2,n+1),delta_{n',n}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="557" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img704.png"
 ALT="$\displaystyle = \left(\frac{m \hbar \omega}{2}\right)\left(-\sqrt{n (n-1)} ...
...n' n-2}-\sqrt{(n+1) (n+2)} \delta_{n' n+2}+ (2 n+1) \delta_{n',n}\right).$">|; 

$key = q/|phirangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2137.png"
 ALT="$ \vert\phi\rangle$">|; 

$key = q/displaystyleV_{ni}^dagger;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2005.png"
 ALT="$\displaystyle V_{ni}^\dagger$">|; 

$key = q/displaystyleP_l(cosgamma)=frac{4pi}{2,l+1}sum_{m=-l,+l}Y_{l,m}^ast(theta_1,varphi_1),Y_{l,m}(theta_2,varphi_2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="367" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2515.png"
 ALT="$\displaystyle P_l(\cos\gamma) = \frac{4\pi}{2 l+1}\sum_{m=-l,+l} Y_{l m}^\ast(\theta_1,\varphi_1) Y_{l m}(\theta_2,\varphi_2).$">|; 

$key = q/B_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1834.png"
 ALT="$ B_1$">|; 

$key = q/displaystyleleftlanglefrac{1}{r}rightrangle=frac{1}{n^2,a_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1005.png"
 ALT="$\displaystyle \left\langle \frac{1}{r}\right\rangle = \frac{1}{n^2 a_0}
$">|; 

$key = q/m_1=m_2=pm1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1410.png"
 ALT="$ m_1=m_2=\pm 1/2$">|; 

$key = q/displaystyleleft(frac{1}{a_2}+frac{alpha}{y}right)g-left(frac{d}{dy}-frac{1}{a}-frac{k}{y}right)f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2873.png"
 ALT="$\displaystyle \left(\frac{1}{a_2}+\frac{\alpha}{y}\right)g - \left(\frac{d}{d y}- \frac{1}{a}-\frac{k}{y}\right)f$">|; 

$key = q/displaystylesigma_l=frac{4pi}{k^2},(2,l+1),sin^2delta_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2263.png"
 ALT="$\displaystyle \sigma_l = \frac{4\pi}{k^2}  (2 l+1)  \sin^2\delta_l$">|; 

$key = q/x^0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2538.png"
 ALT="$ x^0$">|; 

$key = q/displaystylefrac{d^2c_2}{dt^2}+{rmi},(omega-omega_{21}),frac{dc_2}{dt}+frac{gamma^2}{hbar^2},c_2=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1809.png"
 ALT="$\displaystyle \frac{d^2 c_2}{dt^2} + {\rm i} (\omega-\omega_{21}) \frac{d c_2}{dt} + \frac{\gamma^2}{\hbar^2}  c_2 = 0.$">|; 

$key = q/displaystylefrac{{mitGamma}_i}{hbar}=sum_{mneqi}w_{irightarrowm}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1986.png"
 ALT="$\displaystyle \frac{{\mit\Gamma}_i}{\hbar} = \sum_{m\neq i} w_{i\rightarrow m}.$">|; 

$key = q/j_1,j_1-1,cdots,-j_1+1,-j_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1319.png"
 ALT="$ j_1, j_1-1,\cdots,
-j_1+1, -j_1$">|; 

$key = q/displaystylepsi_B(x')=f(x'),psi_A(x'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img318.png"
 ALT="$\displaystyle \psi_B(x') = f(x')  \psi_A(x'),$">|; 

$key = q/m_1+m_2neqm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1408.png"
 ALT="$ m_1+m_2\neq m$">|; 

$key = q/displaystyle||Arangleranglerightarrowleft(!{array}{c}psi_+({bfx}')psi_-({bfx}'){array}!right)equivchi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1187.png"
 ALT="$\displaystyle \vert\vert A\rangle\rangle \rightarrow \left(\! \begin{array}{c}\psi_+({\bf x}') \ \psi_-({\bf x}')\end{array}\!\right)\equiv \chi,$">|; 

$key = q/displaystyle[u,v]_{cl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img256.png"
 ALT="$\displaystyle [u, v]_{cl}$">|; 

$key = q/E=H';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img636.png"
 ALT="$ E= H'$">|; 

$key = q/displaystylefrac{c_n,y}{c_{n-1}}=frac{2,y}{n},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img965.png"
 ALT="$\displaystyle \frac{c_n  y}{c_{n-1}} = \frac{2  y}{n},$">|; 

$key = q/displaystyle=langlel,m|,L_x,|l,m+1rangle-{rmi},langlel,m|,L_y,|l,m+1rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img820.png"
 ALT="$\displaystyle = \langle l, m\vert L_x \vert l,m+1 \rangle - {\rm i}  \langle l, m\vert  L_y \vert l,m+1 \rangle$">|; 

$key = q/displaystyleA_l(r)=exp(,{rmi},delta_l),left[cosdelta_l,j_l(k,r)-sindelta_l,eta_l(k,r)right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2267.png"
 ALT="$\displaystyle A_l(r) = \exp( {\rm i}  \delta_l)  \left[ \cos\delta_l  j_l(k r) -\sin\delta_l  \eta_l(k r)\right].$">|; 

$key = q/hbarrightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img298.png"
 ALT="$ \hbar\rightarrow 0$">|; 

$key = q/E_{200}=-e^2slash(32pi,epsilon_0,a_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1573.png"
 ALT="$ E_{200} = -e^2/(32\pi \epsilon_0  a_0)$">|; 

$key = q/displaystyleA^dag,gamma^0,gamma^{,mu},A=a^mu_{~nu},gamma^0,gamma^nu.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2700.png"
 ALT="$\displaystyle A^\dag \gamma^0 \gamma^{ \mu} A = a^\mu_{&nbsp;\nu} \gamma^0 \gamma^\nu.$">|; 

$key = q/m<m_{rmmin};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img831.png"
 ALT="$ m&lt;m_{\rm min}$">|; 

$key = q/Z=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2489.png"
 ALT="$ Z=2$">|; 

$key = q/P_1(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1829.png"
 ALT="$ P_1(t)$">|; 

$key = q/displaystyle{mitDelta}E,{mitDelta}tstackrel{_{normalsize<}}{_{normalsizesim}}h,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1905.png"
 ALT="$\displaystyle {\mit\Delta} E  {\mit\Delta} t \stackrel {_{\normalsize &lt;}}{_{\normalsize\sim}}h,$">|; 

$key = q/displaystyle[J_2^{,2},J_{2,i}];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1329.png"
 ALT="$\displaystyle [J_2^{ 2}, J_{2 i} ]$">|; 

$key = q/displaystyleH,psi=E,psi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img932.png"
 ALT="$\displaystyle H \psi = E \psi,$">|; 

$key = q/j,l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1263.png"
 ALT="$ j, l$">|; 

$key = q/k=p_xslashhbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img518.png"
 ALT="$ k=p_x/\hbar$">|; 

$key = q/displaystyleA_0(r)=exp(,{rmi},delta_0),left[j_0(k,r)cosdelta_0-eta_0(k,r)sindelta_0right]=frac{exp(,{rmi},delta_0),sin(k,r+delta_0)}{k,r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="559" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2315.png"
 ALT="$\displaystyle A_0(r) = \exp( {\rm i}  \delta_0) \left[ j_0(k r) \cos\delta_...
...elta_0\right] = \frac{ \exp( {\rm i}  \delta_0)  \sin(k r+\delta_0)}{k r},$">|; 

$key = q/displaystylem^2leqf(l,m).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img792.png"
 ALT="$\displaystyle m^2 \leq f(l,m).$">|; 

$key = q/omega_{21};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1823.png"
 ALT="$ \omega_{21}$">|; 

$key = q/A,B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img218.png"
 ALT="$ A B$">|; 

$key = q/1,0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2462.png"
 ALT="$ 1,0$">|; 

$key = q/displaystyleL^4,z-2,L^2,z,L^2+z,L^4-2,hbar^2,(L^2,z+z,L^2)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1519.png"
 ALT="$\displaystyle L^4  z - 2  L^2  z  L^2 + z  L^4 - 2  \hbar^2  (L^2  z +z  L^2) = 0.$">|; 

$key = q/j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img741.png"
 ALT="$ j$">|; 

$key = q/displaystylepsi(x')=int_{-infty}^{+infty}dp_x',langlex'|p_x'ranglelanglep_x'|Araty}^{+infty}dp_x',{mitPsi}(p_x'),exp(,{rmi},p_x',x'slashhbar);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="532" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img433.png"
 ALT="$\displaystyle \psi(x') = \int_{-\infty}^{+\infty} dp_x'  \langle x'\vert p_x'\...
...{-\infty}^{+\infty} dp_x' {\mit\Psi}(p_x')   \exp( {\rm i}  p_x' x'/\hbar)$">|; 

$key = q/displaystylelangleS_zrangle_t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1127.png"
 ALT="$\displaystyle \langle S_z\rangle_t$">|; 

$key = q/displaystylebeta^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2593.png"
 ALT="$\displaystyle \beta^{ 2}$">|; 

$key = q/displaystyleexpleft(-{rmi},{bfS}cdot{bfn},{mitDelta}varphislashhbarright)rightarmi},mbox{boldmathsigma}cdot{bfn},{mitDelta}varphislash2right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1215.png"
 ALT="$\displaystyle \exp\left(-{\rm i}  {\bf S}\cdot{\bf n} {\mit\Delta}\varphi/\hb...
...t(-{\rm i}  \mbox{\boldmath$\sigma$}\cdot{\bf n} {\mit\Delta}\varphi/2\right)$">|; 

$key = q/H^{,4}=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img234.png"
 ALT="$ H^{ 4}=1$">|; 

$key = q/displaystyle|A_trangle=T^{dag}(t,t_0),|Arangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img562.png"
 ALT="$\displaystyle \vert A_t \rangle = T^{\dag }(t, t_0)  \vert A\rangle.$">|; 

$key = q/{rmi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="10" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2924.png"
 ALT="$ {\rm i}$">|; 

$key = q/displaystyle=frac{1}{hbar}left(frac{1-exp[,{rmi},(omega_{ni}+omega),t]}{omega_{nega_{ni}-omega),t]}{omega_{ni}-omega},V_{ni}^{,dagger}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="463" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2002.png"
 ALT="$\displaystyle = \frac{1}{\hbar} \left(\frac{1-\exp[ {\rm i} (\omega_{ni} + \o...
...,(\omega_{ni}-\omega ) t]} {\omega_{ni} - \omega}  V_{ni}^{ \dagger}\right),$">|; 

$key = q/displaystylefrac{c_{s-1}'}{a_2}+alpha,c_s'-(s-k),c_s+frac{c_{s-1}}{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2879.png"
 ALT="$\displaystyle \frac{c_{s-1}'}{a_2}+\alpha c_s' -(s-k) c_s + \frac{c_{s-1}}{a}$">|; 

$key = q/w_{2rightarrow1}^{rmspn};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2102.png"
 ALT="$ w_{2\rightarrow 1}^{\rm spn}$">|; 

$key = q/rrightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img964.png"
 ALT="$ r\rightarrow 0$">|; 

$key = q/displaystylephi({bfx}_1,{bfx}_2)=psi_{nlm}({bfx}_1),psi_{nlm}({bfx}_2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2499.png"
 ALT="$\displaystyle \phi({\bf x}_1,{\bf x}_2)=\psi_{nlm}({\bf x}_1) \psi_{nlm}({\bf x}_2).$">|; 

$key = q/delta_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2247.png"
 ALT="$ \delta_l$">|; 

$key = q/-infty<x'<+infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img299.png"
 ALT="$ -\infty&lt; x'&lt;+\infty$">|; 

$key = q/displaystylelangle({mitDelta}xi)^2ranglelangle({mitDelta}eta)^2ranglegeq|langle{mitDelta}xi,{mitDelta}etarangle|^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img177.png"
 ALT="$\displaystyle \langle ({\mit\Delta} \xi)^2\rangle \langle ({\mit\Delta} \eta)^2\rangle \geq \vert\langle {\mit\Delta} \xi  {\mit\Delta} \eta\rangle \vert^{ 2},$">|; 

$key = q/displaystyleO(epsilon),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1482.png"
 ALT="$\displaystyle O(\epsilon),$">|; 

$key = q/displaystyleleft({array}{cc}E_1-E+e_{11}&e_{12}e_{12}^{,ast}&E_2-E+e_{22}{array}ngle2|Erangle{array}!right)=left(!{array}{c}00{array}!right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="375" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1446.png"
 ALT="$\displaystyle \left( \begin{array}{c c} E_1 -E + e_{11} &amp; e_{12} \ e_{12}^{ \...
...ngle\end{array} \!\right)= \left(\!\begin{array}{c}0\ 0 \end{array}\! \right).$">|; 

$key = q/displaystylec_n(t)=langlen|,U_I(t_0,t),|irangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1865.png"
 ALT="$\displaystyle c_n(t) = \langle n\vert  U_I(t_0, t)  \vert i\rangle,$">|; 

$key = q/P_2(t)=|c_2|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1818.png"
 ALT="$ P_2(t) = \vert c_2\vert^{ 2}$">|; 

$key = q/s,s-1,cdots,-s+1,-s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1255.png"
 ALT="$ s, s-1, \cdots, -s+1, -s$">|; 

$key = q/sigma_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1272.png"
 ALT="$ \sigma_y$">|; 

$key = q/displaystyle-left(frac{hbar^2}{2,m}right)nabla'^{,2}psi_0({bfx'})+[V({bfx'})-E],psi_0({bfx'})=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img634.png"
 ALT="$\displaystyle -\left(\frac{\hbar^2}{2 m}\right)\nabla'^{ 2}\psi_0({\bf x'}) + [V({\bf x'})-E] \psi_0({\bf x'}) =0,$">|; 

$key = q/displaystyle{calE}=left(1+frac{alpha^2}{s^2}right)^{-1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2913.png"
 ALT="$\displaystyle {\cal E} = \left(1+\frac{\alpha^2}{s^2}\right)^{-1/2}.$">|; 

$key = q/displaystylec^+_{l,m},c^{-}_{l,,m+1}=[l,(l+1)-m,(m+1)].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img818.png"
 ALT="$\displaystyle c^+_{l m}  c^{-}_{l  m+1} = [l (l+1) - m (m+1)].$">|; 

$key = q/omeganeqomega_{21};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1827.png"
 ALT="$ \omega\neq \omega_{21}$">|; 

$key = q/f(l,m);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img784.png"
 ALT="$ f(l,m)$">|; 

$key = q/y,psi_arightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2885.png"
 ALT="$ y \psi_a\rightarrow 0$">|; 

$key = q/m_2=-1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2476.png"
 ALT="$ m_2=-1/2$">|; 

$key = q/displaystyle{mitDelta}E_{n,l,m_l,m_s}=frac{hbar^2,m_l,m_s}{2,m_e^{,2},c^2}leftlanglefrac{1}{r}frac{dV}{dr}rightrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1721.png"
 ALT="$\displaystyle {\mit\Delta} E_{n l  m_l  m_s} = \frac{\hbar^2  m_l  m_s} {2  m_e^{ 2} c^2} \left\langle \frac{1}{r} \frac{d V}{dr}\right\rangle.$">|; 

$key = q/mu=nu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2566.png"
 ALT="$ \mu=\nu$">|; 

$key = q/murightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2203.png"
 ALT="$ \mu \rightarrow 0$">|; 

$key = q/displaystylesum_{j=j_1-j_2,j_1+j_2}(2,j+1)equiv(2,j_1+1),(2,j_2+1).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="318" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1380.png"
 ALT="$\displaystyle \sum_{j=j_1-j_2,j_1+j_2} (2 j+1) \equiv (2 j_1+1) (2 j_2+1).$">|; 

$key = q/displaystyle=r,p_r+{rmi},(beta,zeta-hbar),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2836.png"
 ALT="$\displaystyle = r p_r + {\rm i} (\beta \zeta-\hbar),$">|; 

$key = q/displaystyle{bfx}_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1588.png"
 ALT="$\displaystyle {\bf x}_3$">|; 

$key = q/y=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img961.png"
 ALT="$ y=0$">|; 

$key = q/displaystylebeta_{l-}=beta_{l+}=infty,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2284.png"
 ALT="$\displaystyle \beta_{l-} = \beta_{l+} = \infty,$">|; 

$key = q/displaystylep_x(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img718.png"
 ALT="$\displaystyle p_x(t)$">|; 

$key = q/displaystylesigma_{rmtotal}=ointd{mitOmega},frac{dsigma}{d{mitOmega}}=4pi,a^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2301.png"
 ALT="$\displaystyle \sigma_{\rm total} = \oint d{\mit\Omega} \frac{d\sigma}{d{\mit\Omega}} = 4\pi  a^2$">|; 

$key = q/G;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1089.png"
 ALT="$ G$">|; 

$key = q/langleAd|=langleA|,D^{dag};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img451.png"
 ALT="$ \langle Ad\vert = \langle A\vert D^{\dag }$">|; 

$key = q/displaystyleH=-e,phi+c,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2727.png"
 ALT="$\displaystyle H =-e \phi + c $">|; 

$key = q/displaystyle=-[v,u]_{cl},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img257.png"
 ALT="$\displaystyle = - [v, u]_{cl},$">|; 

$key = q/exp(2,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img967.png"
 ALT="$ \exp(2 y)$">|; 

$key = q/displaystylecdot{bfL})=L^{,2}-hbar,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2791.png"
 ALT="$\displaystyle \cdot{\bf L}) = L^{ 2} -\hbar $">|; 

$key = q/displaystyle|k',k'',k'''rangle_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2445.png"
 ALT="$\displaystyle \vert k' k'' k'''\rangle_+$">|; 

$key = q/displaystyleleft[A,gamma^{,mu},A^{-1},a^nu_{~mu}left(p_{nu'}-frac{e}{c},{mitPhi}_{nu'}right)-m_e,cright]psi'=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="319" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2675.png"
 ALT="$\displaystyle \left[A \gamma^{ \mu} A^{-1} a^\nu_{&nbsp;\mu}\left(p_{\nu'}- \frac{e}{c} {\mit\Phi}_{\nu'}\right)-m_e c\right]\psi' = 0.$">|; 

$key = q/displaystylelangleB|,X^{dag},|Arangle=langleA|,X,|Brangle^ast.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img215.png"
 ALT="$\displaystyle \langle B\vert X^{\dag } \vert A\rangle = \langle A\vert X  \vert B\rangle^\ast.
$">|; 

$key = q/x_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img585.png"
 ALT="$ x_i$">|; 

$key = q/cotdelta_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2344.png"
 ALT="$ \cot \delta_l$">|; 

$key = q/displaystylesigma_{rmtotal}=frac{4pi}{k^2}sin^2delta_0=4pi,a^2left(frac{1}{k^2,a^2}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2340.png"
 ALT="$\displaystyle \sigma_{\rm total} = \frac{4\pi}{k^2} \sin^2\delta_0 = 4\pi  a^2 \left(\frac{1}{k^2  a^2}\right).$">|; 

$key = q/V({bfx'});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img626.png"
 ALT="$ V({\bf x'})$">|; 

$key = q/H_0,L^2,S^{,2},J^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1620.png"
 ALT="$ H_0, L^2, S^{ 2}, J^{ 2},$">|; 

$key = q/displaystyle{tabular}{||c|c||c|c|c||c|}hlinem_1&m_2&&&&hlinehline1slash2&1slash2j&1&1&1&0hlinescriptstylej_2=1slash2&m&1&0&-1&0hline{tabular};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="193" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1403.png"
 ALT="$\displaystyle \begin{tabular}{\vert\vert c\vert c\vert\vert c\vert c\vert c\ver...
... \hline
$\scriptstyle j_2=1/2$ &amp; $m$ &amp; 1 &amp; 0 &amp; -1 &amp; 0 \ \hline
\end{tabular}$">|; 

$key = q/E_2>E_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2100.png"
 ALT="$ E_2&gt; E_1$">|; 

$key = q/w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img269.png"
 ALT="$ w$">|; 

$key = q/displaystylej_l(k,r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2293.png"
 ALT="$\displaystyle j_l(k r)$">|; 

$key = q/displaystylea_0=frac{4pi,epsilon_0,hbar^2}{m_e,e^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1534.png"
 ALT="$\displaystyle a_0 = \frac{4\pi  \epsilon_0  \hbar^2}{m_e  e^2}$">|; 

$key = q/{bfS}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2455.png"
 ALT="$ {\bf S}_2$">|; 

$key = q/c,|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="$ c  \vert A\rangle$">|; 

$key = q/displaystyle=a^0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2545.png"
 ALT="$\displaystyle = a^0,$">|; 

$key = q/displaystyle=s_z,hbar,|s,s_zrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1021.png"
 ALT="$\displaystyle = s_z  \hbar  \vert s, s_z\rangle,$">|; 

$key = q/c_{s_0-1}=c_{s_0-1}'=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2893.png"
 ALT="$ c_{s_0-1}=c_{s_0-1}'=0$">|; 

$key = q/displaystylef({bfk}',{bfk})=-frac{(2pi)^2,m}{hbar^2}intd^3{bfx}',frac{exp(-{rmi}bfx}')=-frac{(2pi)^2,m}{hbar^2},langle{bfk}'|,H_1,|psirangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="592" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2174.png"
 ALT="$\displaystyle f({\bf k}', {\bf k}) = - \frac{(2\pi)^2  m}{\hbar^2} \int d^3{\b...
... - \frac{(2\pi)^2  m}{\hbar^2}  \langle {\bf k}'\vert H_1 \vert\psi\rangle.$">|; 

$key = q/s=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1284.png"
 ALT="$ s=1$">|; 

$key = q/displaystylephi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2039.png"
 ALT="$\displaystyle \phi$">|; 

$key = q/displaystyle(H_0+H_1),|A,t_0,trangle=sum_mc_m(t)exp[-{rmi},E_m,(t-t_0)slashhbar],(E_m+H_1),|mrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="501" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1789.png"
 ALT="$\displaystyle (H_0+H_1)  \vert A,t_0,t\rangle = \sum_m c_m(t) \exp[-{\rm i} E_m  (t-t_0)/\hbar]  (E_m + H_1) \vert m\rangle.$">|; 

$key = q/|2,0,0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1575.png"
 ALT="$ \vert 2,0,0\rangle$">|; 

$key = q/z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="$ z$">|; 

$key = q/|xi'_brangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img147.png"
 ALT="$ \vert\xi'_b\rangle$">|; 

$key = q/displaystylefrac{hbar}{2},dot{Sigma}_1=-c,gamma^5,(Sigma_2,p_z-Sigma_3,p_y).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2774.png"
 ALT="$\displaystyle \frac{\hbar}{2} \dot{\Sigma}_1 = -c \gamma^5 (\Sigma_2 p_z-\Sigma_3 p_y).$">|; 

$key = q/|x',y',z',+rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1050.png"
 ALT="$ \vert x', y', z', +\rangle$">|; 

$key = q/N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="$ N$">|; 

$key = q/displaystylepsi({bfx})=frac{1}{(2pi)^{3slash2}}sum_{l=0,infty}left[A_l,j_l(k,r)+B_l,eta_l(k,r)right]P_l(costheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="394" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2242.png"
 ALT="$\displaystyle \psi({\bf x}) = \frac{1}{(2\pi)^{3/2}} \sum_{l=0,\infty}\left[ A_l j_l(k r) + B_l \eta_l(k r)\right] P_l(\cos\theta),$">|; 

$key = q/phi({bfx});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2190.png"
 ALT="$ \phi({\bf x})$">|; 

$key = q/displaystylerightarrow|+rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1846.png"
 ALT="$\displaystyle \rightarrow \vert+\rangle,$">|; 

$key = q/displaystylechi(0,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2472.png"
 ALT="$\displaystyle \chi(0,0)$">|; 

$key = q/f(E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1992.png"
 ALT="$ f(E)$">|; 

$key = q/{rmi},hbar,d_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img482.png"
 ALT="$ {\rm i} \hbar  d_x$">|; 

$key = q/displaystylef(theta,varphi)=f(theta).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2218.png"
 ALT="$\displaystyle f(\theta, \varphi) = f(\theta).$">|; 

$key = q/z'+dz';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1068.png"
 ALT="$ z'+dz'$">|; 

$key = q/s_z=+1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1069.png"
 ALT="$ s_z = +1/2$">|; 

$key = q/displaystyle={rmi},hbar,{bfJ}_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1302.png"
 ALT="$\displaystyle = {\rm i} \hbar {\bf J}_1,$">|; 

$key = q/displaystyle=gamma^0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2598.png"
 ALT="$\displaystyle =\gamma^0,$">|; 

$key = q/a_0=4pi,epsilon_0,hbar^2slash(m_e,e^2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img209.png"
 ALT="$ a_0=4\pi \epsilon_0 \hbar^2/(m_e e^2)$">|; 

$key = q/[cdots,cdots]_{cl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img491.png"
 ALT="$ [\cdots,\cdots]_{cl}$">|; 

$key = q/displaystylesigma_{mu,nu}=frac{{rmi}}{2},[gamma_mu,gamma_nu].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2693.png"
 ALT="$\displaystyle \sigma_{\mu \nu} = \frac{{\rm i}}{2} [\gamma_\mu,\gamma_\nu].$">|; 

$key = q/displaystyleA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2694.png"
 ALT="$\displaystyle A$">|; 

$key = q/|eta'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img154.png"
 ALT="$ \vert\eta'\rangle$">|; 

$key = q/psi^dag;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2635.png"
 ALT="$ \psi^\dag$">|; 

$key = q/e>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1116.png"
 ALT="$ e&gt;0$">|; 

$key = q/A^i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2630.png"
 ALT="$ A^i$">|; 

$key = q/m_l+2,m_s=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1711.png"
 ALT="$ m_l+2 m_s = 2$">|; 

$key = q/displaystyleint!int!intdx'dy'dz'left(,|x',y',z',+ranglelanglex',y',z',+|+|x',y',z',-ranglelanglex',y',z',-|,right)=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="573" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1049.png"
 ALT="$\displaystyle \int\!\int\!\int dx'dy'dz' \left(  \vert x',y',z',+\rangle\langle x', y', z',+\vert+\vert x',y',z',-\rangle\langle x', y', z',-\vert  \right)= 1.$">|; 

$key = q/displaystylesqrt{frac{2,m,|V_0|,a^2}{hbar^2}}=frac{pi}{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2342.png"
 ALT="$\displaystyle \sqrt{\frac{2 m \vert V_0 \vert a^2}{\hbar^2} } = \frac{\pi}{2}$">|; 

$key = q/displaystyler^2frac{d^2R_l}{dr^2}+2,rfrac{dR_l}{dr}+[k^2,r^2-l,(l+1)],R_l=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2225.png"
 ALT="$\displaystyle 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.$">|; 

$key = q/displaystyleeta,|xi'eta'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img158.png"
 ALT="$\displaystyle \eta \vert\xi' \eta'\rangle$">|; 

$key = q/|Rdrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img444.png"
 ALT="$ \vert Rd\rangle$">|; 

$key = q/displaystylec_1,|k'rangle,|k''rangle+c_2,|k''rangle,|k'rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2378.png"
 ALT="$\displaystyle c_1 \vert k'\rangle \vert k''\rangle + c_2 \vert k''\rangle \vert k'\rangle,$">|; 

$key = q/displaystylerightarrowfrac{e,hbar,B_1}{2,m_e}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1850.png"
 ALT="$\displaystyle \rightarrow \frac{e \hbar  B_1}{2 m_e}.$">|; 

$key = q/displaystyle=+frac{1}{2,m_e^{,2},c^2}leftlanglefrac{1}{r}frac{dV}{dr}rightranglefrac{l,hbar^2}{2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1677.png"
 ALT="$\displaystyle = +\frac{1}{2 m_e^{ 2} c^2} \left\langle \frac{1}{r}\frac{dV}{dr} \right\rangle \frac{l \hbar^2}{2},$">|; 

$key = q/y=rslasha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img948.png"
 ALT="$ y=r/a$">|; 

$key = q/displaystyleH_1simeqfrac{e,{bfA}cdot{bfp}}{m_e},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2047.png"
 ALT="$\displaystyle H_1 \simeq \frac{e {\bf A}\cdot{\bf p}}{m_e},$">|; 

$key = q/displaystyle=s,(s+1),hbar^2,|s,s_zrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1023.png"
 ALT="$\displaystyle = s (s+1) \hbar^2  \vert s, s_z\rangle.$">|; 

$key = q/k_a=2pislasha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img523.png"
 ALT="$ k_a=2\pi/a$">|; 

$key = q/sigma_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2607.png"
 ALT="$ \sigma_i$">|; 

$key = q/displaystylecdot({bfa}times{bfb});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1207.png"
 ALT="$\displaystyle \cdot ({\bf a} \times {\bf b} )$">|; 

$key = q/psi_0({bfx'})equivpsi({bfx'},t_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img635.png"
 ALT="$ \psi_0({\bf x'}) \equiv \psi({\bf x'}, t_0)$">|; 

$key = q/displaystyle=frac{|xi_b'rangle-langlexi_a'|xi_b'rangle|xi_a'rangle}{[1-|langlexi_a'|xi_b'rangle|^{,2}]^{1slash2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img151.png"
 ALT="$\displaystyle =\frac{ \vert\xi_b'\rangle - \langle \xi_a'\vert\xi_b'\rangle \vert\xi_a'\rangle}{[1-\vert\langle \xi_a'\vert\xi_b'\rangle\vert^{ 2}]^{1/2}}.$">|; 

$key = q/langle{bfp}rangle_t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img654.png"
 ALT="$ \langle {\bf p} \rangle_t$">|; 

$key = q/dot{c_i}slashc_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1965.png"
 ALT="$ \dot{c_i}/c_i$">|; 

$key = q/m_l,m_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1722.png"
 ALT="$ m_l,m_s$">|; 

$key = q/J^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1324.png"
 ALT="$ J^{ 2}$">|; 

$key = q/displaystyleR_z({mitDelta}varphi_z),|0,0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img880.png"
 ALT="$\displaystyle R_z({\mit\Delta} \varphi_z)  \vert,0\rangle$">|; 

$key = q/displaystylepsi({bfx})=frac{1}{(2pi)^{3slash2}}sum_{l=0,infty}{rmi}^l,(2,l+1),A_l(r),P_l(costheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2266.png"
 ALT="$\displaystyle \psi({\bf x}) = \frac{1}{(2\pi)^{3/2}} \sum_{l=0,\infty} {\rm i}^l  (2 l+1)   A_l(r)  P_l(\cos\theta),$">|; 

$key = q/displaystyle=-frac{rmi}{hbar}int_{t_0}^tdt',langlen|,H_I(t_0,t'),|irangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1877.png"
 ALT="$\displaystyle = -\frac{\rm i}{\hbar} \int_{t_0}^t dt' \langle n \vert H_I(t_0, t') \vert i\rangle,$">|; 

$key = q/displaystyleJ^{pm}|j_1,j_2;j,mrangle=(J_1^pm+J_2^pm)sum_{m_1'}sum_{m_2'}langlej_1,j_2;m_1',m_2'|j_1,j_2;j,mrangle|j_1,j_2;m_1',m_2'rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="562" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1385.png"
 ALT="$\displaystyle J^{\pm} \vert j_1,j_2;j,m\rangle = (J_1^\pm + J_2^\pm )\sum_{m_1'...
..., j_2; m_1', m_2'\vert j_1, j_2; j, m\rangle \vert j_1, j_2; m_1', m_2'\rangle.$">|; 

$key = q/displaystyle|BranglelangleA|~?;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img105.png"
 ALT="$\displaystyle \vert B \rangle \langle A\vert&nbsp;?$">|; 

$key = q/displaystyle{sigma_i,sigma_j}=2,delta_{ij}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1179.png"
 ALT="$\displaystyle \{ \sigma_i, \sigma_j \} = 2  \delta_{ij}.$">|; 

$key = q/k,Rll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2357.png"
 ALT="$ k R\ll 1$">|; 

$key = q/displaystyleH_1=frac{e,B_1}{m_e}left[cos(omega,t),S_x+sin(omega,t),S_yright].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1838.png"
 ALT="$\displaystyle H_1 = \frac{e  B_1}{m_e} \left[\cos(\omega  t)  S_x + \sin(\omega  t)  S_y\right].$">|; 

$key = q/ineqj;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img754.png"
 ALT="$ i\neq j$">|; 

$key = q/displaystyle|Arangle=psi_A(x)rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img321.png"
 ALT="$\displaystyle \vert A\rangle = \psi_A(x) \rangle,$">|; 

$key = q/displaystyleint_{-infty}^{+infty}dp_x',|phi(p_x')|^{,2}=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img410.png"
 ALT="$\displaystyle \int_{-\infty}^{+\infty}dp_x'  \vert\phi(p_x')\vert^{ 2} = 1.$">|; 

$key = q/displaystyle=E,|k'',k''',k'rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2418.png"
 ALT="$\displaystyle = E \vert k'' k''' k'\rangle,$">|; 

$key = q/displaystyleX,|x'rangle=x',|x'rangle,~~~X,|x''rangle=x'',|x''rangle,~~dots,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img113.png"
 ALT="$\displaystyle X \vert x'\rangle = x' \vert x'\rangle,&nbsp;&nbsp;&nbsp;X \vert x''\rangle = x'' \vert x''\rangle,&nbsp;&nbsp;\dots,$">|; 

$key = q/{bfj}({bfx},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2645.png"
 ALT="$ {\bf j}({\bf x},t)$">|; 

$key = q/displaystylefrac{dp_i}{dt}=frac{1}{{rmi},hbar},[p_i,V({bfx})]=-frac{partialV({bfx})}{partialx_i},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img610.png"
 ALT="$\displaystyle \frac{d p_i}{dt} = \frac{1}{{\rm i} \hbar}  [p_i, V({\bf x})] = - \frac{\partial V({\bf x})}{\partial x_i},$">|; 

$key = q/displaystyle[L_z,z]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1500.png"
 ALT="$\displaystyle [L_z, z] = 0.$">|; 

$key = q/displaystyleE_2';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1459.png"
 ALT="$\displaystyle E_2'$">|; 

$key = q/displaystyleleft(frac{ddelta_l(E)}{dE}right)_{E=E_0}=frac{2}{{mitGamma}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2346.png"
 ALT="$\displaystyle \left(\frac{d \delta_l(E)}{d E} \right)_{E=E_0} = \frac{2}{{\mit\Gamma}},$">|; 

$key = q/displaystyle=[u_1,v]_{cl}+[u_2,v]_{cl},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img260.png"
 ALT="$\displaystyle = [u_1, v]_{cl} + [u_2, v]_{cl},$">|; 

$key = q/V(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img517.png"
 ALT="$ V(x)$">|; 

$key = q/{bfp}cdot{bfA}={bfA}cdot{bfp};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2122.png"
 ALT="$ {\bf p}\cdot{\bf A}={\bf A}\cdot{\bf p}$">|; 

$key = q/H_{nm},H_{mi}neq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1938.png"
 ALT="$ H_{nm} H_{mi}\neq 0$">|; 

$key = q/displaystyleint_{-1}^1dmu,P_n(mu),P_m(mu)=frac{delta_{n,m}}{n+1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2236.png"
 ALT="$\displaystyle \int_{-1}^1 d\mu P_n(\mu)  P_m(\mu) = \frac{\delta_{n m}}{n+1/2},$">|; 

$key = q/L^2,S^{,2},L_z,S_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1705.png"
 ALT="$ L^2, S^{ 2}, L_z, S_z$">|; 

$key = q/j_1=l,j_2=1slash2,j=l+1slash2,m_1=m-1slash2,m_2=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="407" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1642.png"
 ALT="$ j_1=l, j_2 = 1/2, j = l+1/2, m_1=m-1/2, m_2=1/2$">|; 

$key = q/|m|leql<n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img984.png"
 ALT="$ \vert m\vert \leq l&lt; n$">|; 

$key = q/J_{1,i};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1330.png"
 ALT="$ J_{1 i}$">|; 

$key = q/displaystylelanglem-1slash2,1slash2|l+1slash2,mrangle=sqrt{frac{l+m+1slash2}{l+mh2}{l+m+5slash2}}langlem+3slash2,1slash2|l+1slash2,m+2rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="664" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1646.png"
 ALT="$\displaystyle \langle m-1/2, 1/2\vert l+1/2, m\rangle =\sqrt{\frac{l+m+1/2}{l+m+3/2}} \sqrt{\frac{l+m+3/2}{l+m+5/2}} \langle m+3/2, 1/2\vert l+1/2, m+2\rangle.$">|; 

$key = q/displaystyle=(X,Y),Z=X,Y,Z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img91.png"
 ALT="$\displaystyle = (X Y)  Z = X Y Z.$">|; 

$key = q/displaystyledot{x}(t)=c,alpha_1(t)=frac{{rmi},hbar,c}{2},dot{alpha}_1(0),expleft(frac{-2,{rmi},H,t}{hbar}right)H^{,-1}+c^2,p_x,H^{,-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="450" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2720.png"
 ALT="$\displaystyle \dot{x}(t) = c \alpha_1(t) = \frac{{\rm i} \hbar c}{2} \dot{\...
...,\exp\left(\frac{-2 {\rm i} H t}{\hbar}\right)H^{ -1} + c^2 p_x H^{ -1},$">|; 

$key = q/displaystylelanglelpm1slash2,m|,S_z,|lpm1slash2,mrangle=frac{hbar}{2,(2,l+1)}left[(lpmm+1slash2)-(lmpm+1slash2)right]=pmfrac{m,hbar}{2,l+1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="628" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1700.png"
 ALT="$\displaystyle \langle l\pm 1/2, m\vert  S_z \vert l\pm 1/2, m\rangle = \frac{...
...+1)} \left[(l\pm m+1/2) - (l\mp m + 1/2) \right] = \pm \frac{m \hbar}{2 l+1}.$">|; 

$key = q/displaystyleomega=frac{g_n,e,B}{m_p}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1135.png"
 ALT="$\displaystyle \omega = \frac{g_n  e B}{m_p}.$">|; 

$key = q/zeta^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2828.png"
 ALT="$ \zeta^{ 2}$">|; 

$key = q/1slash(2pi,137);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1114.png"
 ALT="$ 1/(2\pi  137)$">|; 

$key = q/a_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img468.png"
 ALT="$ a_x$">|; 

$key = q/displaystylembox{boldmath{{epsilon{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2042.png"
 ALT="$\displaystyle \mbox{\boldmath$\epsilon$}$">|; 

$key = q/displaystyleH,A,|Erangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img677.png"
 ALT="$\displaystyle H A \vert E\rangle$">|; 

$key = q/displaystyleint_{-infty}^{+infty},dx',langlephi,frac{d}{dx},|x'rangle,psi(x')=-int_{-infty}^{+infty}dx',frac{dphi(x')}{dx'},psi(x'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="403" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img341.png"
 ALT="$\displaystyle \int_{-\infty}^{+\infty}  dx'  \langle \phi  \frac{d}{dx}  \v...
... \psi(x') = -\int_{-\infty}^{+\infty}dx'   \frac{d \phi(x')}{dx'}  \psi(x'),$">|; 

$key = q/displaystyleq_i={rmi},hbar,frac{partial}{partialp_i}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img415.png"
 ALT="$\displaystyle q_i = {\rm i} \hbar  \frac{\partial}{\partial p_i}.$">|; 

$key = q/displaystyleA^{-1}=1+frac{{rmi}}{4},sigma_{mu,nu},{mitDelta}omega^{,mu,nu}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2686.png"
 ALT="$\displaystyle A^{-1} = 1 + \frac{{\rm i}}{4} \sigma_{\mu \nu} {\mit\Delta}\omega^{ \mu \nu}.$">|; 

$key = q/cdot{bfL};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1200.png"
 ALT="$ \cdot {\bf L}$">|; 

$key = q/T(t_0,t_0)=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img564.png"
 ALT="$ T(t_0, t_0)=1$">|; 

$key = q/displaystyle=langle1slash2,1slash2|1,1rangle=langle1slash2,1slash2|1,-1rangle=pm1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="322" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1416.png"
 ALT="$\displaystyle = \langle 1/2,1/2\vert 1,1\rangle = \langle 1/2,1/2\vert 1,-1\rangle = \pm 1.$">|; 

$key = q/displaystylee_{11}=e_{22}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1453.png"
 ALT="$\displaystyle e_{11} = e_{22} = 0,$">|; 

$key = q/rho({bfx'},t),d^{,3}x';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img645.png"
 ALT="$ \rho({\bf x'}, t) d^{ 3} x'$">|; 

$key = q/displaystyles=i+(k^2-alpha^2)^{1slash2}=i+[(j+1slash2)^2-alpha^2]^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2914.png"
 ALT="$\displaystyle s = i+ (k^2-\alpha^2)^{1/2}= i+[(j+1/2)^2-\alpha^2]^{1/2}.$">|; 

$key = q/displaystyle{bfj}_{rmsca}=frac{hbar}{(2pi)^{3},m}frac{|f({bfk}',{bfk})|^{,2}}{r^2},k,{bfe}_r.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2181.png"
 ALT="$\displaystyle {\bf j}_{\rm sca}=\frac{\hbar}{(2\pi)^{3} m} \frac{\vert f( {\bf k}', {\bf k})\vert^{ 2}}{r^2}   k  {\bf e}_r.$">|; 

$key = q/-langleB|ArangleslashlangleB|Brangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img171.png"
 ALT="$ -\langle B\vert A\rangle/\langle B\vert B\rangle$">|; 

$key = q/displaystyle{bfx}_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1590.png"
 ALT="$\displaystyle {\bf x}_4$">|; 

$key = q/displaystyleint_0^inftydr,r^2,[R_{n,l}(r)]^{,2}=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img982.png"
 ALT="$\displaystyle \int_0^\infty dr r^2 [R_{n l}(r)]^{ 2}= 1.$">|; 

$key = q/displaystyleintd^{,3}x',{bfj}({bfx}',t)=frac{langle{bfp}rangle_t}{m},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img653.png"
 ALT="$\displaystyle \int d^{ 3} x' {\bf j}({\bf x}', t) = \frac{\langle {\bf p }\rangle_t}{m},$">|; 

$key = q/g_n=-1.91;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1136.png"
 ALT="$ g_n = -1.91$">|; 

$key = q/chi_+,chi_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2474.png"
 ALT="$ \chi_+ \chi_-$">|; 

$key = q/displaystyleleft(langleB|,Xright)|Arangle=langleB|left(X,|Arangleright);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img95.png"
 ALT="$\displaystyle \left(\langle B\vert X\right) \vert A\rangle = \langle B\vert\left( X \vert A\rangle\right)$">|; 

$key = q/displaystyleH_0=frac{p^2}{2,m_e}+V_0(r).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2022.png"
 ALT="$\displaystyle H_0 = \frac{p^2}{2  m_e} + V_0(r).$">|; 

$key = q/displaystyle=gamma^0,gamma^i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2600.png"
 ALT="$\displaystyle = \gamma^0 \gamma^i,$">|; 

$key = q/displaystyle[zeta,beta]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2821.png"
 ALT="$\displaystyle [\zeta,\beta]=0.$">|; 

$key = q/displaystyleH=frac{{bfp}_1^{,2}}{2,m_1}+frac{{bfp}_2^{,2}}{Z,m_2}-frac{Z,e^2}{4pac{Z,e^2}{4pi,epsilon_0,r_2}+frac{e^2}{4pi,epsilon_0,r_{12}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="390" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2488.png"
 ALT="$\displaystyle H = \frac{{\bf p}_1^{ 2}}{2 m_1} + \frac{{\bf p}_2^{ 2}}{Z m_...
...} -\frac{Z e^2}{4\pi \epsilon_0 r_2} + \frac{e^2}{4\pi \epsilon_0 r_{12}},$">|; 

$key = q/displaystyleexpleft(frac{,{rmi},sigma_3,{mitDelta}varphi}{2}right)sigma_1expleft}right)=sigma_1cos{mitDelta}varphi-sigma_2sin{mitDelta}varphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="440" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1235.png"
 ALT="$\displaystyle \exp\left(\frac{ {\rm i} \sigma_3  {\mit\Delta}\varphi}{2}\rig...
...{2}\right) = \sigma_1 \cos{\mit\Delta}\varphi -\sigma_2 \sin{\mit\Delta}\varphi$">|; 

$key = q/m';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2496.png"
 ALT="$ m'$">|; 

$key = q/displaystylelangle{bfS}_pcdot{bfS}_erangle=-frac{3}{4},hbar^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1752.png"
 ALT="$\displaystyle \langle {\bf S}_p\cdot{\bf S}_e\rangle = - \frac{3}{4} \hbar^2$">|; 

$key = q/displaystyle=frac{1}{sqrt{3!}}left(|k'rangle,|k''rangle,|k'''rangle+|k'''rangle,|k'rangle,|k''rangle+|k''rangle,|k'''rangle,|k'rangleright.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2446.png"
 ALT="$\displaystyle = \frac{1}{\sqrt{3!}}\left(\vert k'\rangle \vert k''\rangle \ve...
... \vert k''\rangle+\vert k''\rangle \vert k'''\rangle \vert k'\rangle \right.$">|; 

$key = q/displaystyle=frac{1}{l,(l+1slash2),(l+1),n^3}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1770.png"
 ALT="$\displaystyle = \frac{1}{l (l+1/2) (l+1) n^3}.$">|; 

$key = q/displaystyle=j_2,(j_2+1),hbar^2,|j_1,j_2;m_1,m_2rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1345.png"
 ALT="$\displaystyle = j_2 (j_2+1) \hbar^2 \vert j_1,j_2; m_1,m_2\rangle,$">|; 

$key = q/|n,l^{(1)}rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1556.png"
 ALT="$ \vert n,l^{(1)}\rangle$">|; 

$key = q/displaystylecdot{bfB},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1690.png"
 ALT="$\displaystyle \cdot {\bf B},$">|; 

$key = q/displaystylepsi^dagleft[left(-{rmi},hbar,partial_mu-frac{e}{c},{mitPhi}_muright)gamma^0,gamma^{,mu}-m_e,c,gamma^0,right]psi=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2633.png"
 ALT="$\displaystyle \psi^\dag\left[\left(-{\rm i} \hbar \partial_\mu- \frac{e}{c} {\mit\Phi}_\mu\right)\gamma^0 \gamma^{ \mu}-m_e c \gamma^0 \right]\psi=0,$">|; 

$key = q/displaystyle{mitDelta}E_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1606.png"
 ALT="$\displaystyle {\mit\Delta} E_4$">|; 

$key = q/displaystylefrac{2,m}{hbar^2}frac{|V_0|}{mu^2}ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2214.png"
 ALT="$\displaystyle \frac{2 m}{\hbar^2} \frac{\vert V_0\vert}{\mu^2} \ll 1$">|; 

$key = q/R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="$ R$">|; 

$key = q/1:alpha^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1687.png"
 ALT="$ 1 : \alpha^2$">|; 

$key = q/displaystylefrac{dv}{dt}=[v,H]_{cl}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img576.png"
 ALT="$\displaystyle \frac{dv}{dt} = [v,H]_{cl}.$">|; 

$key = q/displaystyle{bfJ}_1times{bfJ}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1301.png"
 ALT="$\displaystyle {\bf J}_1\times {\bf J}_1$">|; 

$key = q/2times1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1249.png"
 ALT="$ 2\times 1$">|; 

$key = q/displaystylelangleB|,X^{dag},|Arangle=langleA|,X,|Brangle^ast,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img99.png"
 ALT="$\displaystyle \langle B\vert X^{\dag } \vert A\rangle = \langle A\vert X  \vert B\rangle^\ast,$">|; 

$key = q/displaystylefrac{hbar}{2}left({rme}^{,{rmi},{mitDelta}varphislash2},|+ranglelang-ranglelangle+|,{rme}^{,-{rmi},{mitDelta}varphislash2}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1084.png"
 ALT="$\displaystyle \frac{\hbar}{2} \left( {\rm e}^{ {\rm i} {\mit\Delta}\varphi/2}...
...ert-\rangle \langle +\vert {\rm e}^{ -{\rm i} {\mit\Delta}\varphi/2}\right),$">|; 

$key = q/displaystyle{rmi},hbar,frac{dT}{dt}=H(t),T.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img560.png"
 ALT="$\displaystyle {\rm i}  \hbar  \frac{dT}{dt} = H(t)  T.$">|; 

$key = q/displaystyleint_{-infty}^{infty}dx,,{rmsinc}^2(x)=pi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1923.png"
 ALT="$\displaystyle \int_{-\infty}^{\infty} dx  {\rm sinc}^2(x) = \pi.$">|; 

$key = q/displaystyledelta_0simeqk,aleft[frac{tan(k',a)}{k',a}-1right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2329.png"
 ALT="$\displaystyle \delta_0 \simeq k a \left[ \frac{\tan( k' a)}{k' a} -1\right].$">|; 

$key = q/p^{,i}rightarrowp^{,i}+e,A^{,i};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2621.png"
 ALT="$ p^{ i}\rightarrow p^{ i} +e A^{ i}$">|; 

$key = q/P_l(costheta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2223.png"
 ALT="$ P_l(\cos\theta)$">|; 

$key = q/displaystyleomega_{ni}-omegasimeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2011.png"
 ALT="$\displaystyle \omega_{ni} - \omega \simeq 0$">|; 

$key = q/displaystyle|Rrangle=c_1,|Arangle+c_2,|Brangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="$\displaystyle \vert R\rangle = c_1  \vert A\rangle + c_2  \vert B\rangle.$">|; 

$key = q/theta=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2261.png"
 ALT="$ \theta=0$">|; 

$key = q/displaystyle[u_1,u_2,v_1,v_2]_{qm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img271.png"
 ALT="$\displaystyle [u_1  u_2, v_1  v_2]_{qm}$">|; 

$key = q/displaystyles,c_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2899.png"
 ALT="$\displaystyle s c_s$">|; 

$key = q/psi_{n00}({bfx}')=psi_{n00}(r');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1527.png"
 ALT="$ \psi_{n00}({\bf x}') = \psi_{n00}(r')$">|; 

$key = q/displaystyleleft(frac{1}{a_1}-frac{alpha}{y}right)f-left(frac{d}{dy}-frac{1}{a}+frac{k}{y}right)g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2872.png"
 ALT="$\displaystyle \left(\frac{1}{a_1}-\frac{\alpha}{y}\right)f - \left(\frac{d}{d y}- \frac{1}{a}+\frac{k}{y}\right)g$">|; 

$key = q/left{{mitDelta}xi,{mitDelta}etaright};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img181.png"
 ALT="$ \left\{ {\mit\Delta} \xi, {\mit\Delta} \eta\right\}$">|; 

$key = q/displaystyle={rmi},hbar,delta_{ij}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img296.png"
 ALT="$\displaystyle = {\rm i}  \hbar  \delta_{ij}.$">|; 

$key = q/displaystylebeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2597.png"
 ALT="$\displaystyle \beta$">|; 

$key = q/P_n(t)=|c_n(t)|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1786.png"
 ALT="$ P_n(t) = \vert c_n(t)\vert^{ 2}$">|; 

$key = q/i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="10" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="$ i$">|; 

$key = q/k=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1469.png"
 ALT="$ k=1$">|; 

$key = q/displaystylelangleA|,D^{dag},D,|Arangle=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img452.png"
 ALT="$\displaystyle \langle A\vert  D^{\dag }  D \vert A\rangle = 1.$">|; 

$key = q/xi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img101.png"
 ALT="$ \xi$">|; 

$key = q/sigma_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1163.png"
 ALT="$ \sigma_k$">|; 

$key = q/displaystyle{bfE}=frac{nablaV}{e}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1612.png"
 ALT="$\displaystyle {\bf E} = \frac{\nabla V}{e}.$">|; 

$key = q/displaystyle=left(frac{-{rmi}}{hbar}right)^2sum_mH_{nm}H_{mi}int_0^tdt',exp(,{rmi},omega_{nm},t',)int_0^{t'},dt'',exp(,{rmi},omega_{mi},t,);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="474" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1930.png"
 ALT="$\displaystyle = \left(\frac{-{\rm i}}{\hbar}\right)^2 \sum_m H_{nm} H_{mi} \int...
...i} \omega_{nm} t' ) \int_0^{t'}   dt''  \exp( {\rm i}  \omega_{mi} t )$">|; 

$key = q/langleL_x^{,2}rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img997.png"
 ALT="$ \langle L_x^{ 2}\rangle$">|; 

$key = q/m_2,hbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1316.png"
 ALT="$ m_2 \hbar$">|; 

$key = q/langleB|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img93.png"
 ALT="$ \langle B\vert$">|; 

$key = q/displaystyle=frac{1}{sqrt{2}},(chi_+,chi_-+chi_-,chi_+),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2469.png"
 ALT="$\displaystyle = \frac{1}{\sqrt{2}} (\chi_+ \chi_-+\chi_- \chi_+),$">|; 

$key = q/-s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1264.png"
 ALT="$ -s$">|; 

$key = q/tau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img548.png"
 ALT="$ \tau$">|; 

$key = q/displaystylegamma^{,mu,dag}=gamma^0,gamma^{,mu},gamma^0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2612.png"
 ALT="$\displaystyle \gamma^{ \mu \dag } = \gamma^0 \gamma^{ \mu} \gamma^0,$">|; 

$key = q/displaystyle[q_i,p_j]_{cl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img253.png"
 ALT="$\displaystyle [q_i, p_j]_{cl}$">|; 

$key = q/displaystyle[a,alpha-a_2,(s-k)],c_s+[a_2,alpha+a,(s+k)],c_{s}'=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="354" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2884.png"
 ALT="$\displaystyle [a \alpha-a_2 (s-k)] c_s + [a_2 \alpha+a (s+k)] c_{s}' = 0.$">|; 

$key = q/displaystylerightarrow-frac{cos(y-l,pislash2)}{y}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2233.png"
 ALT="$\displaystyle \rightarrow - \frac{\cos(y-l \pi/2)}{y}.$">|; 

$key = q/c_{n_{rmmin}},y^{,n_{rmmin}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img958.png"
 ALT="$ c_{n_{\rm min}} y^{ n_{\rm min}}$">|; 

$key = q/displaystyle{rmi},hbar,frac{partial}{partialt},|A,t_0,trangle=sum_mleft({rmi},hbt}+c_m(t),E_mright)exp[-{rmi},E_m,(t-t_0)slashhbar],|mrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="499" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1790.png"
 ALT="$\displaystyle {\rm i} \hbar  \frac{\partial}{\partial t} \vert A,t_0,t\rangl...
...{dt}+ c_m(t)  E_m\right) \exp[-{\rm i} E_m  (t-t_0)/\hbar]  \vert m\rangle,$">|; 

$key = q/displaystylelanglexi''|xi'rangle=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img125.png"
 ALT="$\displaystyle \langle \xi''\vert\xi'\rangle = 0,$">|; 

$key = q/2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2099.png"
 ALT="$ 2$">|; 

$key = q/{bfL}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img751.png"
 ALT="$ {\bf L}_i$">|; 

$key = q/displaystylefrac{|e_{mk}|}{E_m-E_k}simO(epsilon),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1473.png"
 ALT="$\displaystyle \frac{\vert e_{mk}\vert}{E_m - E_k} \sim O(\epsilon),$">|; 

$key = q/R(r)rightarrowexp(rslasha)slashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img970.png"
 ALT="$ R(r) \rightarrow
\exp(r/a) /r $">|; 

$key = q/l=0,infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2239.png"
 ALT="$ l=0, \infty$">|; 

$key = q/displaystylep_i=-{rmi},hbarfrac{partial}{partialx_i}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img622.png"
 ALT="$\displaystyle p_i = -{\rm i} \hbar\frac{\partial}{\partial x_i}.$">|; 

$key = q/y^{,n-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img953.png"
 ALT="$ y^{ n-2}$">|; 

$key = q/displaystyle=left.frac{2pi}{hbar},overline{|V_{ni}^dagger|^{,2}},rho(E_n)right|_{E_n=E_i+hbar,omega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2016.png"
 ALT="$\displaystyle =\left. \frac{2\pi}{\hbar}  \overline{ \vert V_{ni}^\dagger\vert^{ 2}} \rho(E_n)\right\vert _{E_n = E_i+\hbar \omega}.$">|; 

$key = q/displaystylelanglep_1'cdotsp_N'|,q_i={rmi},hbar,frac{partial}{partialp_i'},langlep_1'cdotsp_N'|.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img417.png"
 ALT="$\displaystyle \langle p_1' \cdots p_N'\vert  q_i = {\rm i} \hbar  \frac{\partial}{\partial p_i'}  \langle p_1'\cdots p_N'\vert.$">|; 

$key = q/displaystyle(sigma_k)_{j,l}=frac{langles,j|,S_k,|s,lrangle}{s,hbar}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1262.png"
 ALT="$\displaystyle (\sigma_k)_{j l} = \frac{ \langle s, j\vert S_k  \vert s, l\rangle}{s \hbar}.$">|; 

$key = q/sin^2alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="$ \sin^2\alpha$">|; 

$key = q/V=V_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2310.png"
 ALT="$ V=V_0$">|; 

$key = q/displaystylefrac{d}{dx},x-x,frac{d}{dx}=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img347.png"
 ALT="$\displaystyle \frac{d}{dx}  x - x  \frac{d}{dx} = 1.$">|; 

$key = q/displaystyleH_0,|2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1440.png"
 ALT="$\displaystyle H_0  \vert 2\rangle$">|; 

$key = q/displaystyle=frac{1}{sqrt{2}}left(!{array}{rrr}0&-{rmi}&0{rmi}&0&{-rmi}0&{rmi}&0{array}!right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="88" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1286.png"
 ALT="$\displaystyle = \frac{1}{\sqrt{2}} \left(\!\begin{array}{rrr} 0 &amp;-{\rm i}&amp;0\ {\rm i}&amp;0&amp;{-\rm i}\ 0&amp;{\rm i}&amp; 0\end{array}\!\right),$">|; 

$key = q/hbar,{bfk}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2186.png"
 ALT="$ \hbar { \bf k}'$">|; 

$key = q/displaystyle=a_nu^{~mu},p^{nu'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2668.png"
 ALT="$\displaystyle = a_\nu^{&nbsp;\mu} p^{\nu'},$">|; 

$key = q/{bfJ}={bfL}+{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1621.png"
 ALT="$ {\bf J} = {\bf L} + {\bf S}$">|; 

$key = q/alpha_i=-beta,alpha_i,beta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2932.png"
 ALT="$ \alpha_i=-\beta \alpha_i \beta$">|; 

$key = q/displaystylep^{,mu'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2665.png"
 ALT="$\displaystyle p^{ \mu'}$">|; 

$key = q/displaystyleL_z,L^+|l,mrangle=(L^+L_z+[L_z,L^+]),|l,mrangle=(L^+L_z+hbar,L^+),|l,mrangle=(m+1),hbar,L^+|l,mrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="596" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img795.png"
 ALT="$\displaystyle L_z  L^+ \vert l, m\rangle = (L^+ L_z + [L_z, L^+]) \vert l,m\r...
...^+ L_z + \hbar  L^+)  \vert l,m\rangle = (m+1) \hbar  L^+\vert l, m\rangle,$">|; 

$key = q/displaystyle|Brangle=sum_{i=1,N}beta_i,|irangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="$\displaystyle \vert B\rangle = \sum_{i=1,N} \beta_i  \vert i\rangle$">|; 

$key = q/displaystyle=j,(j+1),hbar^2,|j_1,j_2;j,mrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1357.png"
 ALT="$\displaystyle =j (j+1) \hbar^2 \vert j_1,j_2; j,m\rangle,$">|; 

$key = q/epsilon^2=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2857.png"
 ALT="$ \epsilon^2=1$">|; 

$key = q/displaystyle={rmi},hbar,(-y,p_x+x,p_y)={rmi},hbar,L_z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img744.png"
 ALT="$\displaystyle = {\rm i} \hbar (-y  p_x+ x p_y) = {\rm i} \hbar  L_z.$">|; 

$key = q/displaystyle=-hbar,L^+,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img775.png"
 ALT="$\displaystyle = -\hbar L^+,$">|; 

$key = q/displaystylep_x',langlex'|p_x'rangle=langlex'|,p_x,|p_x'rangle=-{rmi},hbar,frac{d}{dx'},langlex'|p_x'rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img420.png"
 ALT="$\displaystyle p_x'  \langle x'\vert p_x'\rangle = \langle x'\vert p_x \vert p_x'\rangle = -{\rm i} \hbar \frac{d}{dx'}  \langle x'\vert p_x'\rangle,$">|; 

$key = q/c_{l,m}^+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img827.png"
 ALT="$ c_{l m}^+$">|; 

$key = q/displaystyleP=frac{e^2,a^2}{6pi,epsilon_0,c^3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img203.png"
 ALT="$\displaystyle P = \frac{e^2 a^2}{6\pi \epsilon_0 c^3},
$">|; 

$key = q/displaystylelangleF|(|Arangle)=phi_A.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="$\displaystyle \langle F\vert(\vert A\rangle) = \phi_A.$">|; 

$key = q/|psi({bfx},t)|rightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2646.png"
 ALT="$ \vert\psi({\bf x},t)\vert\rightarrow 0$">|; 

$key = q/A_l(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2269.png"
 ALT="$ A_l(r)$">|; 

$key = q/j,(j+1),hbar^2+(1slash4),hbar^2=(j+1slash2)^2,hbar^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2829.png"
 ALT="$ j (j+1) \hbar^2 + (1/4) \hbar^2 = (j+1/2)^2 \hbar^2$">|; 

$key = q/F({bfx},{bfp});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img595.png"
 ALT="$ F({\bf x},{\bf p})$">|; 

$key = q/displaystyle{bfE};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2028.png"
 ALT="$\displaystyle {\bf E}$">|; 

$key = q/displaystyle{rmi},hbar,frac{d|Atrangle}{dt}=H',|Atrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img631.png"
 ALT="$\displaystyle {\rm i} \hbar  \frac{d \vert At\rangle}{dt} = H'  \vert At\rangle.$">|; 

$key = q/displaystyle{bfU},{bfx}=lambda,{bfx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1561.png"
 ALT="$\displaystyle {\bf U}  {\bf x} = \lambda  {\bf x},$">|; 

$key = q/displaystylew_{irightarrown}=frac{2pi}{hbar}frac{e^2}{m_e^{,2}},|A_0|^{,2},|langlen|,exp[,{rmi},(omegaslashc),{bfn}cdot{bfx}],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2052.png"
 ALT="$\displaystyle w_{i\rightarrow n} = \frac{2\pi}{\hbar} \frac{e^2}{m_e^{ 2}}  \...
... 2}  \vert\langle n\vert  \exp[ {\rm i} (\omega/c) {\bf n}\cdot{\bf x}] $">|; 

$key = q/{rmsinc}(0)=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1909.png"
 ALT="$ {\rm sinc}(0) = 1$">|; 

$key = q/displaystylechi(1,-1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2470.png"
 ALT="$\displaystyle \chi(1,-1)$">|; 

$key = q/F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="$ F$">|; 

$key = q/[cdots]_{qm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img575.png"
 ALT="$ [\cdots]_{qm}$">|; 

$key = q/displaystyleH_B=frac{e,B}{2,m_e},(L_z+2,S_z).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1692.png"
 ALT="$\displaystyle H_B = \frac{e  B}{2  m_e}  (L_z + 2  S_z).$">|; 

$key = q/displaystyle=-frac{1}{2,m_e^{,2},c^2}leftlanglefrac{1}{r}frac{dV}{dr}rightranglefrac{(l+1),hbar^2}{2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1679.png"
 ALT="$\displaystyle =- \frac{1}{2 m_e^{ 2} c^2} \left\langle \frac{1}{r}\frac{dV}{dr} \right\rangle \frac{(l+1) \hbar^2}{2},$">|; 

$key = q/displaystyle=-a^3.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2551.png"
 ALT="$\displaystyle = -a^3.$">|; 

$key = q/displaystylelanglex'|x''rangle=delta(x'-x'').;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img301.png"
 ALT="$\displaystyle \langle x' \vert x''\rangle = \delta(x'-x'').$">|; 

$key = q/langlem_l,m_s|j,m_jrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1641.png"
 ALT="$ \langle m_l, m_s\vert j, m_j\rangle$">|; 

$key = q/displaystyletan(k,a+delta_0)=frac{k}{k'},tan(k',a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2323.png"
 ALT="$\displaystyle \tan(k a+\delta_0) = \frac{k}{k'}  \tan (k' a)$">|; 

$key = q/displaystyleexpleft(-{rmi},mbox{boldmathsigma}!cdot!{bfn},{mitDelta}varphislash2right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1223.png"
 ALT="$\displaystyle \exp\left(-{\rm i}  \mbox{\boldmath$\sigma$}\!\cdot\!{\bf n} {\mit\Delta}\varphi/2\right)$">|; 

$key = q/displaystyle=exp(-{rmi},L_y,{mitDelta}varphi_yslashhbar),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img871.png"
 ALT="$\displaystyle = \exp(-{\rm i} L_y  {\mit\Delta}\varphi_y/\hbar),$">|; 

$key = q/langlephi,dslashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img338.png"
 ALT="$ \langle \phi  d/dx$">|; 

$key = q/k,a+delta_0=pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2338.png"
 ALT="$ k a+\delta_0 = \pi/2$">|; 

$key = q/displaystyle=frac{hbar}{2}left(,|+ranglelangle-|+|-ranglelangle+|,right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1037.png"
 ALT="$\displaystyle = \frac{\hbar}{2} \left(  \vert+\rangle \langle -\vert + \vert-\rangle \langle +\vert  \right),$">|; 

$key = q/psi^pm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2172.png"
 ALT="$ \psi^\pm$">|; 

$key = q/displaystylephi(p_x')=langlep_x'|phirangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img407.png"
 ALT="$\displaystyle \phi(p_x') = \langle p_x'\vert\phi \rangle.$">|; 

$key = q/displaystylerangle_{rmnew}={rme}^{-{rmi},gamma}rangle_{rmold},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img358.png"
 ALT="$\displaystyle \rangle_{\rm new} = {\rm e}^{-{\rm i}  \gamma} \rangle_{\rm old},$">|; 

$key = q/left[p^0+(p^1,p^1+p^2,p^2+p^3,p^3+m_e^2,c^2)^{1slash2}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2584.png"
 ALT="$ \left[p^0 +(p^1 p^1+p^2 p^2+p^3 p^3+m_e^2 c^2)^{1/2}\right]$">|; 

$key = q/displaystyle=-frac{(l+1),hbar^2}{2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1671.png"
 ALT="$\displaystyle = -\frac{(l+1)  \hbar^2}{2},$">|; 

$key = q/displaystyleA^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2696.png"
 ALT="$\displaystyle A^{-1}$">|; 

$key = q/c_i(0)=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1976.png"
 ALT="$ c_i(0) = 1$">|; 

$key = q/f(x')=x'^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img320.png"
 ALT="$ f(x') = x'^{ 2}$">|; 

$key = q/j=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1011.png"
 ALT="$ j=0$">|; 

$key = q/{bfA}={bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2754.png"
 ALT="$ {\bf A}={\bf0}$">|; 

$key = q/r_>;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2511.png"
 ALT="$ r_&gt;$">|; 

$key = q/displaystyleP_l(costheta)=sqrt{frac{4pi}{2,l+1}},Y_{l,0}(theta,varphi).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2224.png"
 ALT="$\displaystyle P_l(\cos\theta) = \sqrt{\frac{4\pi}{2 l+1}}  Y_{l 0}(\theta,\varphi).$">|; 

$key = q/displaystyle=frac{{rmi},t}{hbar}exp(,{rmi},omega_{ni},tslash2),left[left(H_{ni}+nm},H_{mi}}{E_m-E_i}right),{rmsinc}(omega_{ni},tslash2)right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="416" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1934.png"
 ALT="$\displaystyle = \frac{{\rm i} t}{\hbar} \exp( {\rm i} \omega_{ni} t/2)  \l...
... \frac{H_{nm} H_{mi}}{E_m - E_i}\right)  {\rm sinc} (\omega_{ni} t/2)\right.$">|; 

$key = q/displaystylecdot{bfp},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2808.png"
 ALT="$\displaystyle \cdot{\bf p},$">|; 

$key = q/displaystyle=langlen|,V,|irangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2004.png"
 ALT="$\displaystyle = \langle n\vert V \vert i\rangle,$">|; 

$key = q/displaystylea_{mu}=g_{mu,nu},a^nu.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2561.png"
 ALT="$\displaystyle a_{\mu} = g_{\mu \nu} a^\nu.$">|; 

$key = q/c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="$ c$">|; 

$key = q/displaystylee_{22};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1449.png"
 ALT="$\displaystyle e_{22}$">|; 

$key = q/psi_A(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img322.png"
 ALT="$ \psi_A(x)$">|; 

$key = q/displaystyleR_{1,0}(r)=frac{2}{a_0^{,3slash2}},expleft(-frac{r}{a_0}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1006.png"
 ALT="$\displaystyle R_{1 0}(r) = \frac{2}{a_0^{ 3/2}} \exp\left(-\frac{r}{a_0}\right).
$">|; 

$key = q/displaystyleH=frac{p^2}{2,m_e}-frac{e,{bfA}cdot{bfp}}{m_e}+frac{e^2A^2}{2,m_e}-e,phi+V_0(r).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2038.png"
 ALT="$\displaystyle 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).$">|; 

$key = q/displaystylefrac{dsigma}{d{mitOmega}}simeqleft(frac{2,m,V_0}{hbar^2,mu}right)^2frac{1}{[4,k^2,sin^2(thetaslash2)+mu^2]^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2202.png"
 ALT="$\displaystyle \frac{d\sigma}{d {\mit\Omega}} \simeq \left(\frac{2 m  V_0}{ \hbar^2 \mu}\right)^2 \frac{1}{[4 k^2 \sin^2(\theta/2) + \mu^2]^{ 2}}.$">|; 

$key = q/psi_A(x');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img306.png"
 ALT="$ \psi_A(x')$">|; 

$key = q/displaystylelangle{bfx}|{bfk}rangle=frac{exp(,{rmi},{bfk}cdot{bfx})}{(2pi)^{3slash2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2158.png"
 ALT="$\displaystyle \langle {\bf x} \vert {\bf k}\rangle = \frac{ \exp( {\rm i} {\bf k}\cdot{\bf x}) }{(2\pi)^{3/2}}.$">|; 

$key = q/displaystyleL_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img732.png"
 ALT="$\displaystyle L_z$">|; 

$key = q/displaystylepsi(x')=psi_0,expleft[-frac{(x'-x_0)^{,2}}{4,sigma^2}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img504.png"
 ALT="$\displaystyle \psi(x') =\psi_0 \exp\left[-\frac{(x'-x_0)^{ 2}}{4 \sigma^2}\right],
$">|; 

$key = q/w_{2rightarrow1}^{rmarm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2103.png"
 ALT="$ w_{2\rightarrow 1}^{\rm arm}$">|; 

$key = q/dslashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img335.png"
 ALT="$ d/dx$">|; 

$key = q/displaystylep_r={bfx}cdot{bfp}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2783.png"
 ALT="$\displaystyle p_r = {\bf x}\cdot{\bf p}.$">|; 

$key = q/psi(theta,varphi)=Y_{l,m}(theta,varphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img994.png"
 ALT="$ \psi(\theta,\varphi)=Y_{l m}(\theta,\varphi)$">|; 

$key = q/{bfL}cdot{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1627.png"
 ALT="$ {\bf L}\cdot {\bf S}$">|; 

$key = q/displaystylefrac{E}{m_e,c^2}=1-frac{alpha^2}{2,n^2}-frac{alpha^4}{2,n^4}left(frac{n}{j+1slash2}-frac{3}{4}right)+{calO}(alpha^6),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2917.png"
 ALT="$\displaystyle \frac{E}{m_e c^2} = 1 - \frac{\alpha^2}{2 n^2}- \frac{\alpha^4}{2 n^4}\left(\frac{n}{j+1/2}-\frac{3}{4}\right)+{\cal O}(\alpha^6),$">|; 

$key = q/displaystyleH_1(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1888.png"
 ALT="$\displaystyle H_1(t)$">|; 

$key = q/displaystyleL^-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img772.png"
 ALT="$\displaystyle L^-$">|; 

$key = q/epsilon_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img205.png"
 ALT="$ \epsilon_0$">|; 

$key = q/epsilon>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1973.png"
 ALT="$ \epsilon &gt;0$">|; 

$key = q/displaystylea=(a_1,a_2)^{1slash2}=frac{alpha}{sqrt{1-{calE}^{,2}}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2871.png"
 ALT="$\displaystyle a = (a_1 a_2)^{1/2} = \frac{\alpha}{\sqrt{1-{\cal E}^{ 2}}},$">|; 

$key = q/p^{,i};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2580.png"
 ALT="$ p^{ i}$">|; 

$key = q/displaystyle=r,costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img893.png"
 ALT="$\displaystyle = r \cos\theta,$">|; 

$key = q/f_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="$ f_i$">|; 

$key = q/displaystyle=left(frac{-{rmi}}{hbar}right)^2int_{t_0}^tdt'int_{t_0}^{t'}dt'',langlen|,H_I(t_0,t'),H_I(t_0,t''),|irangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="366" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1879.png"
 ALT="$\displaystyle = \left(\frac{-{\rm i}}{\hbar}\right)^2 \int_{t_0}^t dt' \int_{t_0}^{t'}dt''  \langle n\vert  H_I(t_0, t' ) H_I(t_0, t'' ) \vert i\rangle.$">|; 

$key = q/|q_1'cdotsq_N'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img372.png"
 ALT="$ \vert q_1'\cdots q_N'\rangle$">|; 

$key = q/displaystyle=sqrt{2},langle-1slash2,1slash2|1,0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1415.png"
 ALT="$\displaystyle = \sqrt{2}  \langle -1/2, 1/2\vert 1, 0\rangle$">|; 

$key = q/e^2slash(4pi,epsilon_0,r_{12});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2506.png"
 ALT="$ e^2/(4\pi \epsilon_0 r_{12})$">|; 

$key = q/displaystylelangle({mitDelta}xi)^{,2}rangle,langle({mitDelta}eta)^{,2}ranglegeq|ngle|^{,2}geqfrac{1}{4},|langleleft[xi,etaright]rangle|^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img190.png"
 ALT="$\displaystyle \langle ({\mit\Delta} \xi)^{ 2}\rangle  \langle ({\mit\Delta} \...
...{ 2}\geq \frac{1}{4}   \vert\langle\left[ \xi, \eta\right]\rangle\vert^{ 2},$">|; 

$key = q/displaystyleL^-,L^+=L_x^{,2}+L_y^{,2}+{rmi},[L_x,L_y]=L^2-L_z^{,2}-hbar,L_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img805.png"
 ALT="$\displaystyle L^-  L^+ = L_x^{ 2} + L_y^{ 2} + {\rm i} [L_x, L_y] = L^2 - L_z^{ 2} - \hbar  L_z,$">|; 

$key = q/displaystylecdot{bfp}}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2811.png"
 ALT="$\displaystyle \cdot {\bf p}\} = 0.$">|; 

$key = q/L_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img735.png"
 ALT="$ L_x$">|; 

$key = q/langle{mitDelta}xirangle=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img189.png"
 ALT="$ \langle {\mit\Delta}\xi\rangle = 0$">|; 

$key = q/langlex'|rangle=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img357.png"
 ALT="$ \langle x'\vert\rangle = 1$">|; 

$key = q/displaystylecdot{bff}_{in}|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2080.png"
 ALT="$\displaystyle \cdot{\bf f}_{in}\vert^{ 2}$">|; 

$key = q/j,j-1,cdots,-j+1,-j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1326.png"
 ALT="$ j, j-1,\cdots,
-j+1, -j$">|; 

$key = q/displaystylecdot{bfp})+(;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2806.png"
 ALT="$\displaystyle \cdot{\bf p}) + ($">|; 

$key = q/deltaxrightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img462.png"
 ALT="$ \delta x\rightarrow 0$">|; 

$key = q/displaystylef(y)=sum_{n}c_n,y^{,n}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img951.png"
 ALT="$\displaystyle f(y) = \sum_{n} c_n  y^{ n}.$">|; 

$key = q/x_0=(hbarslashm,omega)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img694.png"
 ALT="$ x_0=(\hbar/m \omega)^{1/2}$">|; 

$key = q/displaystyle=left([u_1,v_1]_{qm},v_2+v_1,[u_1,v_2]_{qm}right)u_2+u_1left([u_2,v_1]_{qm},v_2+v_1,[u_2,v_2]_{qm}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="496" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img273.png"
 ALT="$\displaystyle =\left([u_1, v_1]_{qm} v_2 + v_1 [u_1, v_2]_{qm}\right) u_2 +u_1\left([u_2, v_1]_{qm} v_2 + v_1 [u_2, v_2]_{qm}\right)$">|; 

$key = q/delta_0simeqpislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2339.png"
 ALT="$ \delta_0 \simeq \pi/2$">|; 

$key = q/displaystyleH_t=T^{dag},H,T.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img572.png"
 ALT="$\displaystyle H_t = T^{\dag }  H  T.$">|; 

$key = q/l=3slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img838.png"
 ALT="$ l=3/2$">|; 

$key = q/displaystylew_{irightarrown}=frac{2pi}{hbar},|H_{ni}|^{,2},delta(E_n-E_i).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1957.png"
 ALT="$\displaystyle w_{i\rightarrow n} = \frac{2\pi}{\hbar}  \vert H_{ni}\vert^{ 2}  \delta(E_n - E_i).$">|; 

$key = q/displaystyletimeslanglej_1,j_2;m_1,m_2mp1|j_1,j_2;j,mrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1394.png"
 ALT="$\displaystyle \times\langle j_1, j_2; m_1, m_2\mp 1\vert j_1,j_2;j, m\rangle.$">|; 

$key = q/displaystylephi({bfx}_1,{bfx}_2)=psi_{100}({bfx}_1),psi_{100}({bfx}_2)=frac{Z^{,3}}{pi,a_0^{,3}},expleft[frac{-Z,(r_1+r_2)}{a_0}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="428" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2502.png"
 ALT="$\displaystyle \phi({\bf x}_1,{\bf x}_2) =\psi_{100}({\bf x}_1) \psi_{100}({\bf...
...2)= \frac{Z^{ 3}}{\pi a_0^{ 3}} \exp\left[\frac{-Z (r_1+r_2)}{a_0}\right],$">|; 

$key = q/psi_n(x')=langlex'|E_nrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img691.png"
 ALT="$ \psi_n(x')=\langle x'\vert E_n\rangle$">|; 

$key = q/s,s-1cdots-s+1,-s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1026.png"
 ALT="$ s, s-1 \cdots -s+1, -s$">|; 

$key = q/displaystyle|+rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1142.png"
 ALT="$\displaystyle \vert+\rangle$">|; 

$key = q/lambda_2=-3,e,a_0,|{bfE}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1581.png"
 ALT="$ \lambda_2 = - 3 e a_0 \vert{\bf E}\vert$">|; 

$key = q/|Btrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img529.png"
 ALT="$ \vert Bt\rangle$">|; 

$key = q/displaystyle{bfB}=B_0,cos(omega,t),{bfe}_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1296.png"
 ALT="$\displaystyle {\bf B} = B_0 \cos(\omega t) {\bf e}_z,
$">|; 

$key = q/0leqjleq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1401.png"
 ALT="$ 0\leq j\leq 1$">|; 

$key = q/displaystylefrac{dot{c}_i}{c_i};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1967.png"
 ALT="$\displaystyle \frac{\dot{c}_i}{c_i}$">|; 

$key = q/|k''rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2372.png"
 ALT="$ \vert k''\rangle$">|; 

$key = q/displaystyleP_{23},|k',k'',k'''rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2405.png"
 ALT="$\displaystyle P_{23} \vert k' k'' k'''\rangle$">|; 

$key = q/displaystylea_mu^{~nu}=g_mu^{~nu}+{mitDelta}omega_mu^{~nu},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2680.png"
 ALT="$\displaystyle a_\mu^{&nbsp;\nu} = g_\mu^{&nbsp;\nu} + {\mit\Delta}\omega_\mu^{&nbsp;\nu},$">|; 

$key = q/displaystylezeta=betaleft(mbox{boldmathSigma}cdot{bfL}+hbarright).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2817.png"
 ALT="$\displaystyle \zeta = \beta\left(\mbox{\boldmath$\Sigma$}\cdot{\bf L} + \hbar\right).$">|; 

$key = q/displaystylec_n,[n,(n-1)-l,(l+1)]=c_{n-1}left[2,(n-1)-frac{2,mu,e^2,a}{4pi,epsilon_0,hbar^2}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="403" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img954.png"
 ALT="$\displaystyle c_n [n (n-1) - l (l+1)] = c_{n-1} \left [2 (n-1) - \frac{2 \mu  e^2  a}{4\pi  \epsilon_0  \hbar^2}\right].$">|; 

$key = q/displaystyle=langleS_xrangle_{t=0}cos(omega,t)-langleS_yrangle_{t=0}sin(omega,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1124.png"
 ALT="$\displaystyle = \langle S_x\rangle_{t=0} \cos(\omega  t) - \langle S_y\rangle_{t=0} \sin(\omega  t),$">|; 

$key = q/displaystyled_x=lim_{deltaxrightarrow0}frac{D-1}{deltax},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img464.png"
 ALT="$\displaystyle d_x = \lim_{\delta x\rightarrow 0 }\frac{D-1}{\delta x},$">|; 

$key = q/J_{2,i};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1331.png"
 ALT="$ J_{2 i}$">|; 

$key = q/displaystyle[p_i,G({bfx},{bfp})];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img593.png"
 ALT="$\displaystyle [p_i, G({\bf x},{\bf p})]$">|; 

$key = q/displaystyle[x_i,p_j]={rmi},hbar,delta_{ij}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img586.png"
 ALT="$\displaystyle [x_i, p_j] = {\rm i} \hbar  \delta_{ij}.$">|; 

$key = q/displaystyle=-3,e,a_0,|{bfE}|,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1604.png"
 ALT="$\displaystyle = -3 e a_0  \vert{\bf E}\vert,$">|; 

$key = q/displaystyle{bfL}times{bfL};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img756.png"
 ALT="$\displaystyle {\bf L} \times {\bf L}$">|; 

$key = q/S_{2,z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2478.png"
 ALT="$ S_{2 z}$">|; 

$key = q/displaystylecotdelta_l(E)=-frac{2}{{mitGamma}},(E-E_0)+cdots.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2347.png"
 ALT="$\displaystyle \cot\delta_l(E) = - \frac{2}{{\mit\Gamma}}  (E-E_0) + \cdots.$">|; 

$key = q/displaystyleP_{31},|k',k'',k'''rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2407.png"
 ALT="$\displaystyle P_{31} \vert k' k'' k'''\rangle$">|; 

$key = q/G({bfx},{bfp});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img596.png"
 ALT="$ G({\bf x},{\bf p})$">|; 

$key = q/c_{l,m}^+exp(,{rmi},gamma);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img828.png"
 ALT="$ c_{l  m}^+\exp( {\rm i} \gamma)$">|; 

$key = q/displaystylelangleq_1'cdotsq_N'|q_1''cdotsq_N''rangle=delta(q_1'-q_1''),delta(q_2'-q_2'')cdotsdelta(q_N'-q_N'').;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="448" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img373.png"
 ALT="$\displaystyle \langle q_1'\cdots q_N'\vert q_1''\cdots q_N''\rangle = \delta(q_1'-q_1'') \delta(q_2'-q_2'')\cdots \delta(q_N'-q_N'').$">|; 

$key = q/displaystylefrac{dA^dag}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img716.png"
 ALT="$\displaystyle \frac{dA^\dag }{dt}$">|; 

$key = q/yrightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img969.png"
 ALT="$ y\rightarrow \infty$">|; 

$key = q/langlephi(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img337.png"
 ALT="$ \langle \phi(x)$">|; 

$key = q/displaystyleE_1,|1rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1439.png"
 ALT="$\displaystyle E_1  \vert 1\rangle,$">|; 

$key = q/displaystyleH_1=frac{e,A_0,mbox{boldmathepsilon}cdot{bfp}}{m_e}left(exp[,{rmi},(exp[-{rmi},(omegaslashc),{bfn}cdot{bfx}+{rmi},omega,t]right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="538" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2050.png"
 ALT="$\displaystyle H_1 = \frac{e  A_0  \mbox{\boldmath$\epsilon$}\cdot{\bf p} }{m_...
...\exp[-{\rm i} (\omega/c)  {\bf n}\cdot{\bf x} + {\rm i}  \omega  t]\right).$">|; 

$key = q/rgeqa;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2312.png"
 ALT="$ r\geq a$">|; 

$key = q/displaystyle{bfJ}times{bfJ}={rmi},hbar,{bfJ}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1323.png"
 ALT="$\displaystyle {\bf J} \times {\bf J} = {\rm i} \hbar  {\bf J}.$">|; 

$key = q/displaystyleE_{n00}=-frac{e^2}{8pi,epsilon_0,a_0,n^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1533.png"
 ALT="$\displaystyle E_{n00} = - \frac{e^2}{8\pi \epsilon_0  a_0 n^2}$">|; 

$key = q/displaystyle|i,trangle=exp[-{rmi},({mitDelta}_i+E_i),tslashhbar],|irangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1978.png"
 ALT="$\displaystyle \vert i, t\rangle = \exp[-{\rm i} ({\mit\Delta}_i + E_i) t/\hbar]  \vert i\rangle.$">|; 

$key = q/T_z({mitDelta}varphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1076.png"
 ALT="$ T_z({\mit\Delta}\varphi)$">|; 

$key = q/displaystyleP_{irightarrow}(t)=intdE_n,P_{irightarrown}(t),rho(E_n),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1915.png"
 ALT="$\displaystyle P_{i\rightarrow} (t) = \int dE_n P_{i\rightarrow n}(t)  \rho(E_n),$">|; 

$key = q/displaystyleH=-frac{hbar^2}{2,M}left[frac{1}{r^2}frac{partial}{partialr},r^2,fra1}{r^2sin^2theta}frac{partial^2}{partialvarphi^2}right]+V(r).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="529" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img927.png"
 ALT="$\displaystyle H= -\frac{\hbar^2}{2 M}\left[ \frac{1}{r^2}\frac{\partial}{\part...
...+ \frac{1}{r^2\sin^2\theta} \frac{\partial^2}{\partial\varphi^2}\right] + V(r).$">|; 

$key = q/displaystyle{mitDelta}_i=H_{ii}+lim_{etarightarrow0}sum_{mneqi}frac{|H_{mi}|^{,2}}{E_i-E_m+{rmi},hbar,eta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1971.png"
 ALT="$\displaystyle {\mit\Delta}_i = H_{ii} + \lim_{\eta\rightarrow 0} \sum_{m\neq i} \frac{\vert H_{mi}\vert^{ 2}}{E_i - E_m+ {\rm i} \hbar  \eta}$">|; 

$key = q/90^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="$ 90^\circ$">|; 

$key = q/E_{nl}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1548.png"
 ALT="$ E_{nl}'$">|; 

$key = q/displaystylecdot{bfL}-frac{hbar^2}{4},Sigma^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2793.png"
 ALT="$\displaystyle \cdot{\bf L} -\frac{\hbar^2}{4} \Sigma^{ 2}.$">|; 

$key = q/{bfp}=hbar,{bfk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2156.png"
 ALT="$ {\bf p} = \hbar  {\bf k}$">|; 

$key = q/displaystylechi'=p_k,chi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1193.png"
 ALT="$\displaystyle \chi' = p_k  \chi,$">|; 

$key = q/displaystylelangle-|A'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1159.png"
 ALT="$\displaystyle \langle - \vert A'\rangle$">|; 

$key = q/displaystyle|Brangle+{rmi},|Crangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="$\displaystyle \vert B\rangle + {\rm i}  \vert C\rangle$">|; 

$key = q/displaystyle[L_x,L_y];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img742.png"
 ALT="$\displaystyle [L_x, L_y]$">|; 

$key = q/displaystyle|{bfx}-{bfx}'|simeqr-{bfe}_rcdot{bfx}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2163.png"
 ALT="$\displaystyle \vert{\bf x} - {\bf x}'\vert \simeq r - {\bf e}_r\cdot{\bf x}'$">|; 

$key = q/m_l+2,m_s=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1712.png"
 ALT="$ m_l+2 m_s = 1$">|; 

$key = q/displaystyleint_{-infty}^{+infty}dx',langlephi,frac{d}{dx},|x'rangle,psi(x')=int_{-infty}^{+infty}dx',phi(x'),frac{dpsi(x')}{dx'}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="385" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img340.png"
 ALT="$\displaystyle \int_{-\infty}^{+\infty}dx'  \langle \phi  \frac{d}{dx}  \vert...
...e   \psi(x') = \int_{-\infty}^{+\infty} dx' \phi(x')  \frac{d\psi(x')}{dx'}.$">|; 

$key = q/t_0=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1808.png"
 ALT="$ t_0=0$">|; 

$key = q/E_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2057.png"
 ALT="$ E_0$">|; 

$key = q/displaystyle={rmi},hbar,frac{partialF}{partialp_i},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img592.png"
 ALT="$\displaystyle = {\rm i} \hbar  \frac{\partial F}{\partial p_i},$">|; 

$key = q/displaystyle=(X+Y)+Z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img87.png"
 ALT="$\displaystyle = (X + Y ) + Z.$">|; 

$key = q/displaystyleH_1=frac{e,B_1}{2,m_e}left[exp(,{rmi},omega,t),S^-+exp(-{rmi},omega,t),S^+right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="336" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1841.png"
 ALT="$\displaystyle H_1 = \frac{e  B_1}{2  m_e} \left[\exp( {\rm i} \omega  t)  S^- + \exp(-{\rm i} \omega  t)  S^+\right],$">|; 

$key = q/psi_{s_z}({bfx}');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1257.png"
 ALT="$ \psi_{s_z}
({\bf x}')$">|; 

$key = q/displaystyle|xi''_irangle=|xi'_irangle-sum_{j=1,i-1}frac{langlexi_j''|xi_i'rangle}{langlexi_j''|xi_j''rangle},|xi_j''rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img232.png"
 ALT="$\displaystyle \vert\xi''_i\rangle = \vert\xi'_i\rangle - \sum_{j=1,i-1}\frac{\l...
...'\vert\xi_i'\rangle}{\langle \xi_j''\vert\xi_j''\rangle} \vert\xi_j''\rangle.
$">|; 

$key = q/langleB|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="$ \langle B\vert A \rangle$">|; 

$key = q/displaystyle1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1479.png"
 ALT="$\displaystyle 1,$">|; 

$key = q/displaystyleL_z=x,p_y-y,p_x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1499.png"
 ALT="$\displaystyle L_z = x p_y - y  p_x,$">|; 

$key = q/displaystyleH_0=frac{p^2}{2,m},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2136.png"
 ALT="$\displaystyle H_0 = \frac{p^2}{2 m},$">|; 

$key = q/-{rmi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2923.png"
 ALT="$ -{\rm i}$">|; 

$key = q/displaystylelim_{trightarrowt_0}frac{|Atrangle-|At_0rangle}{t-t_0}=lim_{trightarrowt_0}frac{T-1}{t-t_0},|At_0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img545.png"
 ALT="$\displaystyle \lim_{t\rightarrow t_0} \frac{\vert At\rangle - \vert At_0\rangle}{t-t_0} = \lim_{t\rightarrow t_0}\frac{T-1}{t-t_0} \vert At_0\rangle$">|; 

$key = q/mp;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1133.png"
 ALT="$ \mp$">|; 

$key = q/hbar,omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2013.png"
 ALT="$ \hbar \omega$">|; 

$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="269" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img921.png"
 ALT="$\displaystyle = [l (l+1) - m  (m-1)]^{1/2}  \hbar  Y_{l  m-1}.$">|; 

$key = q/displaystyleR({mitDelta}varphi),R(-{mitDelta}varphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img861.png"
 ALT="$\displaystyle R({\mit\Delta}\varphi) R(-{\mit\Delta}\varphi)$">|; 

$key = q/psi(x')=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img324.png"
 ALT="$ \psi(x') =1$">|; 

$key = q/r'=|{bfx}'|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2166.png"
 ALT="$ r'=\vert{\bf x}'\vert$">|; 

$key = q/L_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img737.png"
 ALT="$ L_z$">|; 

$key = q/displaystyleH,|Atrangle=H',|Atrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img630.png"
 ALT="$\displaystyle H \vert At\rangle = H' \vert At\rangle.$">|; 

$key = q/displaystyle=left(frac{-{rmi}}{hbar}right)^2sum_mint_{t_0}^tdt'int_{t_0}^{t'}dt'_0)],H_{nm}(t'),exp[,{rmi},omega_{mi},(t''-t_0)],H_{mi}(t''),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="623" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1882.png"
 ALT="$\displaystyle = \left(\frac{-{\rm i}}{\hbar}\right)^2 \sum_m \int_{t_0}^t dt'\i...
...t'-t_0)]  H_{nm}(t')   \exp[ {\rm i}  \omega_{mi} (t''-t_0)] H_{mi}(t''),$">|; 

$key = q/displaystyle2,a_1,a_2,s=a,(a_1-a_2),alpha,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2912.png"
 ALT="$\displaystyle 2 a_1 a_2 s = a (a_1-a_2) \alpha,$">|; 

$key = q/4,gammaslashhbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1830.png"
 ALT="$ 4 \gamma/\hbar$">|; 

$key = q/displaystyleA=1-frac{{rmi}}{4},sigma_{mu,nu},{mitDelta}omega^{,mu,nu},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2683.png"
 ALT="$\displaystyle A = 1 - \frac{{\rm i}}{4} \sigma_{\mu \nu} {\mit\Delta}\omega^{ \mu \nu},$">|; 

$key = q/{bfp}equiv(p_1,p_2,p_3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img616.png"
 ALT="$ {\bf p}\equiv (p_1, p_2, p_3)$">|; 

$key = q/displaystylec_i^{(1)}(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1959.png"
 ALT="$\displaystyle c_i^{(1)} (t)$">|; 

$key = q/displaystylea^{,mu}=g^{,mu,nu},a_nu,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2562.png"
 ALT="$\displaystyle a^{ \mu} = g^{ \mu \nu} a_\nu,$">|; 

$key = q/f(y)rightarrowexp(2,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img968.png"
 ALT="$ f(y)\rightarrow \exp(2 y)$">|; 

$key = q/displaystyleR({mitDelta}varphi),|l,0rangle=exp(0),|l,0rangle=|l,0rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img867.png"
 ALT="$\displaystyle R({\mit\Delta}\varphi) \vert l, 0\rangle = \exp(0) \vert l, 0\rangle = \vert l, 0\rangle.$">|; 

$key = q/displaystylelanglen'|,p_x,|nrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img699.png"
 ALT="$\displaystyle \langle n'\vert p_x \vert n\rangle$">|; 

$key = q/displaystyleH=-e,phi(r)+c,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2780.png"
 ALT="$\displaystyle H = - e \phi(r) + c $">|; 

$key = q/d^{,3}x';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img643.png"
 ALT="$ d^{ 3} x'$">|; 

$key = q/displaystylecdot({bfa}times{bfb}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1212.png"
 ALT="$\displaystyle \cdot ({\bf a} \times {\bf b} ).$">|; 

$key = q/langlei|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="$ \langle i\vert$">|; 

$key = q/displaystylea_2,c_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2901.png"
 ALT="$\displaystyle a_2 c_s$">|; 

$key = q/displaystyle{rmi},hbar,frac{dv}{dt}=[v,H]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img582.png"
 ALT="$\displaystyle {\rm i} \hbar  \frac{d v}{dt} = [v, H] = 0.$">|; 

$key = q/q_1'cdotsq_N';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img371.png"
 ALT="$ q_1'\cdots q_N'$">|; 

$key = q/displaystyle=sum_{m_1}sum_{m_2}langlej_1,j_2;j,m|j_1,j_2;m_1,m_2ranglelanglej_1,j_2;m_1,m_2|j_1,j_2;j,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="457" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1383.png"
 ALT="$\displaystyle =\sum_{m_1}\sum_{m_2} \langle j_1, j_2; j, m\vert j_1, j_2; m_1, m_2\rangle \langle j_1, j_2; m_1, m_2\vert j_1, j_2; j, m\rangle$">|; 

$key = q/displaystylecdot{bfp},|irangle|^{,2},delta(E_n-E_i-hbar,omega),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2053.png"
 ALT="$\displaystyle \cdot{\bf p}  \vert i\rangle\vert^{ 2}  \delta(E_n-E_i -\hbar \omega),$">|; 

$key = q/r>a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2265.png"
 ALT="$ r&gt;a$">|; 

$key = q/|x',y',z',pmrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1046.png"
 ALT="$ \vert x', y', z', \pm\rangle$">|; 

$key = q/L^pm=L_xpm{rmi},L_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img847.png"
 ALT="$ L^\pm = L_x \pm {\rm i}  L_y$">|; 

$key = q/omega_A({bfx});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2486.png"
 ALT="$ \omega_A({\bf x})$">|; 

$key = q/P_{l,m}(xi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img908.png"
 ALT="$ P_{l m}(\xi)$">|; 

$key = q/displaystyle=1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img667.png"
 ALT="$\displaystyle = 1,$">|; 

$key = q/H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img224.png"
 ALT="$ H$">|; 

$key = q/|2,1,-1rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1578.png"
 ALT="$ \vert 2,1,-1\rangle$">|; 

$key = q/displaystyle[q_i,p_j];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img295.png"
 ALT="$\displaystyle [q_i, p_j]$">|; 

$key = q/displaystyle(2pi)^{3slash2}[psi({bfx})-phi({bfx})]=frac{exp(,{rmi},k,r)}{r},f(theta).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2250.png"
 ALT="$\displaystyle (2\pi)^{3/2}[\psi({\bf x} )- \phi({\bf x}) ] = \frac{\exp( {\rm i} k r)}{r}  f(\theta).$">|; 

$key = q/displaystyleomega_{ni}=frac{E_n-E_i}{hbar}rightarrowomega_{ni}pmomega.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2007.png"
 ALT="$\displaystyle \omega_{ni} = \frac{E_n-E_i}{\hbar} \rightarrow \omega_{ni}\pm \omega.$">|; 

$key = q/displaystylefrac{partial}{partialq_i},q_j-q_j,frac{partial}{partialq_i}=delta_{ij}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img389.png"
 ALT="$\displaystyle \frac{\partial}{\partial q_i}  q_j - q_j  \frac{\partial}{\partial q_i} = \delta_{ij}.$">|; 

$key = q/|{bfp}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2207.png"
 ALT="$ \vert{\bf p}\vert$">|; 

$key = q/displaystyler=(x^{2}+y^{2}+z^{2})^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2782.png"
 ALT="$\displaystyle r = (x^{2}+y^{2}+z^{2})^{1/2},$">|; 

$key = q/gammaggR^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2364.png"
 ALT="$ \gamma\gg R^{-1}$">|; 

$key = q/displaystyleR_{n,l}(r)={calR}_{n,l}(rslasha),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img977.png"
 ALT="$\displaystyle R_{n l}(r) = {\cal R}_{n l}(r/a),$">|; 

$key = q/displaystyle=[u,v_1]_{cl},v_2+v_1,[u,v_2]_{cl},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img266.png"
 ALT="$\displaystyle = [u, v_1]_{cl}  v_2 + v_1  [u, v_2]_{cl},$">|; 

$key = q/x^1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2535.png"
 ALT="$ x^1$">|; 

$key = q/c'=c'(p_x');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img422.png"
 ALT="$ c' = c'(p_x')$">|; 

$key = q/displaystyle=langle+|,S_k,|+ranglelanglex',y',z'|langle+||Aranglerangle+langle+|,S_k,|-ranglelanglex',y',z'|langle-||Aranglerangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="432" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1183.png"
 ALT="$\displaystyle = \langle + \vert S_k  \vert+\rangle \langle x',y',z'\vert\lang...
... S_k  \vert-\rangle \langle x',y',z'\vert\langle -\vert\vert A\rangle\rangle,$">|; 

$key = q/displaystyleleft[left(p^0+frac{e}{c},{mitPhi}^{,0}right)-alpha_1left(p^1+frac{e}left(p^3+frac{e}{c},{mitPhi}^{,3}right)-beta,m_e,cright]psi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="624" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2920.png"
 ALT="$\displaystyle \left[\left(p^0+\frac{e}{c} {\mit\Phi}^{ 0}\right)-\alpha_1\lef...
...pha_3\left(p^3+\frac{e}{c} {\mit\Phi}^{ 3}\right) -\beta m_e c\right]\psi=0$">|; 

$key = q/displaystyle(|BranglelangleA|)^dag=|AranglelangleB|.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img217.png"
 ALT="$\displaystyle (\vert B\rangle\langle A\vert)^\dag = \vert A\rangle\langle B\vert.
$">|; 

$key = q/displaystylep_x=-{rmi},hbar,left(frac{d}{dx}right)_{rmnew}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img366.png"
 ALT="$\displaystyle p_x = -{\rm i}  \hbar \left(\frac{d}{dx}\right)_{\rm new}.$">|; 

$key = q/displaystyle(|BranglelangleA|)^{dag}=|AranglelangleB|.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img110.png"
 ALT="$\displaystyle (\vert B \rangle \langle A\vert)^{\dag } = \vert A \rangle \langle B\vert.$">|; 

$key = q/displaystyleE=left(8-frac{5}{2}right)E_0=-74.8,{rmeV}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2522.png"
 ALT="$\displaystyle E = \left(8 - \frac{5}{2}\right)E_0 = -74.8 {\rm eV}.$">|; 

$key = q/displaystylelanglej_1,j_2;j,m|j_1,j_2;m_1,m_2rangle=langlej_1,j_2;m_1,m_2|j_1,j_2;j,mrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="413" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1396.png"
 ALT="$\displaystyle \langle j_1, j_2; j, m\vert j_1, j_2; m_1, m_2\rangle = \langle j_1, j_2; m_1, m_2\vert j_1, j_2; j, m\rangle.$">|; 

$key = q/displaystyle[Sigma_1,Sigma_3];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2771.png"
 ALT="$\displaystyle [\Sigma_1,\Sigma_3]$">|; 

$key = q/displaystyleP_{12}^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2383.png"
 ALT="$\displaystyle P_{12}^{ 2}$">|; 

$key = q/displaystylelangle+|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1146.png"
 ALT="$\displaystyle \langle +\vert$">|; 

$key = q/displaystylep_irangle=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img395.png"
 ALT="$\displaystyle p_i \rangle = 0.$">|; 

$key = q/m_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1624.png"
 ALT="$ m_j$">|; 

$key = q/j_x=psi^dagc,alpha_1,psi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2707.png"
 ALT="$ j_x = \psi^\dag c \alpha_1 \psi$">|; 

$key = q/displaystylefrac{partialrho}{partialt}+nabla'cdot{bfj}=frac{2,{rmIm}(V)}{hbar},rho,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img656.png"
 ALT="$\displaystyle \frac{\partial \rho}{\partial t} + \nabla'\cdot {\bf j} = \frac{2 {\rm Im}(V)}{\hbar}  \rho,$">|; 

$key = q/displaystylexi'=xi'^{,ast},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img124.png"
 ALT="$\displaystyle \xi' = \xi'^{ \ast},$">|; 

$key = q/p^1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2570.png"
 ALT="$ p^1$">|; 

$key = q/b^{,mu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2553.png"
 ALT="$ b^{ \mu}$">|; 

$key = q/rho({bfx},t)=|psi|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2644.png"
 ALT="$ \rho({\bf x},t) = \vert\psi\vert^{ 2}$">|; 

$key = q/|f({bfk}',{bfk})|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2184.png"
 ALT="$ \vert f({\bf k}', {\bf k})\vert^{ 2}$">|; 

$key = q/displaystylezeta^{,2}=[beta,(;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2824.png"
 ALT="$\displaystyle \zeta^{ 2} = [\beta ($">|; 

$key = q/displaystyle=j_1,(j_1+1),hbar^2,|j_1,j_2;m_1,m_2rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1343.png"
 ALT="$\displaystyle = j_1 (j_1+1) \hbar^2 \vert j_1,j_2; m_1,m_2\rangle,$">|; 

$key = q/displaystyle[H,P_{12}]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2391.png"
 ALT="$\displaystyle [H,P_{12}] = 0.$">|; 

$key = q/displaystylealpha^2=-s_0^{,2}+k^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2896.png"
 ALT="$\displaystyle \alpha^2 = - s_0^{ 2} + k^2.$">|; 

$key = q/displaystyle|Brangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img174.png"
 ALT="$\displaystyle \vert B\rangle$">|; 

$key = q/J_{1,z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1313.png"
 ALT="$ J_{1 z}$">|; 

$key = q/displaystyle{bfL}cdot{bfS},|l-1slash2,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1666.png"
 ALT="$\displaystyle {\bf L} \cdot{\bf S}  \vert l- 1/2, m\rangle$">|; 

$key = q/displaystylec_n^{(1)}(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1876.png"
 ALT="$\displaystyle c_n^{(1)}(t)$">|; 

$key = q/rho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2650.png"
 ALT="$ \rho$">|; 

$key = q/A^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2048.png"
 ALT="$ A^2$">|; 

$key = q/displaystyleH({bfx},{bfp})=frac{{p}^2}{2,m}+V({bfx}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img608.png"
 ALT="$\displaystyle H({\bf x},{\bf p}) = \frac{{p}^2}{2 m} + V({\bf x}).$">|; 

$key = q/displaystylee_{12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1451.png"
 ALT="$\displaystyle e_{12}$">|; 

$key = q/displaystyle|Arangle=sum_{i=1,N}alpha_i,|irangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="$\displaystyle \vert A\rangle = \sum_{i=1,N} \alpha_i  \vert i\rangle,$">|; 

$key = q/displaystyleH_{nm}(t)=langlen|,H_1(t),|mrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1795.png"
 ALT="$\displaystyle H_{nm}(t) = \langle n \vert H_1(t) \vert m \rangle,$">|; 

$key = q/displaystyle|k'',k'rangle_+=+|k',k''rangle_+,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2398.png"
 ALT="$\displaystyle \vert k'' k'\rangle_+ =+ \vert k' k''\rangle_+,$">|; 

$key = q/displaystyle=frac{1}{(l+1slash2),n^3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1768.png"
 ALT="$\displaystyle = \frac{1}{(l+1/2) n^3},$">|; 

$key = q/phi({bfx}');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2140.png"
 ALT="$ \phi({\bf x}')$">|; 

$key = q/displaystyle{rmi},hbar,frac{partiallangle{bfx'}|Atrangle}{partialt}=langle{bfx'}|,H,|Atrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img621.png"
 ALT="$\displaystyle {\rm i} \hbar  \frac{\partial \langle {\bf x'}\vert At\rangle}{\partial t} = \langle {\bf x'}\vert H \vert At\rangle,$">|; 

$key = q/c_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="$ c_2$">|; 

$key = q/x^{,mu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2541.png"
 ALT="$ x^{ \mu}$">|; 

$key = q/sinalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$ \sin\alpha$">|; 

$key = q/displaystyle=2,{rmi},hbar,(L_x,z-x,L_z),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1510.png"
 ALT="$\displaystyle = 2 {\rm i}  \hbar  ( L_x  z - x  L_z ),$">|; 

$key = q/displaystyleleft[l^{,2},(l+1)^2-2,l,(l+1),l',(l'+1)+l'^{,2},(l'+1)^2-2,l,(l+1)-2,l',(l'+1)right]langlen,l,m|,z,|n',l',m'rangle=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="723" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1521.png"
 ALT="$\displaystyle \left[l^{ 2}  (l+1)^2 - 2  l (l+1) l' (l'+1) + l'^{ 2} (l...
...(l+1) - 2 l' (l'+1)\right] \langle n,l,m\vert z \vert n',l',m' \rangle = 0,$">|; 

$key = q/displaystyle|A,t_0,trangle=U(t_0,t),|Arangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1853.png"
 ALT="$\displaystyle \vert A, t_0, t\rangle = U(t_0, t)  \vert A\rangle.$">|; 

$key = q/displaystylelangleS_yranglerightarrowlangleS_yrangle,cos{mitDelta}varphi+langleS_xrangle,sin{mitDelta}varphi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1093.png"
 ALT="$\displaystyle \langle S_y \rangle \rightarrow \langle S_y\rangle  \cos{\mit\Delta}\varphi + \langle S_x\rangle \sin{\mit\Delta}\varphi.$">|; 

$key = q/displaystyle0=frac{d}{dt},(langle{bfx}cdot{bfp}rangle)=frac{{rmi}}{hbar},langle[H,{bfx}cdot{bfp}]rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1002.png"
 ALT="$\displaystyle 0= \frac{d}{dt} (\langle{\bf x}\cdot{\bf p}\rangle) = \frac{{\rm i}}{\hbar} \langle[H, {\bf x}\cdot{\bf p}]\rangle
$">|; 

$key = q/l=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img836.png"
 ALT="$ l=2$">|; 

$key = q/displaystylelangleS_xranglerightarrowlangleS_xrangle,cos{mitDelta}varphi-langleS_yrangle,sin{mitDelta}varphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1092.png"
 ALT="$\displaystyle \langle S_x \rangle \rightarrow \langle S_x\rangle  \cos{\mit\Delta}\varphi - \langle S_y\rangle \sin{\mit\Delta}\varphi$">|; 

$key = q/|k',k''rangle_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2397.png"
 ALT="$ \vert k' k''\rangle_+$">|; 

$key = q/displaystyle=E,|k'',k',k'''rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2422.png"
 ALT="$\displaystyle = E \vert k'' k' k'''\rangle,$">|; 

$key = q/R_x({mitDelta}varphi_x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img874.png"
 ALT="$ R_x({\mit\Delta}\varphi_x)$">|; 

$key = q/dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1927.png"
 ALT="$ dt$">|; 

$key = q/langleL_y^{,2}rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img998.png"
 ALT="$ \langle L_y^{ 2}\rangle$">|; 

$key = q/dpsislashdxrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img333.png"
 ALT="$ d\psi/dx\rangle$">|; 

$key = q/displaystyle[epsilon,r]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2839.png"
 ALT="$\displaystyle [\epsilon,r] = 0.$">|; 

$key = q/displaystylelangle{bfk}|{bfk}'rangle=intd^3x,langle{bfk}|{bfx}ranglelangle{bfx}|{rmi},{bfx}cdot({bfk}-{bfk}')]}{(2pi)^3}=delta({bfk}-{bfk'}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="526" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2159.png"
 ALT="$\displaystyle \langle {\bf k}\vert{\bf k}'\rangle =\int d^3 x  \langle {\bf k}...
...  {\bf x}\cdot({\bf k} -{\bf k}')]} {(2\pi )^3} = \delta ({\bf k} - {\bf k'}).$">|; 

$key = q/displaystyleA_0(r)=B,frac{sin(k',r)}{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2316.png"
 ALT="$\displaystyle A_0(r) = B  \frac{\sin (k' r)}{r},$">|; 

$key = q/R(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img935.png"
 ALT="$ R(r)$">|; 

$key = q/displaystyle=(E-hbar,omega),A,|Erangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img678.png"
 ALT="$\displaystyle = (E-\hbar \omega) A \vert E\rangle,$">|; 

$key = q/displaystyle=left(!{array}{rr}0&11&0{array}!right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1167.png"
 ALT="$\displaystyle = \left(\!\begin{array}{rr} 0 &amp;1\ 1&amp;0\end{array}\!\right),$">|; 

$key = q/displaystyleL^2,|l,mrangle=l,(l+1),hbar^2,|l,mrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img814.png"
 ALT="$\displaystyle L^2  \vert l, m\rangle = l (l+1)  \hbar^2  \vert l, m\rangle.$">|; 

$key = q/{bfJ}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1300.png"
 ALT="$ {\bf J}_2$">|; 

$key = q/displaystyle=frac{q}{2,m},{bfL}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1110.png"
 ALT="$\displaystyle = \frac{q}{2 m}  {\bf L}.$">|; 

$key = q/|n,l,m_l,m_srangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1708.png"
 ALT="$ \vert n, l,m_l, m_s\rangle$">|; 

$key = q/displaystylefrac{c_{s-1}}{a_1}-alpha,c_s-(s+k),c_s'+frac{c_{s-1}}{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2878.png"
 ALT="$\displaystyle \frac{c_{s-1}}{a_1} -\alpha c_s - (s+k) c_s' + \frac{c_{s-1}}{a}$">|; 

$key = q/displaystylesimeqa,c_s',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2902.png"
 ALT="$\displaystyle \simeq a c_s',$">|; 

$key = q/x^3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2537.png"
 ALT="$ x^3$">|; 

$key = q/langleB|,X,|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img96.png"
 ALT="$ \langle B\vert X \vert A \rangle$">|; 

$key = q/displaystylelangleB|Arangle=sum_{i=1,N}beta_i^{,ast},alpha_i.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="$\displaystyle \langle B\vert A \rangle = \sum_{i=1,N} \beta_i^{ \ast}  \alpha_i.$">|; 

$key = q/displaystyleA,gamma^{,mu},A^{-1},a^nu_{~mu}=gamma^nu,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2676.png"
 ALT="$\displaystyle A \gamma^{ \mu} A^{-1} a^\nu_{&nbsp;\mu} = \gamma^\nu,$">|; 

$key = q/displaystylec_n=c_n^{(0)}+c_n^{(1)}+c_n^{(2)}+cdots,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1872.png"
 ALT="$\displaystyle c_n = c_n^{(0)} + c_n^{(1)} + c_n^{(2)} + \cdots,$">|; 

$key = q/langleS_xrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1079.png"
 ALT="$ \langle S_x\rangle$">|; 

$key = q/displaystylefrac{dx_i}{dt}=frac{1}{{rmi},hbar},[x_i,H]=frac{1}{{rmi},hbar},frac{!left(sum_{j=1,3}p_j^{,2}right)=frac{p_i}{m}=frac{p_i(0)}{m},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="440" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img601.png"
 ALT="$\displaystyle \frac{d x_i}{dt} = \frac{1}{{\rm i} \hbar}  [x_i, H] = \frac{1}...
... p_i}\! \left(\sum_{j=1,3} p_j^{ 2}\right) = \frac{p_i}{m} = \frac{p_i(0)}{m},$">|; 

$key = q/|p'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img419.png"
 ALT="$ \vert p'\rangle$">|; 

$key = q/langlen|,z,|irangle=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2084.png"
 ALT="$ \langle n\vert z \vert i\rangle=0$">|; 

$key = q/displaystylelangle1slash2,-1slash2|1,0rangle+langle-1slash2,1slash2|1,0rangle=sqrt{2},langle1slash2,1slash2|1,1rangle=pmsqrt{2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="491" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1411.png"
 ALT="$\displaystyle \langle 1/2, -1/2\vert 1, 0\rangle + \langle -1/2, 1/2\vert 1, 0\rangle =\sqrt{2} \langle 1/2, 1/2\vert 1,1\rangle=\pm\sqrt{2},$">|; 

$key = q/H_0+H_B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1704.png"
 ALT="$ H_0 + H_B$">|; 

$key = q/{bfx}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2452.png"
 ALT="$ {\bf x}_1$">|; 

$key = q/displaystyleH_{nm}(t)=langlen|,H_1(t),|mrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1884.png"
 ALT="$\displaystyle H_{nm} (t) = \langle n\vert  H_1(t)  \vert m\rangle.$">|; 

$key = q/displaystyle|k'rangle,|k''rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2373.png"
 ALT="$\displaystyle \vert k'\rangle \vert k''\rangle,$">|; 

$key = q/displaystyle{rmi},hbar,frac{d|Atrangle}{dt}=H(t),|Atrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img553.png"
 ALT="$\displaystyle {\rm i} \hbar  \frac{d\vert At\rangle}{dt} = H(t) \vert At\rangle.$">|; 

$key = q/[L^+,L^-]=2,hbar,L_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img992.png"
 ALT="$ [L^+,L^-] = 2 \hbar L_z$">|; 

$key = q/displaystylep_x={rmi},hbar,d_x+f(x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img483.png"
 ALT="$\displaystyle p_x = {\rm i} \hbar  d_x + f(x),$">|; 

$key = q/lambda_{31}=pm1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2440.png"
 ALT="$ \lambda_{31}=\pm 1$">|; 

$key = q/displaystyleleft(frac{1}{a_1}-frac{alpha}{y}right)psi_a-left(frac{d}{dy}+frac{k+1}{y}right)psi_b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2863.png"
 ALT="$\displaystyle \left(\frac{1}{a_1} - \frac{\alpha}{y}\right)\psi_a - \left(\frac{d}{dy} + \frac{k+1}{y}\right)\psi_b$">|; 

$key = q/S_z=hbarslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1428.png"
 ALT="$ S_z=\hbar/2$">|; 

$key = q/aslasha_1=a_2slasha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2883.png"
 ALT="$ a/a_1=a_2/a$">|; 

$key = q/displaystylesum_jsigma_j,a_jsum_ksigma_k,b_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1208.png"
 ALT="$\displaystyle \sum_j \sigma_j  a_j \sum_k \sigma_k  b_k$">|; 

$key = q/displaystyle(nabla^2+k^2),psi({bfx})=frac{2,m}{hbar^2},langle{bfx}|,H_1,|psirangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2146.png"
 ALT="$\displaystyle (\nabla^2 + k^2) \psi({\bf x}) = \frac{2 m}{\hbar^2}  \langle {\bf x} \vert H_1 \vert \psi\rangle,$">|; 

$key = q/displaystylesigma_{rmtotal}=sum_{l=0,infty}sigma_l,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2262.png"
 ALT="$\displaystyle \sigma_{\rm total} = \sum_{l=0,\infty} \sigma_l,$">|; 

$key = q/displaystylealpha=2,e^2sum_{n>1}frac{|langle1,0,0|,z,|n,1,0rangle|^{,2}}{E_{n00}-E_{100}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1531.png"
 ALT="$\displaystyle \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}}.$">|; 

$key = q/displaystyle{bfx}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1584.png"
 ALT="$\displaystyle {\bf x}_1$">|; 

$key = q/|x',y',z'rangle|+rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1059.png"
 ALT="$ \vert x', y', z'\rangle
\vert+\rangle$">|; 

$key = q/displaystyle=left.frac{2pi}{hbar},overline{|V_{ni}|^{,2}},rho(E_n)right|_{E_n=E_i-hbar,omega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2015.png"
 ALT="$\displaystyle =\left. \frac{2\pi}{\hbar}  \overline{\vert V_{ni}\vert^{ 2}} \rho(E_n) \right\vert _{E_n = E_i-\hbar \omega},$">|; 

$key = q/-rmi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img412.png"
 ALT="$ -\rm i$">|; 

$key = q/left[{mitDelta}xi,{mitDelta}etaright];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img180.png"
 ALT="$ \left[ {\mit\Delta} \xi, {\mit\Delta} \eta\right]$">|; 

$key = q/displaystyleC_l=(2,l+1),exp[,{rmi},(delta_l+l,pislash2)].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2251.png"
 ALT="$\displaystyle C_l = (2 l+1) \exp[ {\rm i} (\delta_l + l \pi/2)].$">|; 

$key = q/displaystyle|Atrangle=exp[-{rmi},H'(t-t_0)slashhbar],|At_0rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img632.png"
 ALT="$\displaystyle \vert At\rangle = \exp[ -{\rm i} H'(t-t_0)/\hbar]  \vert At_0\rangle.$">|; 

$key = q/u(x'+a)=u(x');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img527.png"
 ALT="$ u(x'+a)=u(x')$">|; 

$key = q/-3slash2,-1slash2,1slash2,3slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img839.png"
 ALT="$ -3/2, -1/2, 1/2, 3/2$">|; 

$key = q/u_l=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2283.png"
 ALT="$ u_l =0$">|; 

$key = q/displaystyleX+(Y+Z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img86.png"
 ALT="$\displaystyle X + (Y + Z)$">|; 

$key = q/displaystyleleft(!{array}{c}psi_+'({bfx}')psi_-'({bfx}'){array}!right)=left({arrrmi},hbar,partialpsi_-({bfx}')slashpartialx_k'{array}!right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1192.png"
 ALT="$\displaystyle \left(\! \begin{array}{c} \psi_+'({\bf x}')\ \psi_-' ({\bf x}') ...
... -{\rm i} \hbar  \partial \psi_-({\bf x}')/\partial x_k'\end{array}\!\right).$">|; 

$key = q/J_1^{,2},J_2^{,2},J_{1z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1339.png"
 ALT="$ J_1^{ 2}, J_2^{ 2}, J_{1z}$">|; 

$key = q/mneqn;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1483.png"
 ALT="$ m\neq n$">|; 

$key = q/|~rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img176.png"
 ALT="$ \vert&nbsp;\rangle$">|; 

$key = q/displaystylelangle1slash2,-1slash2|1,0rangle+langle-1slash2,1slash2|1,0rangle=sqrt{2},langle-1slash2,-1slash2|1,-1rangle=pmsqrt{2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="527" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1412.png"
 ALT="$\displaystyle \langle 1/2, -1/2\vert 1, 0\rangle + \langle -1/2, 1/2\vert 1, 0\rangle =\sqrt{2}  \langle -1/2, -1/2\vert 1,-1\rangle =\pm\sqrt{2}.$">|; 

$key = q/trightarrowt_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img543.png"
 ALT="$ t\rightarrow t_0$">|; 

$key = q/(p_x,p_y,p_z)equiv{bfp};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img726.png"
 ALT="$ (p_x, p_y, p_z)\equiv {\bf p}$">|; 

$key = q/displaystyleX,|Arangle=|0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img83.png"
 ALT="$\displaystyle X   \vert A\rangle = \vert\rangle$">|; 

$key = q/displaystyle=frac{[s,(s+1)-j,(j-1)]^{1slash2}}{2,s},delta_{j,,l+1}+frac{[s,(s+1)-j,(j+1)]^{1slash2}}{2,s},delta_{j,,l-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="492" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1279.png"
 ALT="$\displaystyle = \frac{[s (s+1) - j (j-1)]^{1/2} }{2 s} \delta_{j   l+1}+ \frac{[s (s+1) - j (j+1)]^{1/2} }{2 s} \delta_{j   l-1},$">|; 

$key = q/V_0<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2314.png"
 ALT="$ V_0&lt;0$">|; 

$key = q/displaystyle_ecdot{bfB},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1738.png"
 ALT="$\displaystyle _e\cdot {\bf B},$">|; 

$key = q/|m_1,m_2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1420.png"
 ALT="$ \vert m_1, m_2\rangle$">|; 

$key = q/displaystyle[L^2,[L^2,z]]=2,hbar^2,(L^2,z+z,L^2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1518.png"
 ALT="$\displaystyle [L^2, [L^2, z]] = 2  \hbar^2  (L^2  z + z  L^2),$">|; 

$key = q/displaystyleR_{2,1}(r)=frac{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="298" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1008.png"
 ALT="$\displaystyle R_{2 1}(r)= \frac{1}{\sqrt{3} (2 a_0)^{3/2}} \frac{r}{a_0} \exp\left(-\frac{r}{2 a_0}\right).
$">|; 

$key = q/rggr';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2162.png"
 ALT="$ r\gg r'$">|; 

$key = q/displaystylecdot{bfp},|irangle|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2066.png"
 ALT="$\displaystyle \cdot{\bf p}  \vert i\rangle\vert^{ 2}$">|; 

$key = q/gammaequivgamma(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img359.png"
 ALT="$ \gamma\equiv \gamma(x)$">|; 

$key = q/S^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1017.png"
 ALT="$ S^2$">|; 

$key = q/E_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1995.png"
 ALT="$ E_i$">|; 

$key = q/displaystyleH({bfx},{bfp})=frac{{p}^2}{2,m}=frac{1}{2,m}sum_{i=1,3}p_i^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img598.png"
 ALT="$\displaystyle H({\bf x},{\bf p}) = \frac{{p}^2}{2 m} = \frac{1}{2 m} \sum_{i=1,3} p_i^{ 2}.$">|; 

$key = q/displaystyle=-y^lleft(-frac{1}{y}frac{d}{dy}right)^lfrac{cosy}{y}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2231.png"
 ALT="$\displaystyle = -y^l\left(-\frac{1}{y}\frac{d}{dy}\right)^l \frac{\cos y}{y}.$">|; 

$key = q/displaystyleP(p_x',dp_x')=|phi(p_x')|^{,2},dp_x'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img409.png"
 ALT="$\displaystyle P(p_x', dp_x') = \vert\phi(p_x')\vert^{ 2} dp_x'.$">|; 

$key = q/displaystylesim;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1481.png"
 ALT="$\displaystyle \sim$">|; 

$key = q/displaystyleE=frac{hbar^2k^2}{2,m}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2147.png"
 ALT="$\displaystyle E = \frac{\hbar^2 k^2}{2 m}.$">|; 

$key = q/displaystyle{rmi},hbar,frac{partiallangle{bfx}'|Atrangle}{partialt}=-left(frac{hla'^{,2}langle{bfx'}|Atrangle+V({bfx}')langle{bfx}'|Atrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="369" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img627.png"
 ALT="$\displaystyle {\rm i} \hbar  \frac{\partial \langle {\bf x}'\vert At\rangle}{...
...}\langle{\bf x'} \vert At\rangle + V({\bf x}') \langle {\bf x}'\vert At\rangle,$">|; 

$key = q/displaystyle=[u_1,v]_{cl},u_2+u_1,[u_2,v]_{cl},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img264.png"
 ALT="$\displaystyle = [u_1, v]_{cl}  u_2 + u_1  [u_2, v]_{cl},$">|; 

$key = q/displaystyle|1rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1592.png"
 ALT="$\displaystyle \vert 1\rangle$">|; 

$key = q/displaystyle[J^{,2},J_{1,z}];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1333.png"
 ALT="$\displaystyle [J^{ 2}, J_{1 z}]$">|; 

$key = q/eta';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img153.png"
 ALT="$ \eta'$">|; 

$key = q/displaystyle={rmi},hbar,[u_2,v_2]_{qm},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img285.png"
 ALT="$\displaystyle = {\rm i}  \hbar  [u_2, v_2]_{qm},$">|; 

$key = q/r<R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2354.png"
 ALT="$ r&lt;R$">|; 

$key = q/displaystylep_x=-{rmi},hbar,frac{d}{dx}+f(x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img351.png"
 ALT="$\displaystyle p_x = -{\rm i} \hbar \frac{d}{dx} + f(x),$">|; 

$key = q/displaystyle=frac{{rmi},hbar}{2}left(,-,|+ranglelangle-|+|-ranglelangle+|,right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1039.png"
 ALT="$\displaystyle = \frac{{\rm i} \hbar}{2}\left(  - \vert+\rangle \langle -\vert + \vert-\rangle \langle +\vert \right),$">|; 

$key = q/displaystyleI;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2529.png"
 ALT="$\displaystyle I$">|; 

$key = q/displaystylesigma_{mu,nu}=-sigma_{nu,mu}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2687.png"
 ALT="$\displaystyle \sigma_{\mu \nu} = -\sigma_{\nu \mu}.$">|; 

$key = q/-l,-l+1,cdotsl-1,l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img842.png"
 ALT="$ -l, -l+1,\cdots
l-1, l$">|; 

$key = q/displaystyle{bfJ}={bfJ}_1+{bfJ}_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1321.png"
 ALT="$\displaystyle {\bf J} = {\bf J}_1 + {\bf J}_2.$">|; 

$key = q/displaystyleleftlanglefrac{p_x^{,2}}{2,m}rightrangle=E-langleVrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img722.png"
 ALT="$\displaystyle \left\langle \frac{p_x^{ 2}}{2 m}\right\rangle = E- \langle V\rangle,
$">|; 

$key = q/(p_1,p_2,p_3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img589.png"
 ALT="$ (p_1, p_2, p_3)$">|; 

$key = q/r_<;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2512.png"
 ALT="$ r_&lt;$">|; 

$key = q/m_{rmmin};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img830.png"
 ALT="$ m_{\rm min}$">|; 

$key = q/lambdabar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1140.png"
 ALT="$ \lambdabar$">|; 

$key = q/displaystylef(l,m_{rmmax})=m_{rmmax}(m_{rmmax}+1).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img808.png"
 ALT="$\displaystyle f(l, m_{\rm max}) = m_{\rm max} (m_{\rm max} + 1).$">|; 

$key = q/displaystylelanglel,m|,L^2-L_z^{,2},|l,mrangle=langlel,m|,f(l,m),hbar^2-m^2,hbar^2,|l,mrangle=[f(l,m)-m^2],hbar^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="535" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img786.png"
 ALT="$\displaystyle \langle l, m \vert  L^2 - L_z^{ 2}  \vert l, m\rangle =\langle...
...(l, m)  \hbar^2 - m^2  \hbar^2  \vert l, m\rangle =[f(l,m) - m^2]  \hbar^2,$">|; 

$key = q/{bfq}equiv{bfk}-{bfk}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2193.png"
 ALT="$ {\bf q} \equiv {\bf k} - {\bf k}'$">|; 

$key = q/displaystyle(xi,eta-eta,xi),|xi'eta'rangle=|0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img160.png"
 ALT="$\displaystyle (\xi \eta -\eta \xi)  \vert\xi' \eta'\rangle = \vert\rangle$">|; 

$key = q/displaystyleu(omega)=frac{hbar}{pi^2,c^3},frac{omega^3}{exp(hbar,omegaslashk_B,T)-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2097.png"
 ALT="$\displaystyle u(\omega) = \frac{\hbar}{\pi^2 c^3} \frac{\omega^3}{\exp(\hbar \omega/k_B T)-1},$">|; 

$key = q/alphasimeq1slash137ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2903.png"
 ALT="$ \alpha\simeq 1/137 \ll 1$">|; 

$key = q/displaystylea_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2546.png"
 ALT="$\displaystyle a_1$">|; 

$key = q/displaystylea=n,a_0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img978.png"
 ALT="$\displaystyle a = n a_0.$">|; 

$key = q/phi(p_x');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img406.png"
 ALT="$ \phi(p_x')$">|; 

$key = q/displaystylembox{hspace{1cm}for{k{odd};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1222.png"
 ALT="$\displaystyle \mbox{\hspace{1cm}for $k$ odd}$">|; 

$key = q/|psiranglerightarrow|phirangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2144.png"
 ALT="$ \vert\psi\rangle \rightarrow \vert\phi\rangle$">|; 

$key = q/displaystylesimeqfrac{(k,r)^l}{(2,l+1)!!},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2294.png"
 ALT="$\displaystyle \simeq \frac{(k r)^l}{(2 l+1)!!},$">|; 

$key = q/jneq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1012.png"
 ALT="$ j\neq 0$">|; 

$key = q/j_1+j_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1374.png"
 ALT="$ j_1+j_2$">|; 

$key = q/displaystyle=E,|k',k'',k'''rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2416.png"
 ALT="$\displaystyle = E \vert k' k'' k'''\rangle,$">|; 

$key = q/displaystylephi({bfx}_1',{bfx}_2')=frac{1}{sqrt{2}}left[omega_A({bfx}_1'),omega_B({bfx}_2')-omega_B({bfx}_1'),omega_A({bfx}_2')right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="379" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2480.png"
 ALT="$\displaystyle \phi({\bf x}_1',{\bf x}_2') = \frac{1}{\sqrt{2}}\left[\omega_A({\bf x}_1') \omega_B({\bf x}_2')-\omega_B({\bf x}_1') \omega_A({\bf x}_2')\right],$">|; 

$key = q/displaystyleg=2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2779.png"
 ALT="$\displaystyle g= 2.$">|; 

$key = q/displaystyle=Y+X,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img85.png"
 ALT="$\displaystyle = Y + X,$">|; 

$key = q/displaystyle[L^2,z];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1505.png"
 ALT="$\displaystyle [L^2, z]$">|; 

$key = q/displaystyleU(t_0,t)=exp[-{rmi},H_0,(t-t_0)slashhbar].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1857.png"
 ALT="$\displaystyle U(t_0, t) = \exp[-{\rm i}   H_0 (t-t_0)/\hbar].$">|; 

$key = q/displaystyle{rmi},hbar,frac{dc_n}{dt}=sum_mH_{nm}(t),exp[,{rmi},omega_{nm},(t-t_0)],c_m(t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1794.png"
 ALT="$\displaystyle {\rm i} \hbar  \frac{d c_n}{dt} = \sum_m H_{nm}(t)  \exp[ {\rm i} \omega_{nm}  (t-t_0)]  c_m(t),$">|; 

$key = q/displaystyleE=frac{E_0}{n^2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img974.png"
 ALT="$\displaystyle E = \frac{E_0}{n^2},$">|; 

$key = q/langleA|Crangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img107.png"
 ALT="$ \langle A \vert C\rangle$">|; 

$key = q/displaystyle{bfB}=frac{mu_0}{4pi,r^3},left[3,(mbox{boldmathmu}_pcdot{bfe}_r),{bfhmu}_pright]+frac{2,mu_0}{3},mbox{boldmathmu}_p,delta({bfx}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="347" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1735.png"
 ALT="$\displaystyle {\bf B} = \frac{\mu_0}{4\pi r^3} \left[3 (\mbox{\boldmath$\mu$...
...$\mu$}_p\right] + \frac{2 \mu_0}{3} \mbox{\boldmath$\mu$}_p \delta({\bf x}),$">|; 

$key = q/displaystyle={rmi},hbar,left(sinvarphi,frac{partial}{partialtheta}+cotthetacosvarphi,frac{partial}{partialvarphi}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img894.png"
 ALT="$\displaystyle = {\rm i} \hbar \left(\sin\varphi  \frac{\partial}{\partial \theta} + \cot\theta \cos\varphi \frac{\partial}{\partial \varphi}\right),$">|; 

$key = q/displaystyle=langlel,m+1|,L_x,|l,mrangle^ast-{rmi},langlel,m+1|,L_y,|l,mrangle^ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="325" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img821.png"
 ALT="$\displaystyle = \langle l, m+1\vert  L_x \vert l,m \rangle^\ast - {\rm i}  \langle l, m+1\vert  L_y \vert l,m \rangle^\ast$">|; 

$key = q/2p(m=0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1608.png"
 ALT="$ 2p(m=0)$">|; 

$key = q/displaystylesin^2alpha=1-frac{l+m+1slash2}{2,l+1}=frac{l-m+1slash2}{2,l+1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="313" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1653.png"
 ALT="$\displaystyle \sin^2\alpha = 1 - \frac{l+m+1/2}{2 l+1} = \frac{l-m+1/2}{2 l+1}.$">|; 

$key = q/displaystyleleft[gamma^{,mu}left(p_mu-frac{e}{c},{mitPhi}_muright)-m_e,cright]psi=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2669.png"
 ALT="$\displaystyle \left[\gamma^{ \mu}\left(p_\mu- \frac{e}{c} {\mit\Phi}_\mu\right)-m_e c\right]\psi = 0.$">|; 

$key = q/{bfL};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img931.png"
 ALT="$ {\bf L}$">|; 

$key = q/gamma^5,gamma^5=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2733.png"
 ALT="$ \gamma^5 \gamma^5=1$">|; 

$key = q/0,0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2464.png"
 ALT="$ 0,0$">|; 

$key = q/L^2,S^{,2},J^{,2},J_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1706.png"
 ALT="$ L^2, S^{ 2}, J^{ 2},
J_z$">|; 

$key = q/displaystyle[Sigma_1,Sigma_2];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2769.png"
 ALT="$\displaystyle [\Sigma_1,\Sigma_2]$">|; 

$key = q/V({bfx}');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img655.png"
 ALT="$ V({\bf x}')$">|; 

$key = q/displaystyle={bfp}cdot{bfL}+{rmi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2802.png"
 ALT="$\displaystyle = {\bf p}\cdot {\bf L} + {\rm i} $">|; 

$key = q/displaystyle{mitDelta}E_{nlmpm}=langlelpm1slash2,m|,H_B,|lpm1slash2,mrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1696.png"
 ALT="$\displaystyle {\mit\Delta} E_{nlm\pm} = \langle l\pm 1/2, m\vert  H_B  \vert l\pm 1/2, m\rangle.$">|; 

$key = q/||Aranglerangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1062.png"
 ALT="$ \vert\vert A\rangle\rangle$">|; 

$key = q/displaystyleepsilon=frac{|e_{12}|}{|E_1-E_2|}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1455.png"
 ALT="$\displaystyle \epsilon = \frac{\vert e_{12}\vert}{\vert E_1-E_2\vert}.$">|; 

$key = q/displaystyleV=lambda,x^4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1758.png"
 ALT="$\displaystyle V = \lambda x^4
$">|; 

$key = q/displaystyle{mitDelta}E_{nlmpm}=frac{e,hbar,B}{2,m_e},mleft(1pmfrac{1}{2,l+1}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1701.png"
 ALT="$\displaystyle {\mit\Delta} E_{nlm\pm} = \frac{e  \hbar  B}{2  m_e} m \left( 1 \pm \frac{1}{2 l+1} \right).$">|; 

$key = q/[[H,x],x];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img707.png"
 ALT="$ [[H,x], x]$">|; 

$key = q/q_1cdotsq_N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img370.png"
 ALT="$ q_1\cdots q_N$">|; 

$key = q/c_{nneqi}(trightarrow-infty)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1947.png"
 ALT="$ c_{n\neq i}(t\rightarrow -\infty) = 0$">|; 

$key = q/displaystyle=pm1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2086.png"
 ALT="$\displaystyle = \pm 1,$">|; 

$key = q/f_{21};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2117.png"
 ALT="$ f_{21}$">|; 

$key = q/displaystylex_mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2662.png"
 ALT="$\displaystyle x_\mu$">|; 

$key = q/i=1,N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img229.png"
 ALT="$ i=1,N$">|; 

$key = q/|Atranglerightarrow|At_0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img544.png"
 ALT="$ \vert At\rangle\rightarrow \vert A t_0\rangle$">|; 

$key = q/displaystyledelta_0=-k,a.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2287.png"
 ALT="$\displaystyle \delta_0 = -k a.$">|; 

$key = q/displaystyle[{bfS},S^2]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1016.png"
 ALT="$\displaystyle [{\bf S}, S^2] = 0.$">|; 

$key = q/trightarrow-infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1945.png"
 ALT="$ t\rightarrow -\infty$">|; 

$key = q/deltaH^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2749.png"
 ALT="$ \delta H^{ 2}$">|; 

$key = q/displaystylecdotlanglen|,{bfp},|irangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2074.png"
 ALT="$\displaystyle \cdot \langle n\vert {\bf p} \vert i\rangle.$">|; 

$key = q/s,(s+1),hbar^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1253.png"
 ALT="$ s (s+1) \hbar^2$">|; 

$key = q/T(t,t_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img538.png"
 ALT="$ T(t, t_0)$">|; 

$key = q/domega,u(omega);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2063.png"
 ALT="$ d\omega u(\omega)$">|; 

$key = q/r_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2514.png"
 ALT="$ r_2$">|; 

$key = q/v_d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img457.png"
 ALT="$ v_d$">|; 

$key = q/cdot{bfn}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2045.png"
 ALT="$ \cdot{\bf n} = 0$">|; 

$key = q/displaystyle{bfj}=frac{hbar}{m},{rmIm}(psi^ast,nablapsi).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2177.png"
 ALT="$\displaystyle {\bf j} = \frac{\hbar}{m}  {\rm Im}(\psi^\ast  \nabla \psi).$">|; 

$key = q/displaystylelangleS_zranglerightarrowlangleS_zrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1094.png"
 ALT="$\displaystyle \langle S_z \rangle \rightarrow\langle S_z\rangle,$">|; 

$key = q/displaystyle=r,sintheta,sinvarphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img891.png"
 ALT="$\displaystyle = r  \sin\theta  \sin\varphi,$">|; 

$key = q/|xi'eta'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img155.png"
 ALT="$ \vert\xi' \eta'\rangle$">|; 

$key = q/{calP};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1974.png"
 ALT="$ {\cal P}$">|; 

$key = q/displaystyle||A'ranglerangle=p_k,||Aranglerangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1189.png"
 ALT="$\displaystyle \vert\vert A'\rangle\rangle = p_k  \vert\vert A\rangle\rangle.$">|; 

$key = q/|2,1,1rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1577.png"
 ALT="$ \vert 2,1,1\rangle$">|; 

$key = q/x'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img115.png"
 ALT="$ x''$">|; 

$key = q/displaystylecdot{bfa}equivsum_ka_k,sigma_k=left(!{array}{cc}+a_3&a_1-{rmi},a_2[0.5ex]a_1+{rmi},a_2&-a_3{array}!right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1203.png"
 ALT="$\displaystyle \cdot {\bf a} \equiv \sum_k a_k  \sigma_k = \left(\!\begin{array...
...3 &amp; a_1 -{\rm i} a_2\ [0.5ex] a_1 + {\rm i} a_2 &amp; -a_3 \end{array}\!\right).$">|; 

$key = q/displaystylelangleA|Arangle=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img448.png"
 ALT="$\displaystyle \langle A\vert A\rangle = 1$">|; 

$key = q/displaystyle|langle{mitDelta}xi,{mitDelta}etarangle|^{,2}=frac{1}{4},|langleleft{4},|langleleft{{mitDelta}xi,{mitDelta}etaright}rangle|^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img188.png"
 ALT="$\displaystyle \vert\langle{\mit\Delta} \xi  {\mit\Delta} \eta\rangle\vert^{ 2...
...rt\langle\left\{ {\mit\Delta} \xi, {\mit\Delta} \eta\right\}\rangle\vert^{ 2},$">|; 

$key = q/displaystylelambda_{12}=lambda_{23}=lambda_{31}=-1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2444.png"
 ALT="$\displaystyle \lambda_{12} = \lambda_{23} = \lambda_{31}= -1.$">|; 

$key = q/displaystyleA^dag,|E_nrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img688.png"
 ALT="$\displaystyle A^\dag \vert E_n\rangle$">|; 

$key = q/displaystyleint_0^piint_0^{2pi}dtheta,dvarphi,sintheta,Y_{l,m}^{,ast}(theta,varphi),Y_{l',m'}(theta,varphi)=delta_{l,l'},delta_{m,m'}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img912.png"
 ALT="$\displaystyle \int_0^\pi \int_0^{2\pi}d\theta d\varphi   \sin\theta Y_{l m}...
...eta,\varphi)  Y_{l' m'}(\theta, \varphi) = \delta_{l  l'}  \delta_{m  m'}.$">|; 

$key = q/displaystyle={rmi},hbar,L_y.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img749.png"
 ALT="$\displaystyle = {\rm i} \hbar  L_y.$">|; 

$key = q/y';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1047.png"
 ALT="$ y'$">|; 

$key = q/displaystyle{bfp};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2023.png"
 ALT="$\displaystyle {\bf p}$">|; 

$key = q/displaystylexi,|xi'rangle=xi',|xi'rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img117.png"
 ALT="$\displaystyle \xi  \vert\xi'\rangle = \xi'  \vert\xi' \rangle,$">|; 

$key = q/P_1(t)=|c_1|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1817.png"
 ALT="$ P_1(t) = \vert c_1\vert^{ 2}$">|; 

$key = q/displaystylex=(E_n-E_i),tslash2,hbar,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1917.png"
 ALT="$\displaystyle x=(E_n-E_i) t/2 \hbar,$">|; 

$key = q/displaystyle[x_i(0),x_j(0)]=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img603.png"
 ALT="$\displaystyle [x_i(0), x_j(0)] = 0,$">|; 

$key = q/displaystyle||A'ranglerangle=S_k,||Aranglerangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1181.png"
 ALT="$\displaystyle \vert\vert A'\rangle\rangle = S_k  \vert\vert A\rangle\rangle.$">|; 

$key = q/langle2|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1445.png"
 ALT="$ \langle 2\vert$">|; 

$key = q/displaystylephi({bfx})simeqfrac{1}{(2pi)^{3slash2}}sum_{l=0,infty}{rmi}^l,(2l+1)2)]-exp[-{rmi},(k,r-l,pislash2)]}{2,{rmi},k,r},P_l(costheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="616" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2249.png"
 ALT="$\displaystyle \phi({\bf x}) \simeq \frac{1}{(2\pi)^{3/2}} \sum_{l=0,\infty} {\r...
...i/2)] -\exp[-{\rm i} (k r - l \pi/2)]}{2 {\rm i} k r}   P_l(\cos\theta),$">|; 

$key = q/displaystyletandelta_l=frac{-(k,a)^{2,l+1}}{(2,l+1),[(2,l-1)!!]^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2298.png"
 ALT="$\displaystyle \tan\delta_l = \frac{-(k a)^{2 l+1}}{(2 l+1)  [(2 l-1)!!]^{ 2}}.$">|; 

$key = q/s_z,hbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1254.png"
 ALT="$ s_z \hbar$">|; 

$key = q/displaystyle=-a^1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2547.png"
 ALT="$\displaystyle =-a^1,$">|; 

$key = q/displaystylelanglel,m|,L^-,|l,m+1rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img819.png"
 ALT="$\displaystyle \langle l, m\vert  L^-  \vert l, m+1\rangle$">|; 

$key = q/displaystyleleft[frac{1}{x^2}frac{d}{dx},x^2,frac{d}{dx}-frac{l,(l+1)}{x^2}+frac{2,n}{x}-1right]{calR}_{n,l}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img981.png"
 ALT="$\displaystyle \left[\frac{1}{x^2} \frac{d}{dx}  x^2  \frac{d}{dx}-\frac{l (l+1)}{x^2} + \frac{2 n}{x} - 1\right] {\cal R}_{n l} = 0$">|; 

$key = q/displaystylecdot{bfL}=J^{,2}-2,hbar,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2792.png"
 ALT="$\displaystyle \cdot{\bf L} = J^{ 2} - 2 \hbar $">|; 

$key = q/displaystyle{rmi},hbarfrac{dv_t}{dt}=T^{dag},v,H,T-T^{dag},H,T,v_t=v_t,H_t-H_t,v_t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img571.png"
 ALT="$\displaystyle {\rm i} \hbar \frac{d v_t}{dt} = T^{\dag }  v  H T - T^{\dag }  H T  v_t = v_t  H_t - H_t  v_t,$">|; 

$key = q/|+rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1052.png"
 ALT="$ \vert+\rangle$">|; 

$key = q/displaystyletandelta_l=frac{k,a,j_l'(k,a)-beta_{l+},j_l(k,a)}{k,a,eta_l'(k,a)-beta_{l+},eta_l(k,a)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2273.png"
 ALT="$\displaystyle \tan \delta_l = \frac{ k a j_l'(k a) - \beta_{l+}  j_l(k a)} {k a \eta_l'(k a) - \beta_{l+}  \eta_l(k a)}.$">|; 

$key = q/displaystyleleft(frac{1}{a_2}+frac{alpha}{y}right)psi_b-left(frac{d}{dy}-frac{k-1}{y}right)psi_a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2864.png"
 ALT="$\displaystyle \left(\frac{1}{a_2} +\frac{\alpha}{y}\right)\psi_b - \left(\frac{d}{dy} - \frac{k-1}{y}\right)\psi_a$">|; 

$key = q/displaystyle=a_nu^{~mu},x^{nu'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2661.png"
 ALT="$\displaystyle = a_\nu^{&nbsp;\mu} x^{\nu'},$">|; 

$key = q/displaystyleSigma_i=left({array}{cc}sigma_i&0[0.5ex]0&sigma_i{array}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2732.png"
 ALT="$\displaystyle \Sigma_i = \left(\begin{array}{cc} \sigma_i&amp; 0\ [0.5ex]0&amp; \sigma_i\end{array}\right).$">|; 

$key = q/-hbarslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1245.png"
 ALT="$ -\hbar/2$">|; 

$key = q/u,v-v,u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img291.png"
 ALT="$ u v-v u$">|; 

$key = q/displaystyle=xi',|xi'eta'rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img157.png"
 ALT="$\displaystyle = \xi'  \vert\xi' \eta'\rangle,$">|; 

$key = q/v_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img281.png"
 ALT="$ v_2$">|; 

$key = q/{mitPsi}(p_x');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img428.png"
 ALT="$ {\mit\Psi}(p_x')$">|; 

$key = q/displaystylecdot{bfb})={bfa}cdot{bfb}+{rmi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1206.png"
 ALT="$\displaystyle \cdot {\bf b}) = {\bf a} \cdot {\bf b} +{\rm i} $">|; 

$key = q/displaystyleS_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1038.png"
 ALT="$\displaystyle S_y$">|; 

$key = q/displaystyle=left({array}{rr}0&sigma_i[0.5ex]sigma_i&0{array}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2608.png"
 ALT="$\displaystyle = \left(\begin{array}{rr} 0&amp; \sigma_i\ [0.5ex]\sigma_i&amp;0\end{array}\right).$">|; 

$key = q/displaystylew_{2rightarrow1}^{rmspn}=frac{N_1}{N_2},w^{rmabs}_{1rightarrow2}-w_{2rightarrow1}^{rmstm}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2109.png"
 ALT="$\displaystyle w_{2\rightarrow 1}^{\rm spn} = \frac{N_1}{N_2} w^{\rm abs}_{1\rightarrow 2}- w_{2\rightarrow 1}^{\rm stm}.$">|; 

$key = q/displaystyletimes({bfL}times{bfL}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2790.png"
 ALT="$\displaystyle \times ({\bf L}\times {\bf L}).$">|; 

$key = q/D(x,x_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img555.png"
 ALT="$ D(x, x_0)$">|; 

$key = q/displaystyle=left(frac{hbar}{2,m,omega}right)left(sqrt{n,(n-1)},delta_{n',n-2}+sqrt{(n+1),(n+2)},delta_{n',n+2}+(2,n+1),delta_{n',n}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="547" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img702.png"
 ALT="$\displaystyle = \left(\frac{\hbar}{2 m \omega}\right)\left(\sqrt{n (n-1)} \...
...' n-2}+\sqrt{(n+1) (n+2)} \delta_{n' n+2}+ (2 n+1) \delta_{n' n}\right),$">|; 

$key = q/hbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img210.png"
 ALT="$ \hbar$">|; 

$key = q/displaystyleleft.+|k'''rangle,|k''rangle,|k'rangle+|k'rangle,|k'''rangle,|k''rangle+|k''rangle,|k'rangle,|k'''rangleright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2447.png"
 ALT="$\displaystyle \left.+ \vert k'''\rangle \vert k''\rangle \vert k'\rangle + \v...
...,\vert k''\rangle+ \vert k''\rangle \vert k'\rangle \vert k'''\rangle\right),$">|; 

$key = q/|V_0|llE=hbar^2,k^2slash2,m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2356.png"
 ALT="$ \vert V_0\vert\ll E=\hbar^2 k^2/2 m$">|; 

$key = q/displaystyleleftlanglefrac{a_0^{,2}}{r^2}rightrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1767.png"
 ALT="$\displaystyle \left\langle \frac{a_0^{ 2}}{r^2}\right\rangle$">|; 

$key = q/displaystyle{Sigma_i,Sigma_j}=2,delta_{ij}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2937.png"
 ALT="$\displaystyle \{\Sigma_i, \Sigma_j\} = 2 \delta_{ij}.
$">|; 

$key = q/displaystyleP_{23}^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2413.png"
 ALT="$\displaystyle P_{23}^{ 2}$">|; 

$key = q/displaystylev_d=(1+deltax,d_x),v,(1-deltax,d_x)=v+deltax,(d_x,v-v,d_x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="384" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img475.png"
 ALT="$\displaystyle v_d = (1+ \delta x d_x)  v  (1- \delta x d_x) = v + \delta x ( d_x  v - v  d_x),$">|; 

$key = q/j=|l+1slash2|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2787.png"
 ALT="$ j=\vert l+1/2\vert$">|; 

$key = q/displaystyle=2,hbar,L_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img779.png"
 ALT="$\displaystyle = 2 \hbar L_z,$">|; 

$key = q/displaystylelangle+|A'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1157.png"
 ALT="$\displaystyle \langle + \vert A'\rangle$">|; 

$key = q/displaystyle=ointd{mitOmega},|f(theta)|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2255.png"
 ALT="$\displaystyle = \oint d{\mit\Omega} \vert f(\theta)\vert^{ 2}$">|; 

$key = q/displaystylepsi_B(x')=langlex'|f(x)int_{-infty}^{+infty}dx'',psi_A(x''),|x''rangt_{-infty}^{+infty}dx'',f(x''),psi_A(x''),langlex'|x''rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="556" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img317.png"
 ALT="$\displaystyle \psi_B(x') = \langle x'\vert f(x) \int_{-\infty}^{+\infty} dx'' ...
...nt_{-\infty}^{+\infty} dx''  f(x'') \psi_A(x'')  \langle x'\vert x''\rangle,$">|; 

$key = q/displaystyleD(a),x,D(a)^{,dag}=x-a.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img513.png"
 ALT="$\displaystyle D(a) x D(a)^{ \dag } = x - a.
$">|; 

$key = q/displaystyle{mitDelta}E_{000}=frac{mu_0,g_p,e^2}{3pi,m_p,m_e,a_0^{,3}},langle{bfS}_pcdot{bfS}_erangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1745.png"
 ALT="$\displaystyle {\mit\Delta} E_{000} = \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.$">|; 

$key = q/displaystyle=left(!{array}{rr}1&00&-1{array}!right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1283.png"
 ALT="$\displaystyle = \left(\!\begin{array}{rr} 1 &amp;0\ 0&amp;-1\end{array}\!\right),$">|; 

$key = q/displaystyleH=c,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2704.png"
 ALT="$\displaystyle H = c $">|; 

$key = q/displaystyle=langlen|,V^dagger,|irangle=langlei|,V,|nrangle^ast.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2006.png"
 ALT="$\displaystyle = \langle n \vert V^\dagger  \vert i\rangle = \langle i\vert V \vert n\rangle^\ast.$">|; 

$key = q/N>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img230.png"
 ALT="$ N&gt;1$">|; 

$key = q/displaystyle[L_z^{,2},L_z];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img766.png"
 ALT="$\displaystyle [L_z^{ 2}, L_z]$">|; 

$key = q/lambda_{n,l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1565.png"
 ALT="$ \lambda_{n l}$">|; 

$key = q/tau=hbarslash{mitGamma};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2352.png"
 ALT="$ \tau = \hbar/
{\mit\Gamma}$">|; 

$key = q/frightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2887.png"
 ALT="$ f\rightarrow 0$">|; 

$key = q/displaystyleexp(,{rmi},k,rcostheta)=sum_{l=0,infty}a_l,j_l(k,r),P_l(costheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2234.png"
 ALT="$\displaystyle \exp( {\rm i} k r \cos\theta) = \sum_{l=0,\infty} a_l  j_l(k r)  P_l(\cos\theta),$">|; 

$key = q/x_2=Z,r_2slasha_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2521.png"
 ALT="$ x_2=Z r_2/a_0$">|; 

$key = q/e,hbar,Bslash2,m_e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1716.png"
 ALT="$ e  \hbar  B/ 2 m_e$">|; 

$key = q/displaystylesum_nfrac{(2,y)^{,n}}{n!},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img966.png"
 ALT="$\displaystyle \sum_n \frac{(2 y)^{ n}}{n!},$">|; 

$key = q/m_l+2,m_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1710.png"
 ALT="$ m_l + 2  m_s$">|; 

$key = q/displaystylew_{irightarrown}=frac{pi}{hbar^2}frac{e^2}{epsilon_0,m_e^{,2},omega_in}),|langlen|,exp[-{rmi},(omega_{in}slashc),{bfn}cdot{bfx}],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="392" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2067.png"
 ALT="$\displaystyle w_{i\rightarrow n} = \frac{\pi}{\hbar^2} \frac{e^2}{\epsilon_0 m...
...  \vert\langle n\vert  \exp[-{\rm i} (\omega_{in}/c) {\bf n}\cdot{\bf x}] $">|; 

$key = q/displaystyle=sin^2(gamma,tslashhbar).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1825.png"
 ALT="$\displaystyle = \sin^2 (\gamma  t/\hbar ).$">|; 

$key = q/displaystyleA^{-1},gamma^nu,A=a^nu_{~mu},gamma^{,mu}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2677.png"
 ALT="$\displaystyle A^{-1} \gamma^\nu A = a^\nu_{&nbsp;\mu} \gamma^{ \mu}.$">|; 

$key = q/k'aneqpislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2341.png"
 ALT="$ k'a\neq \pi/2$">|; 

$key = q/displaystylefrac{d^2u_l}{dr^2}+left[k^2-frac{2m}{hbar^2},V-frac{l,(l+1)}{r^2}right]u_l=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2278.png"
 ALT="$\displaystyle \frac{d^2 u_l}{d r^2} +\left[k^2 - \frac{2m}{\hbar^2}  V - \frac{l (l+1)} {r^2}\right] u_l = 0.$">|; 

$key = q/displaystyleexp(,{rmi},S_z,{mitDelta}varphislashhbar),S_x,exp(-{rmi},S_z,{mitDelta}varphislashhbar).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1081.png"
 ALT="$\displaystyle \exp( {\rm i} S_z  {\mit\Delta}\varphi/\hbar)  S_x   \exp(-{\rm i} S_z  {\mit\Delta}\varphi/\hbar).$">|; 

$key = q/displaystyleH,|k',k''',k''rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2423.png"
 ALT="$\displaystyle H \vert k' k''' k''\rangle$">|; 

$key = q/L_{n,l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1545.png"
 ALT="$ L_{n l}$">|; 

$key = q/displaystyle=m_2,hbar,|j_1,j_2;m_1,m_2rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1349.png"
 ALT="$\displaystyle = m_2 \hbar \vert j_1,j_2; m_1,m_2\rangle.$">|; 

$key = q/displaystylepsi=left({array}{c}psi_a(r)[0.5ex]psi_b(r){array}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2859.png"
 ALT="$\displaystyle \psi = \left(\begin{array}{c} \psi_a(r)\ [0.5ex]\psi_b(r)\end{array}\right),$">|; 

$key = q/displaystyleA,|E_0rangle=|0rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img692.png"
 ALT="$\displaystyle A \vert E_0\rangle = \vert\rangle,
$">|; 

$key = q/displaystylelanglej_1,j_2;j,m|j_1,j_2;j,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1382.png"
 ALT="$\displaystyle \langle j_1, j_2; j, m\vert j_1, j_2; j, m\rangle$">|; 

$key = q/|Erangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img674.png"
 ALT="$ \vert E\rangle$">|; 

$key = q/displaystyleL^-,|l,mrangle=c^-_{l,m},hbar,|l,m-1rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img799.png"
 ALT="$\displaystyle L^-  \vert l ,m\rangle = c^-_{l m} \hbar  \vert l, m-1\rangle.$">|; 

$key = q/displaystyle[zeta,H]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2822.png"
 ALT="$\displaystyle [\zeta, H] =0.$">|; 

$key = q/displaystyleH=H_0+H_1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1433.png"
 ALT="$\displaystyle H = H_0 + H_1.$">|; 

$key = q/q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img236.png"
 ALT="$ q$">|; 

$key = q/displaystylerightarrow|-rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1845.png"
 ALT="$\displaystyle \rightarrow \vert-\rangle,$">|; 

$key = q/displaystyleintdxi',|xi'ranglelanglexi'|=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img197.png"
 ALT="$\displaystyle \int d\xi'  \vert\xi'\rangle\langle \xi'\vert = 1.$">|; 

$key = q/{bfp}={bfv}slashm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1109.png"
 ALT="$ {\bf p} = {\bf v} /m$">|; 

$key = q/displaystyle=chi_-,chi_-,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2471.png"
 ALT="$\displaystyle = \chi_- \chi_-,$">|; 

$key = q/displaystyle=2,{rmi},hbar,(y,L_z-L_y,z).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1512.png"
 ALT="$\displaystyle = 2 {\rm i}  \hbar  ( y  L_z - L_y  z).$">|; 

$key = q/sigma_{mu,nu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2684.png"
 ALT="$ \sigma_{\mu \nu}$">|; 

$key = q/p_mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2672.png"
 ALT="$ p_\mu$">|; 

$key = q/q_1';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img380.png"
 ALT="$ q_1'$">|; 

$key = q/cos^2alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$ \cos^2\alpha$">|; 

$key = q/C;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$ C$">|; 

$key = q/displaystylepsi(x)=left(frac{m,omega}{pi,hbar}right)^{1slash4},expleft(-frac{m,omega^2,x^2}{2,hbar}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1760.png"
 ALT="$\displaystyle \psi(x) = \left(\frac{m \omega}{\pi \hbar}\right)^{1/4} \exp\left(-\frac{m \omega^2 x^2}{2 \hbar}\right).
$">|; 

$key = q/c_1+c_2=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$ c_1+c_2 = 0$">|; 

$key = q/displaystyleleft[frac{hbar^2}{2,M}left(-frac{1}{r^2}frac{d}{dr},r^2,frac{d}{dr}+frac{l,(l+1)}{r^2}right)+V(r)-Eright]R=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="388" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img934.png"
 ALT="$\displaystyle \left[\frac{\hbar^2}{2 M} \left(-\frac{1}{r^2} \frac{d}{dr} r^2 \frac{d}{dr} +\frac{l (l+1)}{r^2}\right) + V(r) - E\right] R = 0.$">|; 

$key = q/Y_{l,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img983.png"
 ALT="$ Y_{l m}$">|; 

$key = q/displaystyle|k',k'',k'''rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2401.png"
 ALT="$\displaystyle \vert k' k'' k'''\rangle,$">|; 

$key = q/k,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2290.png"
 ALT="$ k a$">|; 

$key = q/displaystyle[L_z,L_x];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img748.png"
 ALT="$\displaystyle [L_z, L_x]$">|; 

$key = q/displaystyleR(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img860.png"
 ALT="$\displaystyle R(0)$">|; 

$key = q/a_{mu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2552.png"
 ALT="$ a_{\mu}$">|; 

$key = q/displaystyle=sqrt{frac{hbar}{2,m,omega}}left(sqrt{n},delta_{n',n-1}+sqrt{n+1},delta_{n',n+1}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img698.png"
 ALT="$\displaystyle = \sqrt{\frac{\hbar}{2 m \omega}}\left(\sqrt{n} \delta_{n' n-1}+\sqrt{n+1} \delta_{n' n+1}\right),$">|; 

$key = q/displaystylegamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1849.png"
 ALT="$\displaystyle \gamma$">|; 

$key = q/displaystyle{mitDelta}E_{nlm}=e,|{bfE}|,langlen,l,m|,z,|n,l,mrangle+e^2,|{bfE}|^c{|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="546" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1498.png"
 ALT="$\displaystyle {\mit\Delta} E_{nlm}=e \vert{\bf E}\vert  \langle n,l,m \vert ...
...\langle n,l,m\vert z \vert n,'l',m'\rangle\vert^{ 2}}{ E_{nlm}-E_{n'l'm'} }.$">|; 

$key = q/displaystylec_2(t)=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1812.png"
 ALT="$\displaystyle c_2(t) =$">|; 

$key = q/displaystyleJ_{z},|j_1,j_2;j,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1358.png"
 ALT="$\displaystyle J_{z}  \vert j_1,j_2; j,m\rangle$">|; 

$key = q/displaystylepsi({bfx})=phi({bfx})-frac{m}{2pi,hbar^2}frac{exp(,{rmi},k,r)}{r}intd^3x',exp(-{rmi},{bfk}'cdot{bfx}'),V({bfx}'),psi({bfx}'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="494" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2189.png"
 ALT="$\displaystyle \psi({\bf x}) = \phi({\bf x})-\frac{m}{2\pi \hbar^2} \frac{\exp(...
...x'  \exp(- {\rm i}  {\bf k}' \cdot {\bf x}')  V({\bf x}')  \psi ({\bf x}'),$">|; 

$key = q/t=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img606.png"
 ALT="$ t=0$">|; 

$key = q/langlej_1,j_2;j,m|j_1,j_2;m_1,m_2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1395.png"
 ALT="$ \langle j_1, j_2; j, m\vert j_1, j_2; m_1, m_2\rangle$">|; 

$key = q/displaystyle{mitDelta}E_i={rmRe}({mitDelta}_i)=H_{ii}+{calP}sum_{mneqi}frac{|H_{mi}|^{,2}}{E_i-E_m},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1982.png"
 ALT="$\displaystyle {\mit\Delta} E_i = {\rm Re}({\mit\Delta}_i) = H_{ii} + {\cal P} \sum_{m\neq i} \frac{\vert H_{mi}\vert^{ 2}}{E_i - E_m} ,$">|; 

$key = q/j_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1311.png"
 ALT="$ j_1$">|; 

$key = q/displaystylek,a+delta_0simeqfrac{k}{k'},tan(k',a).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2328.png"
 ALT="$\displaystyle k a + \delta_0 \simeq \frac{k}{k'} \tan (k' a).$">|; 

$key = q/displaystylesum_{m_1}sum_{m_2}|langlej_1,j_2;m_1,m_2|j_1,j_2;j,mrangle|^{,2}=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1381.png"
 ALT="$\displaystyle \sum_{m_1}\sum_{m_2} \vert\langle j_1,j_2;m_1,m_2\vert j_1,j_2;j,m\rangle\vert^{ 2} =1.$">|; 

$key = q/a^nu_{~mu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2678.png"
 ALT="$ a^\nu_{&nbsp;\mu}$">|; 

$key = q/displaystylerho({bfx}',t)=|psi({bfx}',t)|^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img661.png"
 ALT="$\displaystyle \rho({\bf x}',t)= \vert\psi({\bf x}',t)\vert^{ 2},
$">|; 

$key = q/displaystyle=[u_1,v],u_2+u_1,[u_2,v],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img500.png"
 ALT="$\displaystyle = [u_1, v]  u_2 + u_1  [u_2, v],$">|; 

$key = q/displaystylepsi(r,theta)=sum_{l=0,infty}R_l(r),P_l(costheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2222.png"
 ALT="$\displaystyle \psi(r,\theta) = \sum_{l=0,\infty} R_l(r)  P_l(\cos\theta),$">|; 

$key = q/displaystyle(nabla^2+k^2),G({bfx},{bfx}')=delta({bfx}-{bfx}').;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2149.png"
 ALT="$\displaystyle (\nabla^2 + k^2) G({\bf x}, {\bf x}') = \delta({\bf x} -{\bf x}').$">|; 

$key = q/f({bfx},{bfp});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1001.png"
 ALT="$ f({\bf x},{\bf p})$">|; 

$key = q/displaystylelangleA|,1,|Arangle=langleA|Arangle=1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img145.png"
 ALT="$\displaystyle \langle A\vert 1 \vert A\rangle = \langle A\vert A\rangle =1,$">|; 

$key = q/displaystylesimeq1-frac{rmi}{hbar}int_{t_0}^tH_I(t_0,t')left[1-frac{rmi}{hbar}int_{t_0}^{t'}dt',H_I(t_0,t''),U_I(t_0,t'')right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="423" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1870.png"
 ALT="$\displaystyle \simeq 1 - \frac{\rm i}{\hbar} \int_{t_0}^t H_I(t_0, t') \left[ 1 - \frac{\rm i}{\hbar} \int_{t_0}^{t'} dt' H_I(t_0, t'')  U_I(t_0, t'')\right]$">|; 

$key = q/displaystyle=H_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1891.png"
 ALT="$\displaystyle = H_1$">|; 

$key = q/displaystylez;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img892.png"
 ALT="$\displaystyle z$">|; 

$key = q/{rmIm}(V)<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img658.png"
 ALT="$ {\rm Im}(V)&lt;0$">|; 

$key = q/i,j=1,3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2594.png"
 ALT="$ i,j=1,3$">|; 

$key = q/displaystyleequiv{mitDelta}xi,{mitDelta}eta+{mitDelta}eta,{mitDelta}xi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img185.png"
 ALT="$\displaystyle \equiv {\mit\Delta} \xi  {\mit\Delta} \eta + {\mit\Delta}\eta   {\mit\Delta}\xi.$">|; 

$key = q/displaystylealpha<frac{16}{3},4pi,epsilon_0,a_0^{,3}simeq5.3,4pi,epsilon_0,a_0^{,3}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1539.png"
 ALT="$\displaystyle \alpha &lt; \frac{16}{3}  4\pi  \epsilon_0  a_0^{ 3} \simeq 5.3 4\pi  \epsilon_0  a_0^{ 3}.$">|; 

$key = q/displaystyleR({mitDelta}varphi)=exp(-{rmi},L_z,{mitDelta}varphislashhbar).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img859.png"
 ALT="$\displaystyle R({\mit\Delta}\varphi) = \exp(-{\rm i} L_z  {\mit\Delta}\varphi/\hbar).$">|; 

$key = q/displaystyleleft[p^0-(p^1,p^1+p^2,p^2+p^3,p^3+m_e^2,c^2)^{1slash2}right]psi=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2582.png"
 ALT="$\displaystyle \left[p^0 - (p^1 p^1+p^2 p^2+p^3 p^3+m_e^2 c^2)^{1/2}\right]\psi = 0,$">|; 

$key = q/displaystyle[u_1,u_2,v];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img499.png"
 ALT="$\displaystyle [u_1  u_2, v]$">|; 

$key = q/displaystylelanglen,l'^{(1)}|,H_1,|n,l^{(1)}rangle=lambda_{n,l},delta_{l,l'}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1558.png"
 ALT="$\displaystyle \langle n, l'^{(1)} \vert  H_1 \vert n, l^{(1)}\rangle = \lambda_{n l} \delta_{l l'}.$">|; 

$key = q/displaystylehbar,cleft(frac{d}{dr}-frac{k-1}{r}right)psi_a-(E+m_e,c^2+e,phi),psi_b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="322" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2861.png"
 ALT="$\displaystyle \hbar c\left(\frac{d}{d r}-\frac{k-1}{r}\right)\psi_a - (E+m_e c^2+e \phi) \psi_b$">|; 

$key = q/displaystylef({bfk}',{bfk})simeq-frac{m}{2pi,hbar^2}intd^3x',expleft[,{rmi},({bfk}-{bfk}')cdot{bfx}'right]V({bfx}').;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="398" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2192.png"
 ALT="$\displaystyle f({\bf k}', {\bf k}) \simeq - \frac{m}{2\pi  \hbar^2} \int d^3 x' \exp\left[  {\rm i}  ({\bf k} - {\bf k}')\cdot {\bf x}'\right] V({\bf x}').$">|; 

$key = q/displaystylelangleA|=langleA|+ranglelangle+|+langleA|-ranglelangle-|rightarrow(langleA|+rangle,langleA|-rangle).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1151.png"
 ALT="$\displaystyle \langle A\vert =\langle A\vert+\rangle \langle +\vert + \langle A...
...le \langle -\vert \rightarrow (\langle A\vert+\rangle, \langle A\vert-\rangle).$">|; 

$key = q/psi(x')=langlex'|psi(x)rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img369.png"
 ALT="$ \psi(x')= \langle x'\vert
\psi(x)\rangle$">|; 

$key = q/displaystylefrac{N_1}{N_2}=frac{exp(-E_1slashk_B,T)}{exp(-E_2slashk_B,T)}=exp(,hbar,omega_{21}slashk_B,T).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2115.png"
 ALT="$\displaystyle \frac{N_1}{N_2} = \frac{\exp(-E_1/k_B T)}{\exp(-E_2/k_B T)} = \exp( \hbar  \omega_{21}/k_B T).$">|; 

$key = q/|A,t_0,trangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1854.png"
 ALT="$ \vert A, t_0, t\rangle$">|; 

$key = q/displaystyle|i,trangle=exp(-{rmi},[E_i+{rmRe}({mitDelta}_i),],tslashhbar),exp[,{rmIm}({mitDelta}_i),tslashhbar],|irangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1979.png"
 ALT="$\displaystyle \vert i, t\rangle = \exp(-{\rm i} [E_i + {\rm Re}({\mit\Delta}_i) ] t/\hbar)  \exp[ {\rm Im}({\mit\Delta}_i) t/\hbar]  \vert i\rangle.$">|; 

$key = q/epsilon;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2044.png"
 ALT="$ \epsilon$">|; 

$key = q/{mitGamma};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2351.png"
 ALT="$ {\mit\Gamma}$">|; 

$key = q/psi({bfx});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2161.png"
 ALT="$ \psi({\bf x})$">|; 

$key = q/displaystylexi,eta=eta,xi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img162.png"
 ALT="$\displaystyle \xi \eta =\eta \xi.$">|; 

$key = q/f(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img955.png"
 ALT="$ f(y)$">|; 

$key = q/displaystyle|x',y',z'rangle|+rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1053.png"
 ALT="$\displaystyle \vert x', y', z'\rangle \vert+\rangle$">|; 

$key = q/{bfS}_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1731.png"
 ALT="$ {\bf S}_p$">|; 

$key = q/psi^ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2926.png"
 ALT="$ \psi^\ast$">|; 

$key = q/displaystyle=[u,v_1]_{cl}+[u,v_2]_{cl},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img262.png"
 ALT="$\displaystyle = [u, v_1] _{cl}+ [u, v_2]_{cl},$">|; 

$key = q/gamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img367.png"
 ALT="$ \gamma$">|; 

$key = q/{mitDelta}varphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img853.png"
 ALT="$ {\mit\Delta}\varphi$">|; 

$key = q/displaystyleS_z,|s,jrangle=j,hbar,|s,jrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1268.png"
 ALT="$\displaystyle S_z  \vert s, j\rangle = j \hbar  \vert s, j\rangle.$">|; 

$key = q/displaystyle|i,trangle=exp[-{rmi},(E_i+{mitDelta}E_i),tslashhbar]exp(-{mitGamma}_i,tslash2,hbar),|irangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1981.png"
 ALT="$\displaystyle \vert i, t\rangle = \exp[-{\rm i} (E_i + {\mit\Delta} E_i) t/\hbar] \exp( - {\mit\Gamma}_i t/2 \hbar) \vert i\rangle,$">|; 

$key = q/alpha_i^ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="$ \alpha_i^\ast$">|; 

$key = q/{bfx'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img644.png"
 ALT="$ {\bf x'}$">|; 

$key = q/displaystyle=exp(-{rmi},L_x,{mitDelta}varphi_xslashhbar),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img869.png"
 ALT="$\displaystyle = \exp(-{\rm i} L_x  {\mit\Delta}\varphi_x/\hbar),$">|; 

$key = q/displaystylecdot{bfp}+beta,m_e,c^2),psi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2617.png"
 ALT="$\displaystyle \cdot{\bf p} + \beta m_e c^2) \psi,$">|; 

$key = q/displaystyle=-frac{e,g}{2,m_e},{bfS},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2778.png"
 ALT="$\displaystyle = - \frac{e g}{2 m_e} {\bf S},$">|; 

$key = q/2times2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1196.png"
 ALT="$ 2\times 2$">|; 

$key = q/j=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1402.png"
 ALT="$ j=1$">|; 

$key = q/(2,s+1)times(2,s+1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1259.png"
 ALT="$ (2 s+1) \times (2 s+1)$">|; 

$key = q/x^{,mu'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2654.png"
 ALT="$ x^{ \mu'}$">|; 

$key = q/|k'rangle,|k''rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2376.png"
 ALT="$ \vert k'\rangle \vert k''\rangle$">|; 

$key = q/displaystylechi'=expleft(-{rmi},mbox{boldmathsigma}cdot{bfn},{mitDelta}varphislash2right)chi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1229.png"
 ALT="$\displaystyle \chi' = \exp\left(-{\rm i}  \mbox{\boldmath$\sigma$}\cdot{\bf n} {\mit\Delta}\varphi/2\right) \chi,$">|; 

$key = q/d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2176.png"
 ALT="$ d{\mit\Omega}$">|; 

$key = q/S_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1082.png"
 ALT="$ S_x$">|; 

$key = q/displaystylew_{irightarrown}=frac{2pi}{hbar}frac{e^2}{m_e^{,2}},|A_0|^{,2},|langlen|,exp[-{rmi},(omegaslashc),{bfn}cdot{bfx}],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2054.png"
 ALT="$\displaystyle w_{i\rightarrow n} = \frac{2\pi}{\hbar} \frac{e^2}{m_e^{ 2}}  \...
...{ 2}  \vert\langle n\vert  \exp[-{\rm i} (\omega/c) {\bf n}\cdot{\bf x}] $">|; 

$key = q/displaystyle|1rangle+frac{e_{12}^{~ast}}{E_1-E_2},|2rangle+cdots,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1462.png"
 ALT="$\displaystyle \vert 1\rangle + \frac{e_{12}^{&nbsp;\ast}}{E_1-E_2}  \vert 2\rangle + \cdots,$">|; 

$key = q/|V_0|ggE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2325.png"
 ALT="$ \vert V_0\vert\gg E$">|; 

$key = q/displaystyle=frac{1}{n^2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1766.png"
 ALT="$\displaystyle = \frac{1}{n^2},$">|; 

$key = q/10^8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2095.png"
 ALT="$ 10^8$">|; 

$key = q/displaystyle={rmi},hbar,[u_1,v_1]_{qm},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img283.png"
 ALT="$\displaystyle = {\rm i} \hbar  [u_1, v_1]_{qm},$">|; 

$key = q/gamma^5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2814.png"
 ALT="$ \gamma^5$">|; 

$key = q/displaystyle|xi_2'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img150.png"
 ALT="$\displaystyle \vert\xi_2'\rangle$">|; 

$key = q/displaystyleint_{-infty}^{+infty}dp_x',|p_x'ranglelanglep_x'|=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img403.png"
 ALT="$\displaystyle \int_{-\infty}^{+\infty} dp_x'  \vert p_x'\rangle \langle p_x'\vert = 1.$">|; 

$key = q/displaystyle=cos(omega,t),p_x(0)-m,omega,sin(omega,t),x(0),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img719.png"
 ALT="$\displaystyle = \cos(\omega t)  p_x(0)- m \omega \sin(\omega t) x(0),$">|; 

$key = q/displaystyleS_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1036.png"
 ALT="$\displaystyle S_x$">|; 

$key = q/displaystylec_1(t)=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1814.png"
 ALT="$\displaystyle c_1(t)=$">|; 

$key = q/n_{rmmin}=l+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img960.png"
 ALT="$ n_{\rm min} = l+1$">|; 

$key = q/displaystyleP(q_1'cdotsq_N';dq_1'cdotsdq_N')=|psi(q_1'cdotsq_N')|^{,2},dq_1'cdotsdq_N'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="405" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img384.png"
 ALT="$\displaystyle P(q_1'\cdots q_N'; dq_1'\cdots dq_N') = \vert\psi(q_1'\cdots q_N')\vert^{ 2} dq_1'\cdots dq_N'.$">|; 

$key = q/langleB|Arangle=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img72.png"
 ALT="$ \langle B\vert A \rangle=0$">|; 

$key = q/langlexi'|xi''rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img129.png"
 ALT="$ \langle \xi'\vert\xi''\rangle$">|; 

$key = q/J_1^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1307.png"
 ALT="$ J_1^{ 2}$">|; 

$key = q/displaystyle[u,v_1+v_2]_{cl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img261.png"
 ALT="$\displaystyle [u, v_1 + v_2]_{cl}$">|; 

$key = q/displaystyle={rmi},frac{2^7}{3^5},a_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2132.png"
 ALT="$\displaystyle = {\rm i} \frac{2^7}{3^5} a_0,$">|; 

$key = q/(L_x,L_y,L_z)equiv{bfL};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img734.png"
 ALT="$ (L_x, L_y, L_z)\equiv {\bf L}$">|; 

$key = q/J_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1098.png"
 ALT="$ J_k$">|; 

$key = q/displaystyle|A,trangle={rme}^{-{rmi},omega,tslash2},langle+|A,0rangle|+rangle+{rme}^{,{rmi},omega,tslash2},langle-|A,0rangle|-rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="354" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1129.png"
 ALT="$\displaystyle \vert A, t\rangle = {\rm e}^{-{\rm i} \omega  t/2}  \langle +\...
...+ {\rm e}^{ {\rm i} \omega  t/2}  \langle -\vert A, 0\rangle \vert-\rangle.$">|; 

$key = q/H_0+H_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1436.png"
 ALT="$ H_0+H_1$">|; 

$key = q/displaystylepsi({bfx})rightarrowphi({bfx})=frac{exp(,{rmi},{bfk}cdot{bfx})}{(2pi)^{3slash2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2191.png"
 ALT="$\displaystyle \psi({\bf x}) \rightarrow \phi({\bf x}) = \frac{\exp( {\rm i} {\bf k} \cdot {\bf x} ) }{(2\pi)^{3/2}}.$">|; 

$key = q/H(t_0)={rmi},hbar,tau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img551.png"
 ALT="$ H(t_0)= {\rm i} \hbar  \tau$">|; 

$key = q/displaystylelangle{bfx'}|,V({bfx})=V({bfx'}),langle{bfx'}|,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img625.png"
 ALT="$\displaystyle \langle {\bf x'}\vert V({\bf x}) = V({\bf x'}) \langle {\bf x'} \vert,$">|; 

$key = q/sigma_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1267.png"
 ALT="$ \sigma_z$">|; 

$key = q/displaystyleP_1(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1821.png"
 ALT="$\displaystyle P_1(t)$">|; 

$key = q/displaystyle=frac{|2,0,0rangle-|2,1,0rangle}{sqrt{2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1595.png"
 ALT="$\displaystyle = \frac{\vert 2,0,0\rangle - \vert 2,1,0\rangle}{\sqrt{2}},$">|; 

$key = q/t>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1911.png"
 ALT="$ t&gt;0$">|; 

$key = q/displaystylesigma_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1166.png"
 ALT="$\displaystyle \sigma_1$">|; 

$key = q/|x',y',z'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1051.png"
 ALT="$ \vert x', y', z'\rangle$">|; 

$key = q/|xi'_arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img146.png"
 ALT="$ \vert\xi'_a\rangle$">|; 

$key = q/u_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img280.png"
 ALT="$ u_2$">|; 

$key = q/P_{irightarrown}(t)=|c_n^{(1)}|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2008.png"
 ALT="$ P_{i\rightarrow n}(t)=\vert c_n^{(1)}\vert^{ 2}$">|; 

$key = q/displaystylecdot{bfL}+hbar,,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2810.png"
 ALT="$\displaystyle \cdot {\bf L}+ \hbar, $">|; 

$key = q/displaystyleV=-frac{e,A_0,mbox{boldmathepsilon}cdot{bfp}}{m_e},exp[-{rmi},(omegaslashc),{bfn}cdot{bfx},];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2051.png"
 ALT="$\displaystyle V = - \frac{e  A_0  \mbox{\boldmath$\epsilon$}\cdot{\bf p} }{m_e}  \exp[-{\rm i} (\omega/c)  {\bf n}\cdot{\bf x} ]$">|; 

$key = q/displaystyle[u,c]_{cl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img258.png"
 ALT="$\displaystyle [u, c]_{cl}$">|; 

$key = q/|nrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img696.png"
 ALT="$ \vert n\rangle$">|; 

$key = q/displaystyle=frac{q}{2},{bfx}times{bfv},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1107.png"
 ALT="$\displaystyle = \frac{q}{2}  {\bf x} \times {\bf v},$">|; 

$key = q/displaystylepsi_{100}({bfx})=frac{1}{sqrt{4pi}}left(frac{Z}{a_0}right)^{3slash2}2,expleft(frac{-Z,r}{a_0}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="289" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2503.png"
 ALT="$\displaystyle \psi_{100}({\bf x})= \frac{1}{\sqrt{4\pi}}\left(\frac{Z}{a_0}\right)^{3/2}2 \exp\left(\frac{-Z r}{a_0}\right).$">|; 

$key = q/displaystylelangleA|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img136.png"
 ALT="$\displaystyle \langle A\vert A\rangle$">|; 

$key = q/displaystylelangle({mitDelta}x)^2rangle,langle({mitDelta}p_x)^2ranglegeqfrac{hbar^2}{4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img442.png"
 ALT="$\displaystyle \langle ({\mit\Delta} x)^2\rangle  \langle ({\mit\Delta} p_x)^2\rangle \geq \frac{\hbar^2}{4}$">|; 

$key = q/displaystylesum_{m_1}sum_{m_2}|j_1,j_2;m_1,m_2ranglelanglej_1,j_2;m_1,m_2|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1360.png"
 ALT="$\displaystyle \sum_{m_1}\sum_{m_2 }\vert j_1,j_2; m_1, m_2\rangle \langle j_1,j_2; m_1, m_2\vert$">|; 

$key = q/displaystyle{mitPsi}(p_x')=int_{-infty}^{+infty}dx',langlep_x'|x'ranglelanglex'|int_{-infty}^{+infty}dx',psi(x'),exp(-{rmi},p_x'x'slashhbar),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="532" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img434.png"
 ALT="$\displaystyle {\mit\Psi}(p_x') = \int_{-\infty}^{+\infty} dx'  \langle p_x'\ve...
...{1/2}} \int_{-\infty}^{+\infty} dx' \psi(x')  \exp(-{\rm i}  p_x' x'/\hbar),$">|; 

$key = q/displaystylep^{,i}=-{rmi},hbar,frac{partial}{partialx^{,i}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2573.png"
 ALT="$\displaystyle p^{ i} = -{\rm i} \hbar \frac{\partial}{\partial x^{ i}},$">|; 

$key = q/langlen,l,m|,z,|n',l',m'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1503.png"
 ALT="$ \langle n,l, m\vert z \vert n',l',m'\rangle$">|; 

$key = q/displaystyle={mitDelta}xi,|~rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img173.png"
 ALT="$\displaystyle = {\mit\Delta} \xi  \vert&nbsp;\rangle,$">|; 

$key = q/displaystyle=sum_{xi'}|xi'ranglelanglexi'|Arangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img133.png"
 ALT="$\displaystyle = \sum_{\xi'} \vert\xi'\rangle \langle\xi'\vert A\rangle,$">|; 

$key = q/displaystyleH,|Erangle=E,|Erangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img676.png"
 ALT="$\displaystyle H \vert E\rangle = E \vert E\rangle.
$">|; 

$key = q/displaystylec_1,|Brangle+c_2,|Crangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="$\displaystyle c_1  \vert B\rangle + c_2  \vert C\rangle,$">|; 

$key = q/psi(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img330.png"
 ALT="$ \psi(x)$">|; 

$key = q/displaystylelanglex'|p_x'rangle=h^{-1slash2},exp(,{rmi},p_x',x'slashhbar).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img427.png"
 ALT="$\displaystyle \langle x'\vert p_x'\rangle = h^{-1/2}  \exp( {\rm i}  p_x'  x'/\hbar).$">|; 

$key = q/m=-1,0,1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img988.png"
 ALT="$ m=-1, 0, 1$">|; 

$key = q/r>R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2355.png"
 ALT="$ r&gt;R$">|; 

$key = q/displaystyleV_0-E=frac{hbar^2kappa^2}{2,m}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2321.png"
 ALT="$\displaystyle V_0 - E = \frac{\hbar^2 \kappa^2}{2 m}.$">|; 

$key = q/zeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2823.png"
 ALT="$ \zeta$">|; 

$key = q/(2s)_{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1774.png"
 ALT="$ (2s)_{1/2}$">|; 

$key = q/gamma^0,gamma^{,mu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2611.png"
 ALT="$ \gamma^0 \gamma^{ \mu}$">|; 

$key = q/displaystylew_{irightarrown}=4pi^2,alpha,frac{c}{hbar},u(omega_{in}),|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2079.png"
 ALT="$\displaystyle w_{i\rightarrow n} = 4\pi^2 \alpha \frac{c}{\hbar} u(\omega_{in}) \vert$">|; 

$key = q/langlen|mrangle=delta_{nm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1797.png"
 ALT="$ \langle n\vert m\rangle
=\delta_{nm}$">|; 

$key = q/{mitGamma}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1988.png"
 ALT="$ {\mit\Gamma}_i$">|; 

$key = q/displaystylep^{,i}={rmi},hbar,frac{partial}{partialx_i},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2577.png"
 ALT="$\displaystyle p^{ i} = {\rm i} \hbar \frac{\partial}{\partial x_i},$">|; 

$key = q/displaystylep_x=-{rmi},hbar,frac{d}{dx}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img368.png"
 ALT="$\displaystyle p_x = -{\rm i} \hbar \frac{d}{dx}.$">|; 

$key = q/displaystyle=-{rmi},hbar,frac{partial}{partialvarphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img993.png"
 ALT="$\displaystyle = -{\rm i} \hbar \frac{\partial}{\partial\varphi},$">|; 

$key = q/displaystyleexpleft(frac{,{rmi},S_z,{mitDelta}varphi}{hbar}right)S_xexpleft(frac{hbar}right)=S_x,cos{mitDelta}varphi-S_y,sin{mitDelta}varphi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="445" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1236.png"
 ALT="$\displaystyle \exp\left(\frac{ {\rm i} S_z  {\mit\Delta}\varphi}{\hbar}\righ...
...}{\hbar}\right) = S_x  \cos{\mit\Delta}\varphi -S_y  \sin{\mit\Delta}\varphi.$">|; 

$key = q/j_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1312.png"
 ALT="$ j_2$">|; 

$key = q/displaystyle[x_i,p_j];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img739.png"
 ALT="$\displaystyle [x_i, p_j]$">|; 

$key = q/displaystylelanglephi,frac{d}{dx},|x'rangle=-frac{dphi(x')}{dx'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img342.png"
 ALT="$\displaystyle \langle \phi  \frac{d}{dx}  \vert x'\rangle = - \frac{d \phi(x')}{dx'},$">|; 

$key = q/displaystyle={rmi},hbar,delta_{ij},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img740.png"
 ALT="$\displaystyle = {\rm i} \hbar  \delta_{ij},$">|; 

$key = q/displaystyle{mitDelta}E_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1605.png"
 ALT="$\displaystyle {\mit\Delta} E_3$">|; 

$key = q/displaystyle=1-P_2(t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1822.png"
 ALT="$\displaystyle = 1 - P_2(t).$">|; 

$key = q/displaystyleexp(,{rmi},k,rcostheta)=sum_{l=0,infty}{rmi}^l,(2,l+1),j_l(k,r),P_l(costheta).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="371" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2241.png"
 ALT="$\displaystyle \exp( {\rm i} k r \cos\theta) = \sum_{l=0,\infty} {\rm i}^l  (2 l+1)  j_l(k r)  P_l(\cos\theta).$">|; 

$key = q/displaystyleE=2,Z^{,2},E_0=-108.8,{rmeV},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2504.png"
 ALT="$\displaystyle E = 2 Z^{ 2} E_0=-108.8 {\rm eV},$">|; 

$key = q/1slashmu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2199.png"
 ALT="$ 1/\mu$">|; 

$key = q/l,(l+1),hbar^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img840.png"
 ALT="$ 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="167" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2204.png"
 ALT="$ V_0/\mu \rightarrow
Z Z'  e^2 / 4\pi \epsilon_0$">|; 

$key = q/|psi_-(x',y',z')|^{,2},dx'dy'dz';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1072.png"
 ALT="$ \vert\psi_- (x', y', z')\vert^{ 2} dx' dy' dz'$">|; 

$key = q/w^{rmabs}_{1rightarrow2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2106.png"
 ALT="$ w^{\rm abs}_{1\rightarrow 2}$">|; 

$key = q/displaystyle=sum_{s}c_s,y^s,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2875.png"
 ALT="$\displaystyle = \sum_{s} c_s y^s,$">|; 

$key = q/displaystylefrac{dA}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img714.png"
 ALT="$\displaystyle \frac{dA}{dt}$">|; 

$key = q/alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$ \alpha$">|; 

$key = q/{bfp};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img590.png"
 ALT="$ {\bf p}$">|; 

$key = q/displaystyle=cos(omega,t),x(0)+frac{sin(omega,t)}{m,omega},p_x(0),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img721.png"
 ALT="$\displaystyle = \cos(\omega t) x(0) + \frac{\sin(\omega t)}{m \omega} p_x(0),$">|; 

$key = q/l'=lpm1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1524.png"
 ALT="$ l' = l\pm 1$">|; 

$key = q/psi(x');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img331.png"
 ALT="$ \psi(x')$">|; 

$key = q/Y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img81.png"
 ALT="$ Y$">|; 

$key = q/|mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1791.png"
 ALT="$ \vert m\rangle$">|; 

$key = q/displaystyle[u,v]_{cl}=sum_{i=1,N}left(frac{partialu}{partialq_i}frac{partialv}{}-frac{partialu}{partialp_i}frac{partialv}{partialq_i}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img249.png"
 ALT="$\displaystyle [u, v]_{cl} = \sum_{i=1,N} \left(\frac{\partial u}{\partial q_i}\...
... p_i} - \frac{\partial u}{\partial p_i}\frac{\partial v}{\partial q_i} \right),$">|; 

$key = q/displaystyleintd^3x,j^{,0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2934.png"
 ALT="$\displaystyle \int d^3 x j^{ 0}
$">|; 

$key = q/displaystyleointd{mitOmega},({calY}_{l,m}^+)^dagger,{bfL}cdot{bfS},{calY}_{l,m}^+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1668.png"
 ALT="$\displaystyle \oint d{\mit\Omega} ({\cal Y}_{l m}^+)^\dagger  {\bf L} \cdot{\bf S}  {\cal Y}_{l  m}^+$">|; 

$key = q/displaystylegamma^5=left({array}{cc}0&1[0.5ex]1&0{array}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2731.png"
 ALT="$\displaystyle \gamma^5 = \left(\begin{array}{cc} 0&amp; 1\ [0.5ex]1 &amp; 0\end{array}\right),$">|; 

$key = q/{mitDelta}B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1141.png"
 ALT="$ {\mit\Delta} B$">|; 

$key = q/displaystyle=-frac{partialH}{partialq_i},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img245.png"
 ALT="$\displaystyle = - \frac{\partial H}{\partial q_i},$">|; 

$key = q/displaystyle=z,p_x-x,p_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img731.png"
 ALT="$\displaystyle = z  p_x - x  p_z,$">|; 

$key = q/displaystylembox{boldmath{{mu{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1106.png"
 ALT="$\displaystyle \mbox{\boldmath$\mu$}$">|; 

$key = q/e_{nl'nl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1553.png"
 ALT="$ e_{nl'nl}$">|; 

$key = q/etarightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1954.png"
 ALT="$ \eta\rightarrow 0$">|; 

$key = q/(3slash4),hbar^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1750.png"
 ALT="$ (3/4) \hbar^2$">|; 

$key = q/dot{y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2711.png"
 ALT="$ \dot{y}$">|; 

$key = q/q_1'+dq_1';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img381.png"
 ALT="$ q_1'+dq_1'$">|; 

$key = q/displaystyle|x',y',z',pmrangle=|x',y',z'rangle|pmrangle=|pmrangle|x',y',z'rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1060.png"
 ALT="$\displaystyle \vert x', y', z', \pm\rangle = \vert x', y', z'\rangle\vert \pm\rangle = \vert \pm\rangle \vert x', y', z'\rangle.$">|; 

$key = q/B_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2244.png"
 ALT="$ B_l$">|; 

$key = q/delta_0rightarrowpi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2335.png"
 ALT="$ \delta_0\rightarrow \pi$">|; 

$key = q/displaystyleG({bfx},{bfx}')=-frac{exp(pm{rmi},k,|{bfx}-{bfx}'|,)}{4pi,|{bfx}-{bfx}'|}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2150.png"
 ALT="$\displaystyle G({\bf x}, {\bf x}') = -\frac{\exp(\pm {\rm i} k  \vert{\bf x} - {\bf x}'\vert )}{4\pi \vert{\bf x} - {\bf x}'\vert}.$">|; 

$key = q/displaystyle{bfL}_itimes{bfL}_i={rmi},hbar,{bfL}_i.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img752.png"
 ALT="$\displaystyle {\bf L}_i\times {\bf L}_i = {\rm i} \hbar  {\bf L}_i.$">|; 

$key = q/2,j+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1378.png"
 ALT="$ 2 j+1$">|; 

$key = q/|m_2|leq1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1400.png"
 ALT="$ \vert m_2\vert\leq 1/2$">|; 

$key = q/displaystyle=frac{1}{k^2}ointdvarphiint_{-1}^{1}dmusum_lsum_{l'}(2,l+1),(2,l'+1)a_l-delta_{l'})],sindelta_l,sindelta_{l'},P_l(mu),P_{l'}(mu),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="636" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2256.png"
 ALT="$\displaystyle = \frac{1}{k^2} \oint d\varphi \int_{-1}^{1} d\mu \sum_l \sum_{l'...
...elta_l-\delta_{l'})]  \sin\delta_l  \sin\delta_{l'}  P_l(\mu)  P_{l'}(\mu),$">|; 

$key = q/displaystylecdot{bfx}),(;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2832.png"
 ALT="$\displaystyle \cdot{\bf x}) ($">|; 

$key = q/displaystylelanglep_x'|p_x''rangle=|c'|^{,2},h,delta(p_x'-p_x'').;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img425.png"
 ALT="$\displaystyle \langle p_x'\vert p_x''\rangle = \vert c'\vert^{ 2}   h  \delta(p_x'-p_x'').$">|; 

$key = q/displaystyleH=frac{left(p^2+e,{bfA}cdot{bfp}+e,{bfp}cdot{bfA}+e^2A^2right)}{2,m_e}-e,phi+V_0(r).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="381" HEIGHT="83" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2035.png"
 ALT="$\displaystyle H = \frac{ \left(p^2 +e  {\bf A}\cdot {\bf p} +e  {\bf p}\cdot{\bf A} + e^2 A^2\right)}{2 m_e}- e  \phi + V_0(r).$">|; 

$key = q/{bfj};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img648.png"
 ALT="$ {\bf j}$">|; 

$key = q/displaystyle[L^2,H]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img930.png"
 ALT="$\displaystyle [L^2, H] = 0.$">|; 

$key = q/hbar,k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2206.png"
 ALT="$ \hbar  k$">|; 

$key = q/beta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1295.png"
 ALT="$ \beta$">|; 

$key = q/displaystylew_{irightarrown}=frac{pi}{hbar}frac{e^2}{epsilon_0,m_e^{,2},omega^2},U,|langlen|,exp[,{rmi},(omegaslashc),{bfn}cdot{bfx}],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2060.png"
 ALT="$\displaystyle w_{i\rightarrow n} = \frac{\pi}{\hbar} \frac{e^2}{\epsilon_0 m_e...
...  U  \vert\langle n\vert  \exp[ {\rm i} (\omega/c) {\bf n}\cdot{\bf x}] $">|; 

$key = q/displaystylefrac{c_s}{a_1}+frac{c_s'}{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2908.png"
 ALT="$\displaystyle \frac{c_s}{a_1} + \frac{c_s'}{a}$">|; 

$key = q/displaystyle{bfB}=B_0,{bfe}_z+B_1left[cos(omega,t),{bfe}_x+sin(omega,t),{bfe}_yright],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1833.png"
 ALT="$\displaystyle {\bf B} = B_0  {\bf e}_z + B_1\left[\cos(\omega  t)  {\bf e}_x + \sin(\omega  t)  {\bf e}_y\right],$">|; 

$key = q/displaystyle=cosalpha,|m-1slash2,1slash2rangle+sinalpha,|m+1slash2,-1slash2rangle,~~;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="372" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1634.png"
 ALT="$\displaystyle = \cos\alpha  \vert m-1/2, 1/2\rangle + \sin\alpha  \vert m+1/2, -1/2\rangle,&nbsp;&nbsp;$">|; 

$key = q/displaystyleS^2=frac{3,hbar^2}{4}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1044.png"
 ALT="$\displaystyle S^2 = \frac{3 \hbar^2}{4}.$">|; 

$key = q/displaystylev,|Arangle=|Brangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img455.png"
 ALT="$\displaystyle v  \vert A\rangle = \vert B\rangle,$">|; 

$key = q/S_y=hbarslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1429.png"
 ALT="$ S_y=\hbar/2$">|; 

$key = q/displaystylelanglexi'|xi''rangle=delta(xi'-xi''),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img192.png"
 ALT="$\displaystyle \langle \xi'\vert\xi''\rangle = \delta(\xi'-\xi''),$">|; 

$key = q/l';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2495.png"
 ALT="$ l'$">|; 

$key = q/phi=phi(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2753.png"
 ALT="$ \phi=\phi(r)$">|; 

$key = q/displaystyleU_I(t_0,t_0)=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1862.png"
 ALT="$\displaystyle U_I(t_0, t_0) = 1.$">|; 

$key = q/nabla'equiv(partialslashpartialx',partialslashpartialy',partialslashpartialz');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img624.png"
 ALT="$ \nabla'\equiv(\partial/\partial x',
\partial/\partial y', \partial/\partial z')$">|; 

$key = q/Sigma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2740.png"
 ALT="$ \Sigma$">|; 

$key = q/|lpm1slash2,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1662.png"
 ALT="$ \vert l\pm 1/2, m\rangle$">|; 

$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="162" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1460.png"
 ALT="$\displaystyle E_2 - \frac{\vert e_{12}\vert^{ 2}}{E_1-E_2} + \cdots.$">|; 

$key = q/displaystyle=pmfrac{hbar}{2},|pmrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1028.png"
 ALT="$\displaystyle =\pm \frac{\hbar}{2}  \vert\pm \rangle,$">|; 

$key = q/displaystylelangle+|,H_1,|+rangle=langle-|,H_1,|-rangle=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1843.png"
 ALT="$\displaystyle \langle + \vert H_1  \vert+\rangle = \langle - \vert H_1 \vert-\rangle = 0,$">|; 

$key = q/displaystyle=frac{alpha}{1-{calE}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2866.png"
 ALT="$\displaystyle = \frac{\alpha}{1-{\cal E}},$">|; 

$key = q/displaystyle=exp(-{rmi},L_z,{mitDelta}varphi_zslashhbar),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img873.png"
 ALT="$\displaystyle = \exp(-{\rm i} L_z  {\mit\Delta}\varphi_z/\hbar),$">|; 

$key = q/displaystyle(sigma_2)_{j,l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1280.png"
 ALT="$\displaystyle (\sigma_2)_{j l}$">|; 

$key = q/displaystyleH=frac{p_x^{,2}}{2,m}+V(x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img706.png"
 ALT="$\displaystyle H = \frac{p_x^{ 2}}{2 m} + V(x).
$">|; 

$key = q/displaystyleH=H_0+H_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2135.png"
 ALT="$\displaystyle H= H_0 + H_1,$">|; 

$key = q/displaystyleomega_{nm}=frac{E_n-E_m}{hbar},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1883.png"
 ALT="$\displaystyle \omega_{nm} = \frac{E_n -E_m}{\hbar},$">|; 

$key = q/displaystylep_r=-{rmi},hbar,frac{partial}{partialr}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2785.png"
 ALT="$\displaystyle p_r = -{\rm i} \hbar \frac{\partial}{\partial r}.$">|; 

$key = q/P_{31};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2429.png"
 ALT="$ P_{31}$">|; 

$key = q/(u_1,v_1-v_1,u_1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img286.png"
 ALT="$ (u_1  v_1- v_1  u_1)$">|; 

$key = q/displaystylelangleAt_0|At_0rangle=langleAt|Atrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img540.png"
 ALT="$\displaystyle \langle A t_0\vert A t_0\rangle =\langle A t\vert A t\rangle$">|; 

$key = q/displaystyleH_0=frac{{p}^2}{2,m_e}+V(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1618.png"
 ALT="$\displaystyle H_0 = \frac{{p}^2}{2 m_e} + V(r)$">|; 

$key = q/displaystyleE<lim_{|{bfx}'|rightarrowinfty}V({bfx'}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img640.png"
 ALT="$\displaystyle E &lt; \lim_{\vert{\bf x}'\vert\rightarrow \infty} V({\bf x'}).$">|; 

$key = q/displaystyleW_{2rightarrow1}=N_2,(w_{2rightarrow1}^{rmspn}+w_{2rightarrow1}^{rmstm}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2101.png"
 ALT="$\displaystyle W_{2\rightarrow 1} = N_2 (w_{2\rightarrow 1}^{\rm spn} + w_{2\rightarrow 1}^{\rm stm}),$">|; 

$key = q/displaystyle=-frac{(l+1),hbar^2}{2},|l-1slash2,mrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1667.png"
 ALT="$\displaystyle = -\frac{(l+1)  \hbar^2}{2}  \vert l- 1/2, m\rangle.$">|; 

$key = q/displaystyle[p_i,p_j];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img294.png"
 ALT="$\displaystyle [p_i, p_j]$">|; 

$key = q/displaystylea_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2544.png"
 ALT="$\displaystyle a_0$">|; 

$key = q/displaystyleJ_2^{,2},|j_1,j_2;j,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1354.png"
 ALT="$\displaystyle J_2^{ 2}  \vert j_1,j_2; j,m\rangle$">|; 

$key = q/displaystylea_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2550.png"
 ALT="$\displaystyle a_3$">|; 

$key = q/displaystyle|k'',k'rangle_-=-|k',k''rangle_-.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2400.png"
 ALT="$\displaystyle \vert k'' k'\rangle_- =- \vert k' k''\rangle_-.$">|; 

$key = q/c_pm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1153.png"
 ALT="$ c_\pm$">|; 

$key = q/displaystylefrac{w_{irightarrow[n]}}{rho(E_n)}=frac{w_{nrightarrow[i]}}{rho(E_i)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2021.png"
 ALT="$\displaystyle \frac{w_{i\rightarrow [n]}}{\rho(E_n)} = \frac{w_{n\rightarrow [i]}}{\rho(E_i)}.$">|; 

$key = q/mu=1,3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2610.png"
 ALT="$ \mu=1,3$">|; 

$key = q/-l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img835.png"
 ALT="$ -l$">|; 

$key = q/langle{bfx}'|phirangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2139.png"
 ALT="$ \langle {\bf x}'\vert\phi\rangle$">|; 

$key = q/displaystylelangleA|Aranglegeq0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.png"
 ALT="$\displaystyle \langle A\vert A \rangle \geq 0.$">|; 

$key = q/displaystyle|A,t_0,trangle=sum_nc_n(t)exp[-{rmi},E_n,(t-t_0)slashhbar],|nrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1785.png"
 ALT="$\displaystyle \vert A, t_0, t\rangle = \sum_n c_n(t) \exp[-{\rm i} E_n (t-t_0)/\hbar] \vert n\rangle,$">|; 

$key = q/N_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1546.png"
 ALT="$ N_n$">|; 

$key = q/displaystyle{mitDelta}omega^{,mu,nu}+{mitDelta}omega^{nu,mu}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2682.png"
 ALT="$\displaystyle {\mit\Delta}\omega^{ \mu \nu} + {\mit\Delta}\omega^{\nu \mu} = 0.$">|; 

$key = q/displaystyleS_x,cos{mitDelta}varphi-S_y,sin{mitDelta}varphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1085.png"
 ALT="$\displaystyle S_x \cos{\mit\Delta}\varphi - S_y \sin{\mit\Delta}\varphi,$">|; 

$key = q/sigma_1,sigma_2-sigma_2,sigma_1=2,{rmi},sigma_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1178.png"
 ALT="$ \sigma_1  \sigma_2 - \sigma_2 \sigma_1 = 2 {\rm i}  \sigma_3$">|; 

$key = q/displaystylelanglel,m|L^-,L^+|l,mrangle=langlel,m|L^-,hbar,c^+_{l,m}|l,m+1rangle=hbar^2,c^+_{l,m},c^{-}_{l,,m+1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="428" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img817.png"
 ALT="$\displaystyle \langle l,m\vert L^-  L^+ \vert l,m\rangle = \langle l, m\vert L...
...bar  c^+_{l m} \vert l,m+1\rangle = \hbar^2  c^+_{l m}  c^{-}_{l   m+1},$">|; 

$key = q/{mitDelta}ksim1slash{mitDelta}x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img436.png"
 ALT="$ {\mit\Delta} k \sim 1/{\mit\Delta} x$">|; 

$key = q/displaystyleL_x=y,p_z-z,p_y={rmi},hbarleft(z,frac{partial}{partialy}-y,frac{partial}{partialz}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2756.png"
 ALT="$\displaystyle L_x = y p_z-z p_y = {\rm i} \hbar\left(z \frac{\partial}{\partial y} - y \frac{\partial}{\partial z}\right).$">|; 

$key = q/displaystyle{mitDelta}E_{nlm+};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1676.png"
 ALT="$\displaystyle {\mit\Delta} E_{nlm+}$">|; 

$key = q/p^0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2583.png"
 ALT="$ p^0$">|; 

$key = q/displaystyle[(l+1slash2),(l+3slash2)-m,(m+1)]^{1slash2},langlem-1slash2,1slash2|l+1slash2,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="460" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1643.png"
 ALT="$\displaystyle [(l+1/2) (l+3/2)-m (m+1)]^{1/2}  \langle m-1/2, 1/2\vert l+1/2, m\rangle$">|; 

$key = q/displaystyle[Sigma_1,H]=2,{rmi},c,gamma^5,(Sigma_3,p_y-Sigma_2,p_z),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2773.png"
 ALT="$\displaystyle [\Sigma_1,H] = 2 {\rm i} c \gamma^5 (\Sigma_3 p_y-\Sigma_2 p_z),$">|; 

$key = q/displaystyleL^2=L_x^{,2}+L_y^{,2}+L_z^{,2}=L_z^{,2}+(L^+,L^-+L^-,L^+)slash2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img898.png"
 ALT="$\displaystyle L^2 = L_x^{ 2}+L_y^{ 2}+L_z^{ 2} = L_z^{ 2} + (L^+  L^- + L^-  L^+) /2,$">|; 

$key = q/displaystyle|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img132.png"
 ALT="$\displaystyle \vert A\rangle$">|; 

$key = q/{bfb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2739.png"
 ALT="$ {\bf b}$">|; 

$key = q/displaystyleS_x+left(frac{{rmi},{mitDelta}varphi}{hbar}right)[S_z,S_x]+left(fracc{{rmi},{mitDelta}varphi}{hbar}right)^2[S_z,[S_z,S_x]]+cdots,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="422" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1090.png"
 ALT="$\displaystyle S_x + \left(\frac{{\rm i} {\mit\Delta}\varphi}{\hbar}\right) [S_...
...\frac{{\rm i} {\mit\Delta}\varphi}{\hbar}\right)^2 [S_z, [S_z, S_x]] + \cdots,$">|; 

$key = q/[u,v];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img290.png"
 ALT="$ [u, v]$">|; 

$key = q/displaystyleleftlangle|mbox{boldmathepsilon}cdot{bff}_{21}|^{,2}rightrangle=frac{f_{21}^{,2}}{3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2111.png"
 ALT="$\displaystyle \left\langle \vert\mbox{\boldmath$\epsilon$}\cdot{\bf f}_{21}\vert^{ 2}\right\rangle = \frac{f_{21}^{ 2}}{3},$">|; 

$key = q/A_0(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2322.png"
 ALT="$ A_0(r)$">|; 

$key = q/displaystylea_l,j_l(k,r)=(l+1slash2)int_{-1}^1dmu,exp(,{rmi},k,r,mu),P_l(mu).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2237.png"
 ALT="$\displaystyle a_l  j_l(k r) = (l+1/2)\int_{-1}^1 d\mu \exp( {\rm i} k r  \mu)  P_l(\mu).$">|; 

$key = q/displaystylesigma_{rmtotal}=frac{4pi}{k^2}sum_{l=0,l_{rmmax}}(2,l+1),sin^2delta_l.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2305.png"
 ALT="$\displaystyle \sigma_{\rm total} = \frac{4\pi}{k^2} \sum_{l=0,l_{\rm max}} (2 l+1) \sin^2\delta_l.$">|; 

$key = q/displaystylefrac{dsigma}{d{mitOmega}}=|f({bfk}',{bfk})|^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2183.png"
 ALT="$\displaystyle \frac{d\sigma}{d {\mit\Omega}} = \vert f({\bf k}', {\bf k})\vert^{ 2}.$">|; 

$key = q/2s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1571.png"
 ALT="$ 2s$">|; 

$key = q/O(epsilon^3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1488.png"
 ALT="$ O(\epsilon^3)$">|; 

$key = q/displaystyle=frac{5,Z}{4}=frac{5}{2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2519.png"
 ALT="$\displaystyle = \frac{5 Z}{4} = \frac{5}{2}.$">|; 

$key = q/displaystyleint_0^inftydr',exp(-mu,r'),sin(q,r')=frac{q}{q^2+mu^2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2201.png"
 ALT="$\displaystyle \int_0^\infty dr'  \exp(-\mu  r')  \sin(q r') = \frac{q}{q^2+\mu^2}.$">|; 

$key = q/g^{11}=g^{22}=g^{33}=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2564.png"
 ALT="$ g^{11}=g^{22}=g^{33}=-1$">|; 

$key = q/displaystylew_{irightarrowf}^{rmspn}=frac{4pi^2,alpha,c}{hbar}left(frac{N_1}{N_2tlangle|mbox{boldmathepsilon}cdot{bff}_{21}|^{,2}rightrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="330" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2110.png"
 ALT="$\displaystyle w_{i\rightarrow f}^{\rm spn} =\frac{4\pi^2 \alpha c}{\hbar} \le...
...ngle \vert\mbox{\boldmath$\epsilon$}\cdot {\bf f}_{21}\vert^{ 2}\right\rangle,$">|; 

$key = q/displaystyleu_1,v_1-v_1,u_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img282.png"
 ALT="$\displaystyle u_1  v_1 - v_1  u_1$">|; 

$key = q/displaystyle{rmi},hbarfrac{dv_t}{dt}=[v_t,H_t].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img573.png"
 ALT="$\displaystyle {\rm i} \hbar \frac{d v_t}{dt} = [v_t, H_t].$">|; 

$key = q/s=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1282.png"
 ALT="$ s=1/2$">|; 

$key = q/displaystylefrac{d}{dt}intd^{,3}x',rho({bfx'},t)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img652.png"
 ALT="$\displaystyle \frac{d}{d t} \int d^{ 3} x' \rho({\bf x'}, t) = 0.$">|; 

$key = q/displaystyle{bfL}_itimes{bfL}_j+{bfL}_jtimes{bfL}_i=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img753.png"
 ALT="$\displaystyle {\bf L}_i\times {\bf L}_j + {\bf L}_j\times {\bf L}_i =0,$">|; 

$key = q/displaystyle=a^{,mu}_{~nu},p^nu,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2666.png"
 ALT="$\displaystyle = a^{ \mu}_{&nbsp;\nu} p^\nu,$">|; 

$key = q/displaystyleg_{33};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2560.png"
 ALT="$\displaystyle g_{33}$">|; 

$key = q/displaystyleg_e=2left(1+frac{1}{2pi}frac{e^2}{4pi,epsilon_0,hbar,c}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1113.png"
 ALT="$\displaystyle g_e = 2\left( 1 + \frac{1}{2\pi} \frac{e^2}{4\pi  \epsilon_0 \hbar  c} \right).$">|; 

$key = q/c_n(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1787.png"
 ALT="$ c_n(t)$">|; 

$key = q/displaystyleH_{LS}=-{mbox{boldmathmu}}cdot{bfB}=-frac{e,{bfS}}{m_e,c^2}cdot{bfv}dr}right)=frac{1}{m_e^{,2},c^2,r}frac{dV}{dr},{bfL}cdot{bfS},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="448" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1615.png"
 ALT="$\displaystyle H_{LS} = - {\mbox{\boldmath$\mu$}}\cdot {\bf B} = - \frac{e  {\b...
...dr}\right) = \frac{1}{m_e^{ 2} c^2 r} \frac{d V}{dr}  {\bf L}\cdot {\bf S},$">|; 

$key = q/displaystyle|l-1slash2,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1635.png"
 ALT="$\displaystyle \vert l-1/2, m\rangle$">|; 

$key = q/displaystylepsi({bfx})simeqfrac{1}{(2pi)^{3slash2}}sum_lC_l,frac{sin(k,r-l,pislash2+delta_l)}{k,r},P_l(costheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2246.png"
 ALT="$\displaystyle \psi ({\bf x} ) \simeq \frac{1}{(2\pi)^{3/2}} \sum_l C_l  \frac{\sin(k r - l \pi/2+ \delta_l)}{k r}  P_l(\cos\theta),$">|; 

$key = q/displaystylealpha,c_{s_0}'-(s_0-k),c_{s_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2895.png"
 ALT="$\displaystyle \alpha c_{s_0}' - (s_0-k) c_{s_0}$">|; 

$key = q/alpha^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2916.png"
 ALT="$ \alpha^2$">|; 

$key = q/y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="$ y$">|; 

$key = q/displaystyleD({mitDelta}x)=lim_{Nrightarrowinfty}left(1-{rmi},frac{{mitDelta}x}{N}frac{p_x}{hbar}right)^N,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img487.png"
 ALT="$\displaystyle D({\mit\Delta} x) = \lim_{N\rightarrow \infty} \left(1-{\rm i}  \frac{{\mit\Delta} x}{N} \frac{p_x}{\hbar}\right)^N,$">|; 

$key = q/(2,j_1+1),(2,j_2+1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1377.png"
 ALT="$ (2 j_1+1) (2 j_2+1)$">|; 

$key = q/displaystyle{;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2809.png"
 ALT="$\displaystyle \{$">|; 

$key = q/deltavarphirightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img857.png"
 ALT="$ \delta\varphi\rightarrow 0$">|; 

$key = q/|0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="$ \vert\rangle$">|; 

$key = q/displaystylea^{,mu}_{~nu},a_mu^{~lambda}=g_nu^{~lambda}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2659.png"
 ALT="$\displaystyle a^{ \mu}_{&nbsp;\nu} a_\mu^{&nbsp;\lambda} = g_\nu^{&nbsp;\lambda}.$">|; 

$key = q/displaystyleD(a)=expleft(-{rmi},p_x,aslashhbarright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img512.png"
 ALT="$\displaystyle D(a) = \exp\left(-{\rm i} p_x a/\hbar\right),
$">|; 

$key = q/displaystyle|A_ttrangle=T^{dag}(t,t_0),|Atrangle=T^{dag}(t,t_0),T(t,t_0),|At_0rangle=|A_tt_0rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="394" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img563.png"
 ALT="$\displaystyle \vert A_t t\rangle = T^{\dag } (t, t_0) \vert At\rangle = T^{\dag } (t, t_0) T (t, t_0)  \vert At_0\rangle = \vert A_t t_0\rangle,$">|; 

$key = q/exp(mp{rmi},omega,Tslash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1132.png"
 ALT="$ \exp(\mp{\rm i}  \omega  T/2)$">|; 

$key = q/displaystyleneq0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1334.png"
 ALT="$\displaystyle \neq 0,$">|; 

$key = q/displaystyle=-hbar,omega,A,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img671.png"
 ALT="$\displaystyle =-\hbar \omega A,$">|; 

$key = q/langleA|Brangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img111.png"
 ALT="$ \langle A\vert B \rangle$">|; 

$key = q/displaystylefrac{exp(,{rmi},k,r)}{(2pi)^{3slash2}}frac{f({bfk}',{bfk})}{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2180.png"
 ALT="$\displaystyle \frac{ \exp( {\rm i}  k r)}{(2\pi)^{3/2}}\frac{ f({\bf k}', {\bf k})}{r},$">|; 

$key = q/displaystyle[L_x,p_z];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2762.png"
 ALT="$\displaystyle [L_x,p_z]$">|; 

$key = q/displaystylecdot{bfL}times{bfp}={rmi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2798.png"
 ALT="$\displaystyle \cdot{\bf L}\times {\bf p}= {\rm i} $">|; 

$key = q/displaystylec_{l,,m_{rmmin}}^-=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img833.png"
 ALT="$\displaystyle c_{l   m_{\rm min}}^- = 0.$">|; 

$key = q/displaystyleleft[frac{hbar^2}{2,mu}left(-frac{1}{r^2}frac{d}{dr},r^2,frac{d}{dr}ac{l,(l+1)}{r^2}right)-frac{e^2}{4pi,epsilon_0,r}-Eright]R=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="399" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img939.png"
 ALT="$\displaystyle \left[\frac{\hbar^2}{2 \mu} \left(-\frac{1}{r^2} \frac{d}{dr} r...
... +\frac{l (l+1)}{r^2}\right) -\frac{e^2}{4\pi \epsilon_0 r}- E\right] R = 0.$">|; 

$key = q/s,m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2460.png"
 ALT="$ s, m$">|; 

$key = q/displaystyleL^2,|l,mrangle=m_{rmmax}(m_{rmmax}+1),hbar^2,|l,mrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="268" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img813.png"
 ALT="$\displaystyle L^2  \vert l, m\rangle = m_{\rm max}(m_{\rm max} + 1) \hbar^2  \vert l, m\rangle.$">|; 

$key = q/P_l(1)=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2260.png"
 ALT="$ P_l(1) = 1$">|; 

$key = q/X;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img77.png"
 ALT="$ X$">|; 

$key = q/displaystyle[L^-,L_z];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img776.png"
 ALT="$\displaystyle [L^-, L_z]$">|; 

$key = q/|j,m_jrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1637.png"
 ALT="$ \vert j,m_j\rangle $">|; 

$key = q/displaystyle|E_nrangle=frac{1}{sqrt{n!}},(A^dag)^n,|E_0rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img690.png"
 ALT="$\displaystyle \vert E_n\rangle = \frac{1}{\sqrt{n!}} (A^\dag )^n \vert E_0\rangle.
$">|; 

$key = q/displaystyle[L^2,[L^2,z]]=-hbar^2left[4,(L_x,x+L_y,y+L_z,z),L_z-4,(L_x^{,2}+L_y^{,2}+L_z^{,2}),z+2,(L^2,z-z,L^2)right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="622" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1516.png"
 ALT="$\displaystyle [L^2, [L^2, z]] = - \hbar^2 \left[4 (L_x  x + L_y  y + L_z  z...
...- 4  (L_x^{ 2} + L_y^{ 2} + L_z^{ 2})  z + 2  (L^2  z - z  L^2)\right].$">|; 

$key = q/displaystyle=sqrt{l,(l+1)-m,(m+1)},hbar,|l,m+1rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img844.png"
 ALT="$\displaystyle = \sqrt{l (l+1)-m (m+1)} \hbar \vert l, m+1\rangle$">|; 

$key = q/displaystyles_0=(k^2-alpha^2)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2897.png"
 ALT="$\displaystyle s_0 = (k^2-\alpha^2)^{1/2}.$">|; 

$key = q/t=2pislashomega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1131.png"
 ALT="$ t=2\pi/\omega$">|; 

$key = q/displaystyleexp(pm{rmi},k,|{bfx}-{bfx}'|,)simeqexp(pm{rmi},k,r)exp(mp{rmi},{bfk}'cdot{bfx}').;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2170.png"
 ALT="$\displaystyle \exp(\pm {\rm i}  k \vert{\bf x} - {\bf x}' \vert ) \simeq \exp(\pm {\rm i}  k  r) \exp(\mp {\rm i}  {\bf k}' \cdot {\bf x}').$">|; 

$key = q/langleC|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img109.png"
 ALT="$ \langle C\vert$">|; 

$key = q/displaystyle=y^lleft(-frac{1}{y}frac{d}{dy}right)^lfrac{siny}{y},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2229.png"
 ALT="$\displaystyle = y^l\left(-\frac{1}{y}\frac{d}{dy}\right)^l \frac{\sin y}{y},$">|; 

$key = q/displaystyle=-sinalpha,|m-1slash2,1slash2rangle+cosalpha,|m+1slash2,-1slash2rangle.~~~~;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="394" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1636.png"
 ALT="$\displaystyle = -\sin\alpha  \vert m-1/2, 1/2\rangle + \cos\alpha  \vert m+1/2, -1/2\rangle.&nbsp;&nbsp;&nbsp;&nbsp;$">|; 

$key = q/|psi_{000}(0)|^{,2}=1slash(pi,a_0^{,3});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1744.png"
 ALT="$ \vert\psi_{000}(0)\vert^{ 2}=
1/(\pi a_0^{ 3})$">|; 

$key = q/displaystyle{bfx}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1586.png"
 ALT="$\displaystyle {\bf x}_2$">|; 

$key = q/displaystyle=E_1,|1rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1799.png"
 ALT="$\displaystyle = E_1  \vert 1\rangle,$">|; 

$key = q/t>t_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img539.png"
 ALT="$ t&gt;t_0$">|; 

$key = q/displaystyle(gamma^{,mu},p_mu-m_e,c),psi=({rmi},hbar,gamma^{,mu},partial_mu-m_e,c),psi=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2613.png"
 ALT="$\displaystyle (\gamma^{ \mu} p_\mu - m_e c) \psi = ({\rm i} \hbar \gamma^{ \mu} \partial_\mu-m_e c) \psi=0,$">|; 

$key = q/displaystylecdot{bfp},|iranglesimeq;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2073.png"
 ALT="$\displaystyle \cdot{\bf p}  \vert i\rangle \simeq$">|; 

$key = q/displaystyle=[u_1,v]+[u_2,v],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img496.png"
 ALT="$\displaystyle = [u_1, v] + [u_2, v],$">|; 

$key = q/displaystyle={rmi}sqrt{frac{m,hbar,omega}{2}}left(-sqrt{n},delta_{n',n-1}+sqrt{n+1},delta_{n',n+1}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img700.png"
 ALT="$\displaystyle = {\rm i}\sqrt{\frac{m \hbar \omega}{2}}\left(-\sqrt{n} \delta_{n' n-1}+\sqrt{n+1} \delta_{n' n+1}\right),$">|; 

$key = q/displaystyleS^2,|pmrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1029.png"
 ALT="$\displaystyle S^2  \vert\pm\rangle$">|; 

$key = q/displaystyle=2,{rmi},hbar,(L_y,x-y,L_x)=2,{rmi},hbar,(x,L_y-L_x,y),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="307" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1508.png"
 ALT="$\displaystyle = 2  {\rm i}  \hbar  ( L_y  x - y  L_x ) = 2  {\rm i}  \hbar  ( x  L_y - L_x  y),$">|; 

$key = q/displaystyle=-{rmi},hbar,left(cosvarphi,frac{partial}{partialtheta}-cotthetasinvarphi,frac{partial}{partialvarphi}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="269" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img895.png"
 ALT="$\displaystyle = -{\rm i}  \hbar \left(\cos\varphi  \frac{\partial}{\partial\theta} -\cot\theta \sin\varphi  \frac{\partial}{\partial \varphi}\right),$">|; 

$key = q/displaystyle=sum_jsum_k(delta_{j,k}+{rmi},epsilon_{j,k,l},sigma_l),a_j,b_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1210.png"
 ALT="$\displaystyle = \sum_j \sum_k (\delta_{j k} + {\rm i} \epsilon_{j k l}  \sigma_l) a_j  b_k$">|; 

$key = q/displaystyle=+{rmi},hbar,(L_x,L_y+L_y,L_x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img765.png"
 ALT="$\displaystyle = +{\rm i} \hbar (L_x L_y + L_y  L_x),$">|; 

$key = q/displaystyle[D(a),H]=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img710.png"
 ALT="$\displaystyle [D(a),H] = 0,
$">|; 

$key = q/m=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img866.png"
 ALT="$ m=0$">|; 

$key = q/g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1112.png"
 ALT="$ g$">|; 

$key = q/(S_x,S_y,S_z)equiv{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1013.png"
 ALT="$ (S_x, S_y, S_z)\equiv {\bf S}$">|; 

$key = q/displaystylesimeq1-frac{rmi}{hbar}int_{t_0}^tH_I(t_0,t'),dt'+left(frac{-{rmi}}{h_{t_0}^tdt'int_{t_0}^{t'}dt'',H_I(t_0,t'),H_I(t_0,t'')+cdots.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="555" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1871.png"
 ALT="$\displaystyle \simeq 1 - \frac{\rm i}{\hbar} \int_{t_0}^t H_I(t_0, t') dt' + \...
...\int_{t_0}^t dt' \int_{t_0}^{t'} dt''  H_I(t_0, t' ) H_I(t_0, t'' ) + \cdots.$">|; 

$key = q/displaystylepsi'^{dag},gamma^0,gamma^{,mu},psi'=a^{,mu}_{~nu},psi^dag,gamma^0,gamma^nu,psi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2702.png"
 ALT="$\displaystyle \psi'^{\dag } \gamma^0 \gamma^{ \mu} \psi'= a^{ \mu}_{&nbsp;\nu} \psi^\dag \gamma^0 \gamma^\nu \psi,$">|; 

$key = q/BslashA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2360.png"
 ALT="$ B/A$">|; 

$key = q/s_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2889.png"
 ALT="$ s_0$">|; 

$key = q/displaystyler,epsilon=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2837.png"
 ALT="$\displaystyle r \epsilon =$">|; 

$key = q/displaystylecdot{bfn}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1227.png"
 ALT="$\displaystyle \cdot{\bf n}.$">|; 

$key = q/{bfx}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2454.png"
 ALT="$ {\bf x}_2$">|; 

$key = q/displaystyle=left({array}{c}1slashsqrt{2}1slashsqrt{2}00{array}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="115" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1585.png"
 ALT="$\displaystyle = \left( \begin{array}{c} 1/\sqrt{2} \ 1/\sqrt{2} \ 0 \ 0 \end{array} \right),$">|; 

$key = q/S_x=-hbarslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1430.png"
 ALT="$ S_x=-\hbar/2$">|; 

$key = q/ldots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img116.png"
 ALT="$ \ldots$">|; 

$key = q/Z=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2493.png"
 ALT="$ Z=1$">|; 

$key = q/(x_1,x_2,x_3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img587.png"
 ALT="$ (x_1, x_2, x_3)$">|; 

$key = q/{calR}_{n,l}(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img980.png"
 ALT="$ {\cal R}_{n l}(x)$">|; 

$key = q/displaystyleL_z+2,S_z=J_z+S_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1697.png"
 ALT="$\displaystyle L_z + 2  S_z = J_z + S_z,$">|; 

$key = q/displaystyle=frac{2pi}{hbar},|V_{ni}^dagger|^{,2},delta(E_n-E_i-hbar,omega).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2019.png"
 ALT="$\displaystyle = \frac{2\pi}{\hbar}   \vert V_{ni}^\dagger\vert^{ 2} \delta(E_n -E_i-\hbar \omega).$">|; 

$key = q/{bfa}equiv(a_x,a_y,a_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1202.png"
 ALT="$ {\bf a} \equiv (a_x, a_y, a_z)$">|; 

$key = q/displaystylelangle({mitDelta}xi)^2rangle=langlexi^{,2}rangle-langlexirangle^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img166.png"
 ALT="$\displaystyle \langle({\mit\Delta} \xi)^2\rangle = \langle \xi^{ 2}\rangle - \langle \xi\rangle^2.$">|; 

$key = q/displaystyleR_y({mitDelta}varphi_y),|0,0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img879.png"
 ALT="$\displaystyle R_y({\mit\Delta} \varphi_y)  \vert,0\rangle$">|; 

$key = q/displaystyle(m-m_1-m_2),langlej_1,j_2;m_1,m_2|j_1,j_2;j,mrangle=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="345" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1370.png"
 ALT="$\displaystyle (m-m_1-m_2)  \langle j_1, j_2; m_1, m_2\vert j_1,j_2; j, m\rangle=0,$">|; 

$key = q/displaystylephi(q_1'cdotsq_N')=langlephi|q_1'cdotsq_N'rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img379.png"
 ALT="$\displaystyle \phi(q_1'\cdots q_N') = \langle \phi\vert q_1'\cdots q_N'\rangle.$">|; 

$key = q/displaystyle(;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1204.png"
 ALT="$\displaystyle ($">|; 

$key = q/displaystyle{mitDelta}E_{nlm}=e^2,|{bfE}|^{,2}sum_{n'}sum_{l'=lpm1}frac{|langlen,l,m|,z,|n',l',mrangle|^{,2}}{E_{nlm}-E_{n'l'm}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1528.png"
 ALT="$\displaystyle {\mit\Delta} E_{nlm} = e^2  \vert{\bf E}\vert^{ 2} \sum_{n'}\su...
...t\langle n,l,m\vert z \vert n',l',m\rangle\vert^{ 2}}{E_{nlm} - E_{n'l' m}}.$">|; 

$key = q/displaystyleH_0,|nrangle=E_n,|nrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1777.png"
 ALT="$\displaystyle H_0  \vert n\rangle = E_n  \vert n\rangle.$">|; 

$key = q/displaystyle(J_z-J_{1,z}-J_{2,z}),|j_1,j_2;j,mrangle=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1368.png"
 ALT="$\displaystyle (J_z - J_{1 z} - J_{2 z})  \vert j_1,j_2; j, m\rangle =0.$">|; 

$key = q/displaystylesqrt{j,(j+1)-m,(mpm1)},langlej_1,j_2;m_1,m_2|j_1,j_2;j,mpm1rangle=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="420" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1390.png"
 ALT="$\displaystyle \sqrt{j (j+1)- m (m\pm 1)} \langle j_1, j_2; m_1, m_2\vert j_1,j_2;j, m\pm 1\rangle=$">|; 

$key = q/|Adrangle=D,|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img450.png"
 ALT="$ \vert Ad\rangle = D \vert A\rangle$">|; 

$key = q/cosalpha_1,chi_++sinalpha_1,{rme}^{,{rmi},beta_1},chi_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1431.png"
 ALT="$ \cos\alpha_1 \chi_+
+ \sin\alpha_1 {\rm e}^{ {\rm i} \beta_1} \chi_-$">|; 

$key = q/m_l,m_s,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1623.png"
 ALT="$ m_l, m_s,$">|; 

$key = q/|{bfk}rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2157.png"
 ALT="$ \vert{\bf k}\rangle$">|; 

$key = q/displaystyle[q_i,q_j]_{cl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img250.png"
 ALT="$\displaystyle [q_i, q_j]_{cl}$">|; 

$key = q/displaystyle[u,v_1,v_2];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img501.png"
 ALT="$\displaystyle [u, v_1  v_2]$">|; 

$key = q/xi'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img120.png"
 ALT="$ \xi''$">|; 

$key = q/displaystyle[H,A];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img670.png"
 ALT="$\displaystyle [H,A]$">|; 

$key = q/u(x'-a)=u(x');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img712.png"
 ALT="$ u(x'-a)=u(x')$">|; 

$key = q/L_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1198.png"
 ALT="$ L_k$">|; 

$key = q/displaystylepsi_{s_z}({bfx}')=langle{bfx'}|langles,s_z||Aranglerangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1258.png"
 ALT="$\displaystyle \psi_{s_z}({\bf x}') = \langle {\bf x'}\vert\langle s, s_z\vert \vert A\rangle\rangle,$">|; 

$key = q/displaystyleU_{j,k}=langlen,j|,H_1,|n,krangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1564.png"
 ALT="$\displaystyle U_{j k} = \langle n, j\vert  H_1 \vert n, k\rangle.$">|; 

$key = q/displaystylelanglem|Erangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1480.png"
 ALT="$\displaystyle \langle m\vert E\rangle$">|; 

$key = q/[L^pm,L_z]=mp,hbar,L^pm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img991.png"
 ALT="$ [L^\pm, L_z] = \mp  \hbar L^\pm$">|; 

$key = q/displaystyleH,|k',k'',k'''rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2415.png"
 ALT="$\displaystyle H \vert k' k'' k'''\rangle$">|; 

$key = q/V({bfx});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img607.png"
 ALT="$ V({\bf x})$">|; 

$key = q/langleA|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="$ \langle A\vert$">|; 

$key = q/displaystylex^{,mu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2660.png"
 ALT="$\displaystyle x^{ \mu}$">|; 

$key = q/displaystyle=-{rmi},omega,A,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img715.png"
 ALT="$\displaystyle = -{\rm i} \omega A,$">|; 

$key = q/displaystyle{mitDelta}E_{n,l,m_l,m_s}=langlen,l,m_l,m_s|,H_{LS},|n,l,m_l,m_srangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="313" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1717.png"
 ALT="$\displaystyle {\mit\Delta} E_{n l m_l  m_s} = \langle n,l,m_l, m_s\vert H_{LS} \vert n,l,m_l, m_s\rangle,$">|; 

$key = q/displaystyle=c',|x',y',z'rangle|+rangle+c'',|x'',y'',z''rangle|+rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="281" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1056.png"
 ALT="$\displaystyle = c'  \vert x', y', z'\rangle \vert+\rangle+ c''  \vert x'', y'', z''\rangle \vert+\rangle,$">|; 

$key = q/1s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2093.png"
 ALT="$ 1s$">|; 

$key = q/m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img597.png"
 ALT="$ m$">|; 

$key = q/displaystylefrac{exp(,{rmi},{bfk}cdot{bfx})}{(2pi)^{3slash2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2178.png"
 ALT="$\displaystyle \frac{ \exp( {\rm i}  {\bf k}\cdot {\bf x})}{(2\pi)^{3/2}},$">|; 

$key = q/{mitDelta}x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img435.png"
 ALT="$ {\mit\Delta} x$">|; 

$key = q/displaystyleleft(langlephi,frac{d}{dx}right)psirangle=langlephileft(frac{d}{dx},psirangleright).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img339.png"
 ALT="$\displaystyle \left( \langle \phi  \frac{d}{dx} \right) \psi\rangle = \langle \phi\left( \frac{d}{dx} \psi\rangle\right).$">|; 

$key = q/{calE}=Eslash(m_e,c^2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2868.png"
 ALT="$ {\cal E} = E/(m_e c^2)$">|; 

$key = q/displaystyle(sigma_3)_{j,l}=frac{langles,j|,S_z,|s,lrangle}{s,hbar}=frac{j}{s},delta_{j,l},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1269.png"
 ALT="$\displaystyle (\sigma_3)_{j l} = \frac{\langle s, j\vert S_z  \vert s, l\rangle}{s \hbar} = \frac{j}{s}  \delta_{j l},$">|; 

$key = q/displaystylelambda=21.1,{rmcm}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1757.png"
 ALT="$\displaystyle \lambda = 21.1 {\rm cm}.$">|; 

$key = q/langle1|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1444.png"
 ALT="$ \langle 1\vert$">|; 

$key = q/d{mitOmega}=sintheta,dtheta,dvarphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1672.png"
 ALT="$ d{\mit\Omega} = \sin\theta d\theta d\varphi$">|; 

$key = q/{bfx}_1';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2458.png"
 ALT="$ {\bf x}_1'$">|; 

$key = q/displaystylesigma_{rmtotal}=frac{4pi}{k},{rmIm}left[f(0)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2259.png"
 ALT="$\displaystyle \sigma_{\rm total} = \frac{4\pi}{k}  {\rm Im}\left[f(0)\right],$">|; 

$key = q/k_B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2098.png"
 ALT="$ k_B$">|; 

$key = q/j=3slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1702.png"
 ALT="$ j=3/2$">|; 

$key = q/displaystylecdot{bfp}times{bfL},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2804.png"
 ALT="$\displaystyle \cdot{\bf p}\times{\bf L},$">|; 

$key = q/{mitDeltaE};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2525.png"
 ALT="$ {\mit\Delta E}$">|; 

$key = q/displaystyleH_0=frac{{p}^2}{2,m_e}+V(r),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1493.png"
 ALT="$\displaystyle H_0 = \frac{{p}^2}{2 m_e} + V(r),$">|; 

$key = q/displaystylea_1left[a,alpha-a_2,(s-k)right]=aleft[a_2,alpha+a,(s+k)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2911.png"
 ALT="$\displaystyle a_1\left[a \alpha-a_2 (s-k)\right] = a\left[a_2 \alpha+a (s+k)\right],$">|; 

$key = q/displaystyleL^+|l,mrangle=c^+_{l,m},hbar,|l,m+1rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img797.png"
 ALT="$\displaystyle L^+ \vert l ,m\rangle = c^+_{l m}  \hbar \vert l, m+1\rangle,$">|; 

$key = q/displaystyleE=E_n+O(epsilon),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1477.png"
 ALT="$\displaystyle E= E_n + O(\epsilon),$">|; 

$key = q/displaystylec_{l,m}^-=(c_{l,,m-1}^+)^ast=sqrt{l,(l+1)-m,(m-1)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img829.png"
 ALT="$\displaystyle c_{l  m}^- = (c_{l   m-1}^+)^\ast = \sqrt{l (l+1)- m  (m-1)}.$">|; 

$key = q/psi({bfx})=langle{bfx}|psirangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2188.png"
 ALT="$ \psi({\bf x})=\langle {\bf x}\vert\psi\rangle$">|; 

$key = q/displaystylea=sqrt{frac{-hbar^2}{2,mu,E}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img947.png"
 ALT="$\displaystyle a= \sqrt{\frac{-\hbar^2}{2 \mu  E}},$">|; 

$key = q/displaystyle=intdxi',langleA|xi'ranglelanglexi'|Arangle=intdxi',|langleA|xi'rangle|^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="301" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img200.png"
 ALT="$\displaystyle = \int d\xi' \langle A\vert\xi'\rangle\langle \xi'\vert A\rangle = \int d\xi'  \vert\langle A\vert\xi'\rangle\vert^{ 2}.$">|; 

$key = q/displaystylecdot{bfp}),(;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2800.png"
 ALT="$\displaystyle \cdot {\bf p}) ($">|; 

$key = q/{mitDelta}varphi_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img875.png"
 ALT="$ {\mit\Delta}\varphi_x$">|; 

$key = q/displaystyle-{rmi}left[(mbox{boldmathsigma}cdot{bfn})left(frac{{mitDelta}varphi}fn})^3}{3!}left(frac{{mitDelta}varphi}{2}right)^3+cdotsright];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="310" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1225.png"
 ALT="$\displaystyle - {\rm i} \left[ (\mbox{\boldmath$\sigma$}\cdot {\bf n} )\left( \...
...ot {\bf n})^3}{3!} \left(\frac{{\mit\Delta}\varphi}{2}\right)^3 + \cdots\right]$">|; 

$key = q/displaystyle[L_y^{,2},L_z];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img764.png"
 ALT="$\displaystyle [L_y^{ 2}, L_z]$">|; 

$key = q/displaystylefrac{|e_{mm}|}{E_m}simO(epsilon),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1476.png"
 ALT="$\displaystyle \frac{\vert e_{mm}\vert}{E_m} \sim O(\epsilon),$">|; 

$key = q/p_x'slashhbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img438.png"
 ALT="$ p_x'/\hbar$">|; 

$key = q/t=4pislashomega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1130.png"
 ALT="$ t= 4\pi/\omega$">|; 

$key = q/displaystyleH_1=frac{mu_0,g_p,e^2}{8pi,m_p,m_e}left(frac{3,({bfS}_pcdot{bfe}_r),rac{mu_0,g_p,e^2}{3,m_p,m_e},{bfS}_pcdot{bfS}_e,delta({bfx}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="519" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1742.png"
 ALT="$\displaystyle H_1=\frac{\mu_0 g_p e^2}{8\pi m_p m_e}\left(\frac{3 ({\bf S}...
... \frac{\mu_0 g_p e^2}{3 m_p m_e} {\bf S}_p\cdot{\bf S}_e \delta({\bf x}).$">|; 

$key = q/psi(x'-deltax);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img477.png"
 ALT="$ \psi(x'-\delta x)$">|; 

$key = q/displaystylelanglen,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',m'rangle=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="481" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1520.png"
 ALT="$\displaystyle \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.$">|; 

$key = q/displaystyle=frac{2pi}{hbar},|V_{ni}|^{,2},delta(E_n-E_i+hbar,omega),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2018.png"
 ALT="$\displaystyle = \frac{2\pi}{\hbar}  \vert V_{ni}\vert^{ 2}   \delta(E_n-E_i+\hbar \omega),$">|; 

$key = q/|phi({bfx}_1,{bfx}_2)|^{,2},d^3x_1,d^3x_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2484.png"
 ALT="$ \vert\phi({\bf x}_1,{\bf x}_2)\vert^{ 2} d^3x_1 d^3x_2$">|; 

$key = q/j_1=j_2=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1398.png"
 ALT="$ j_1=j_2=1/2$">|; 

$key = q/deltax;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img461.png"
 ALT="$ \delta x$">|; 

$key = q/|BranglelangleA|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img108.png"
 ALT="$ \vert B\rangle \langle A\vert$">|; 

$key = q/i=1,3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img659.png"
 ALT="$ i=1,3$">|; 

$key = q/q_2'+dq_2';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img383.png"
 ALT="$ q_2'
+dq_2'$">|; 

$key = q/displaystyle=[u_1,v_1,v_2]_{qm},u_2+u_1,[u_2,v_1,v_2]_{qm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img272.png"
 ALT="$\displaystyle = [u_1, v_1  v_2]_{qm} u_2 + u_1 [u_2, v_1  v_2]_{qm}$">|; 

$key = q/displaystyle[L^2,L_z]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img767.png"
 ALT="$\displaystyle [L^2, L_z] = 0.$">|; 

$key = q/(r,theta,varphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img848.png"
 ALT="$ (r, \theta, \varphi)$">|; 

$key = q/displaystyleleftlangle{bfx}'left|frac{{p}^2}{2,m}right|Atrightrangle=-left(frac{hbar^2}{2,m}right)nabla'^{,2}langle{bfx'}|Atrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img623.png"
 ALT="$\displaystyle \left\langle {\bf x}'\left\vert\frac{{p}^2}{2 m}\right\vert At\r...
...left(\frac{\hbar^2}{2 m}\right) \nabla'^{ 2}\langle {\bf x'} \vert At\rangle,$">|; 

$key = q/displaystylesigma_lsimeqfrac{4pi}{k^2},(2,l+1),frac{{mitGamma}^{,2}slash4}{(E-E_0)^{,2}+{mitGamma}^{,2}slash4}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2349.png"
 ALT="$\displaystyle \sigma_l \simeq \frac{4\pi}{k^2}  (2 l+1)  \frac{{\mit\Gamma}^{ 2}/4}{(E-E_0)^{ 2} + {\mit\Gamma}^{ 2}/4}.$">|; 

$key = q/D;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img447.png"
 ALT="$ D$">|; 

$key = q/displaystylem,frac{d^2langle{bfx}rangle}{dt^2}=frac{dlangle{bfp}rangle}{dt}=-langlenablaV({bfx})rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img614.png"
 ALT="$\displaystyle m  \frac{d^2\langle {\bf x}\rangle}{dt^2}= \frac{d\langle{\bf p}\rangle}{dt} = - \langle \nabla V({\bf x})\rangle.$">|; 

$key = q/S_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1043.png"
 ALT="$ S_j$">|; 

$key = q/displaystylepsi({bfx})=frac{1}{(2pi)^{3slash2}}left[exp(,{rmi},{bfk}cdot{bfx})+frac{exp(,{rmi},k,r)}{r}f({bfk}',{bfk})right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="381" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2173.png"
 ALT="$\displaystyle \psi({\bf x}) = \frac{1}{(2\pi)^{3/2}} \left[\exp( {\rm i} {\bf k}\cdot{\bf x}) + \frac{\exp( {\rm i} k r)}{r} f({\bf k}', {\bf k}) \right],$">|; 

$key = q/displaystyleT(t,t_0)=expleft[-frac{{rmi}}{hbar}int_{t_0}^tdt',H(t')right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img561.png"
 ALT="$\displaystyle T(t, t_0) = \exp\left[-\frac{{\rm i}}{\hbar} \int_{t_0}^t dt' H(t') \right],$">|; 

$key = q/|E_0|=13.6,{rmeV};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1756.png"
 ALT="$ \vert E_0\vert=13.6 {\rm eV}$">|; 

$key = q/displaystylerho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2639.png"
 ALT="$\displaystyle \rho$">|; 

$key = q/displaystylegamma^i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2605.png"
 ALT="$\displaystyle \gamma^i$">|; 

$key = q/displaystyle=|xi_a'rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img149.png"
 ALT="$\displaystyle = \vert\xi_a'\rangle,$">|; 

$key = q/displaystyleA,|E_nrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img686.png"
 ALT="$\displaystyle A \vert E_n\rangle$">|; 

$key = q/displaystyle{rmi},hbar,frac{dH}{dt}=[H,H]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img580.png"
 ALT="$\displaystyle {\rm i} \hbar  \frac{dH}{dt} = [H, H] = 0.$">|; 

$key = q/cdot{bfa};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1201.png"
 ALT="$ \cdot {\bf a}$">|; 

$key = q/displaystyle=frac{{rmi},t}{hbar}sum_mfrac{H_{nm}H_{mi}}{E_m-E_i}left[exp(,{rmi},{rmi},omega_{nm},tslash2),{rmsinc}(omega_{nm},tslash2)right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="588" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1932.png"
 ALT="$\displaystyle = \frac{{\rm i} t}{\hbar} \sum_m \frac{H_{nm} H_{mi}}{E_m - E_i}...
...t/2)- \exp( {\rm i} \omega_{nm}  t/2)  {\rm sinc}(\omega_{nm} t/2)\right].$">|; 

$key = q/j^{,mu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2651.png"
 ALT="$ j^{ \mu}$">|; 

$key = q/pi,a^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2302.png"
 ALT="$ \pi  a^2$">|; 

$key = q/displaystyle=sum_jsum_kleft(frac{1}{2},{sigma_j,sigma_k}+frac{1}{2}[sigma_j,sigma_k]right)a_j,b_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="301" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1209.png"
 ALT="$\displaystyle = \sum_j \sum_k \left(\frac{1}{2}  \{\sigma_j, \sigma_k\} + \frac{1}{2} [\sigma_j, \sigma_k]\right) a_j  b_k$">|; 

$key = q/displaystylej_l(y)=frac{(-{rmi})^l}{2}int_{-1}^1dmu,exp(,{rmi},y,mu),P_l(mu),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2238.png"
 ALT="$\displaystyle j_l(y) = \frac{(-{\rm i})^l}{2} \int_{-1}^1 d\mu  \exp( {\rm i}  y \mu)  P_l(\mu),$">|; 

$key = q/displaystyleleftlanglefrac{p^2}{2,m}rightrangle=frac{1}{2}leftlangler,frac{dV}{dr}rightrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1004.png"
 ALT="$\displaystyle \left\langle\frac{p^2}{2 m}\right\rangle = \frac{1}{2}\left\langle r \frac{dV}{dr}\right\rangle
$">|; 

$key = q/N=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1798.png"
 ALT="$ N=2$">|; 

$key = q/displaystyle|nrangle'=|nrangle+sum_{kneqn}frac{e_{kn}}{E_n-E_k},|krangle+O(epsilon^2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1491.png"
 ALT="$\displaystyle \vert n\rangle' = \vert n\rangle +\sum_{k\neq n}\frac{e_{kn}}{E_n - E_k} \vert k\rangle + O(\epsilon^2).$">|; 

$key = q/displaystyle{bfj}_{rminc}=frac{hbar}{(2pi)^{3},m},{bfk}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2179.png"
 ALT="$\displaystyle {\bf j}_{\rm inc} = \frac{\hbar}{(2\pi)^{3} m}  {\bf k}.$">|; 

$key = q/displaystyle=frac{q}{2,m}left({bfL}+g,{bfS}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1111.png"
 ALT="$\displaystyle = \frac{q}{2 m} \left({\bf L} + g  {\bf S}\right),$">|; 

$key = q/displaystylepsi({bfx'},t)=langle{bfx'}|Atrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img620.png"
 ALT="$\displaystyle \psi({\bf x'}, t) = \langle {\bf x'}\vert At\rangle.$">|; 

$key = q/displaystyleomega=frac{e,B}{m_e}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1120.png"
 ALT="$\displaystyle \omega = \frac{e B}{m_e}.$">|; 

$key = q/j^{,mu}=(c,rho,{bfj});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2638.png"
 ALT="$ j^{ \mu} = (c \rho, {\bf j})$">|; 

$key = q/displaystylelanglen|Erangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1478.png"
 ALT="$\displaystyle \langle n\vert E\rangle$">|; 

$key = q/displaystyleL^-,L^+,|l,mrangle=(L^2-L_z^{,2}-hbar,L_z),|l,mrangle=hbar^2,[l,(l+1)-m,(m+1)],|l,mrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="527" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img815.png"
 ALT="$\displaystyle L^-  L^+  \vert l, m\rangle = (L^2 - L_z^{ 2}-\hbar  L_z) \vert l, m \rangle = \hbar^2  [l (l+1) - m (m+1)] \vert l,m\rangle.$">|; 

$key = q/displaystyle|Rdrangle=D,|Rrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img446.png"
 ALT="$\displaystyle \vert R d\rangle = D  \vert R\rangle,$">|; 

$key = q/displaystylep^{,mu}={rmi},hbar,frac{partial}{partialx_{mu}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2578.png"
 ALT="$\displaystyle p^{ \mu} = {\rm i} \hbar \frac{\partial}{\partial x_{\mu}}.$">|; 

$key = q/m_{rmmin}=-l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img834.png"
 ALT="$ m_{\rm min} = -l$">|; 

$key = q/mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img942.png"
 ALT="$ \mu$">|; 

$key = q/E_{nlm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1497.png"
 ALT="$ E_{nlm}$">|; 

$key = q/displaystylepsi({bfx})=frac{1}{(2pi)^{3slash2}}sum_{l=0,infty}{rmi}^l,(2,l+1),R_l(r),P_l(costheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2276.png"
 ALT="$\displaystyle \psi({\bf x}) = \frac{1}{(2\pi)^{3/2}}\sum_{l=0,\infty} {\rm i}^l  (2 l+1) R_l(r) P_l(\cos\theta),$">|; 

$key = q/Y_{0,0}=1slashsqrt{4pi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img922.png"
 ALT="$ Y_{0 0} = 1/\sqrt{4\pi}$">|; 

$key = q/displaystyle=P_{23},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2410.png"
 ALT="$\displaystyle = P_{23},$">|; 

$key = q/displaystyleY_{l,m}(theta,varphi)=sqrt{frac{2,l+1}{4pi}frac{(l-m)!}{(l+m)!}},(-1)^m,{rme}^{,{rmi},m,varphi},P_{l,m}(cosvarphi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="403" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img906.png"
 ALT="$\displaystyle Y_{l m}(\theta, \varphi) = \sqrt{ \frac{2 l+1}{4\pi} \frac{(l-m)!}{(l+m)!}}  (-1)^m  {\rm e}^{ {\rm i}  m \varphi}  P_{l m}(\cos\varphi),$">|; 

$key = q/pm1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1409.png"
 ALT="$ \pm 1$">|; 

$key = q/displaystyleleft[gamma^{,mu}left(p_{mu'}-frac{e}{c},{mitPhi}_{mu'}right)-m_e,cright]psi'=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2679.png"
 ALT="$\displaystyle \left[\gamma^{ \mu}\left(p_{\mu'}- \frac{e}{c} {\mit\Phi}_{\mu'}\right)-m_e c\right]\psi' = 0.$">|; 

$key = q/displaystyleH_1(t)=exp(eta,t),H_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1944.png"
 ALT="$\displaystyle H_1(t) = \exp(\eta t) H_1,$">|; 

$key = q/c^+_{l,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img798.png"
 ALT="$ c^+_{l  m}$">|; 

$key = q/displaystylelanglen'|,x^2,|nrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img701.png"
 ALT="$\displaystyle \langle n'\vert x^2 \vert n\rangle$">|; 

$key = q/V_0>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2313.png"
 ALT="$ V_0&gt;0$">|; 

$key = q/displaystyled_x^{~dag}+d_x=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img473.png"
 ALT="$\displaystyle d_x^{&nbsp;\dag } + d_x = 0.$">|; 

$key = q/r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img207.png"
 ALT="$ r$">|; 

$key = q/displaystyle=32,Zint_0^inftydx_1,x_1^{,2}left(int_0^{x_1}dx_2,frac{x_2^{,2}}{x_1(x_1+x_2)}+int_{x_1}^inftydx_2,x_2,{rme}^{-2,(x_1+x_2)}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="491" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2518.png"
 ALT="$\displaystyle = 32 Z\int_0^\infty d x_1 x_1^{ 2}\left(\int_0^{x_1} d x_2 \f...
...e}^{-2 (x_1+x_2)} +\int_{x_1}^\infty dx_2 x_2 {\rm e}^{-2 (x_1+x_2)}\right)$">|; 

$key = q/E_mneqE_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1937.png"
 ALT="$ E_m\neq E_i$">|; 

$key = q/S^{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2457.png"
 ALT="$ S^{2}$">|; 

$key = q/|x',y',z'rangle|pmrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1061.png"
 ALT="$ \vert x',y', z'\rangle\vert\pm\rangle$">|; 

$key = q/displaystyle|k',k'',k'''rangle_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2448.png"
 ALT="$\displaystyle \vert k' k'' k'''\rangle_-$">|; 

$key = q/i=j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="$ i=j$">|; 

$key = q/|n,m_{rmmax}rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img809.png"
 ALT="$ \vert n, m_{\rm max}\rangle$">|; 

$key = q/phi_A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="$ \phi_A$">|; 

$key = q/{bf1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1195.png"
 ALT="$ {\bf 1}$">|; 

$key = q/psi_Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img328.png"
 ALT="$ \psi_A\rangle$">|; 

$key = q/Rslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="$ R/2$">|; 

$key = q/displaystyle=langlex',y',z'|,p_klangle-||Aranglerangle=-{rmi},hbarfrac{partial}{partialx_k'}langlex',y',z'|langle-||Aranglerangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="373" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1191.png"
 ALT="$\displaystyle = \langle x', y', z'\vert p_k \langle -\vert\vert A\rangle\rangl...
...l} {\partial x_k'} \langle x', y', z'\vert \langle -\vert\vert A\rangle\rangle,$">|; 

$key = q/displaystyle=[u_1,u_2,v_1]_{qm},v_2+v_1,[u_1,u_2,v_2]_{qm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img275.png"
 ALT="$\displaystyle = [u_1  u_2, v_1 ]_{qm} v_2 + v_1 [u_1  u_2, v_2]_{qm}$">|; 

$key = q/displaystyle|Arangle=langle+|Arangle|+rangle+langle-|Arangle|-ranglerightarrowleft(!{array}{c}langle+|Aranglelangle-|Arangle{array}!right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1150.png"
 ALT="$\displaystyle \vert A\rangle = \langle +\vert A\rangle \vert+\rangle + \langle ...
...array}{c}\langle +\vert A\rangle\ \langle -\vert A\rangle\end{array}\!\right),$">|; 

$key = q/displaystylelangleprangle=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img510.png"
 ALT="$\displaystyle \langle p\rangle = 0,$">|; 

$key = q/|E_0|slashhbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2118.png"
 ALT="$ \vert E_0\vert/\hbar$">|; 

$key = q/displaystyleL_z,|l,mrangle=m,hbar,|l,mrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img782.png"
 ALT="$\displaystyle L_z  \vert l, m\rangle = m \hbar  \vert l, m\rangle.$">|; 

$key = q/displaystyleS^pm=S_xpm{rmi},S_y.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1273.png"
 ALT="$\displaystyle S^\pm = S_x \pm {\rm i}  S_y.$">|; 

$key = q/displaystyleepsilon^2=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2842.png"
 ALT="$\displaystyle \epsilon^2 = 1.$">|; 

$key = q/displaystylelangle1slash2,-1slash2|1,0rangle^{,2}+langle-1slash2,1slash2|1,0rangle^{,2}=1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1413.png"
 ALT="$\displaystyle \langle 1/2, -1/2\vert 1, 0\rangle^{ 2} + \langle -1/2, 1/2\vert 1, 0\rangle^{ 2} = 1,$">|; 

$key = q/displaystyleR(2pi),|l,mrangle=exp(-{rmi},L_z,2pislashhbar),|l,mrangle=exp(-{rmi},2pi,m),|l,mrangle=|l,mrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="498" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img865.png"
 ALT="$\displaystyle R(2\pi) \vert l,m\rangle = \exp(-{\rm i} L_z  2\pi/\hbar) \vert l,m\rangle = \exp(-{\rm i}  2\pi m)  \vert l,m\rangle = \vert l,m\rangle.$">|; 

$key = q/displaystyleP_{irightarrowi}=exp(-tslashtau_i).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1990.png"
 ALT="$\displaystyle P_{i\rightarrow i} = \exp(- t/\tau_i).$">|; 

$key = q/displaystyle=pmsqrt{frac{lpmm+1slash2}{2,l+1}},Y_{l,,m-1slash2}(theta,varphi),chac{lmpm+1slash2}{2,l+1}},Y_{l,,m+1slash2}(theta,varphi),chi_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="523" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1657.png"
 ALT="$\displaystyle = \pm \sqrt{\frac{l\pm m+1/2}{2 l+1}}  Y_{l  m-1/2}(\theta, \...
...i_+ +\sqrt{\frac{l\mp m+1/2}{2 l+1}}  Y_{l  m+1/2}(\theta,\varphi)  \chi_-$">|; 

$key = q/displaystylepsi_0({bfx}')rightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img637.png"
 ALT="$\displaystyle \psi_0({\bf x}') \rightarrow 0$">|; 

$key = q/j_1,(j_1+1),hbar^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1309.png"
 ALT="$ j_1 (j_1+1) \hbar^2$">|; 

$key = q/displaystylesimeq-frac{e}{2,m_e}left({bfL}+2,{bfS}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1115.png"
 ALT="$\displaystyle \simeq - \frac{e}{2 m_e} \left({\bf L} + 2  {\bf S}\right)$">|; 

$key = q/displaystyle(H_0+H_1),|Erangle=E,|Erangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1442.png"
 ALT="$\displaystyle (H_0 + H_1)  \vert E\rangle = E \vert E\rangle.$">|; 

$key = q/displaystylepsi^dagleft[gamma^0,gamma^{,mu}left({rmi},hbar,partial_mu-frac{e}{c},{mitPhi}_muright)-gamma^0,m_e,cright]psi=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2632.png"
 ALT="$\displaystyle \psi^\dag\left[\gamma^0 \gamma^{ \mu}\left({\rm i} \hbar \partial_\mu-\frac{e}{c} {\mit\Phi}_\mu\right)-\gamma^0 m_e c\right]\psi=0.$">|; 

$key = q/[H_0,L]=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1544.png"
 ALT="$ [H_0, L] = 0$">|; 

$key = q/|tilde{A}rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img73.png"
 ALT="$ \vert\tilde{A}\rangle$">|; 

$key = q/displaystyle{rmi},hbar,frac{partialpsi}{partialt}=(c,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2615.png"
 ALT="$\displaystyle {\rm i} \hbar \frac{\partial\psi}{\partial t} = (c $">|; 

$key = q/displaystyle{bfA};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2040.png"
 ALT="$\displaystyle {\bf A}$">|; 

$key = q/displaystyle{rmi},hbar,frac{partialpsi({bfx}',t)}{partialt}=-left(frac{hbar^2}{2,m}right)nabla'^{,2}psi({bfx}',t)+V({bfx'}),psi({bfx'},t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img628.png"
 ALT="$\displaystyle {\rm i} \hbar \frac{\partial \psi({\bf x}', t)}{\partial t} = -...
...{2 m}\right) \nabla'^{ 2}\psi({\bf x}', t) + V({\bf x'})  \psi({\bf x'}, t).$">|; 

$key = q/displaystyleS_krightarrows,hbar,sigma_k,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1261.png"
 ALT="$\displaystyle S_k \rightarrow s \hbar  \sigma_k,$">|; 

$key = q/displaystyleU(t_0,t)=exp[-{rmi},H_0,(t-t_0)slashhbar],U_I(t_0,t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1858.png"
 ALT="$\displaystyle U(t_0, t) = \exp[ -{\rm i}   H_0 (t-t_0)/\hbar]  U_I(t_0, t).$">|; 

$key = q/displaystyle=+3,e,a_0,|{bfE}|,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1602.png"
 ALT="$\displaystyle = +3 e a_0  \vert{\bf E}\vert,$">|; 

$key = q/P_n(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1784.png"
 ALT="$ P_n(t)$">|; 

$key = q/psi_B(x');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img313.png"
 ALT="$ \psi_B(x')$">|; 

$key = q/displaystylecdotleft[{bfx},({bfx}cdot{bfp})-({bfx}cdot{bfp}),{bfx}right]={rmi},hbar,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2847.png"
 ALT="$\displaystyle \cdot\left[{\bf x} ({\bf x}\cdot{\bf p})- ({\bf x}\cdot{\bf p}) {\bf x}\right] = {\rm i} \hbar $">|; 

$key = q/displaystylelangleq_1'cdotsq_N'|rangle=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img375.png"
 ALT="$\displaystyle \langle q_1'\cdots q_N'\vert\rangle = 1.$">|; 

$key = q/displaystyle=(E+hbar,omega),A^dag,|Erangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img680.png"
 ALT="$\displaystyle = (E+\hbar \omega) A^\dag \vert E\rangle.$">|; 

$key = q/displaystylefrac{d}{dt}!intd^3x,,rho({bfx},t)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2648.png"
 ALT="$\displaystyle \frac{d}{dt}\!\int d^3 x  \rho({\bf x},t) = 0.$">|; 

$key = q/displaystylembox{boldmath{{alpha{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2616.png"
 ALT="$\displaystyle \mbox{\boldmath$\alpha$}$">|; 

$key = q/displaystyle|x',y',z'rangleleft(c_+,|+rangle+c_-,|-rangleright);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1057.png"
 ALT="$\displaystyle \vert x', y', z'\rangle\left(c_+  \vert+\rangle + c_-  \vert-\rangle\right)$">|; 

$key = q/displaystyleU(t_0,t_0)=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1856.png"
 ALT="$\displaystyle U(t_0, t_0 ) = 1.$">|; 

$key = q/displaystyleX,(Y,Z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img90.png"
 ALT="$\displaystyle X (Y Z)$">|; 

$key = q/v_t=v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img581.png"
 ALT="$ v_t= v$">|; 

$key = q/displaystyleexp[-{rmi},(E_i+{mitDelta}E_i),tslashhbar]exp(-{mitGamma}_i,tslash2,hbar)=intdE,f(E)exp(-{rmi},E,tslashhbar),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="500" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1991.png"
 ALT="$\displaystyle \exp[-{\rm i} (E_i + {\mit\Delta} E_i) t/\hbar] \exp( - {\mit\Gamma}_i t/2 \hbar) = \int dE f(E) \exp(-{\rm i}  E t/\hbar),$">|; 

$key = q/displaystyleP_{31}^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2414.png"
 ALT="$\displaystyle P_{31}^{ 2}$">|; 

$key = q/displaystyle|k''rangle,|k'rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2374.png"
 ALT="$\displaystyle \vert k''\rangle \vert k'\rangle,$">|; 

$key = q/displaystyle=left(!{array}{rr}1&00&-1{array}!right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1171.png"
 ALT="$\displaystyle = \left(\!\begin{array}{rr} 1 &amp;0\ 0&amp;-1\end{array}\!\right).$">|; 

$key = q/j_2,j_2-1,cdots,-j_2+1,-j_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1320.png"
 ALT="$ j_2, j_2-1,\cdots,
-j_2+1, -j_2$">|; 

$key = q/1,1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2461.png"
 ALT="$ 1,1$">|; 

$key = q/displaystyle({bfx}_{n,l})_k=langlen,k|n,l^{(1)}rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1567.png"
 ALT="$\displaystyle ({\bf x}_{n l})_k = \langle n, k\vert n, l^{(1)}\rangle,$">|; 

$key = q/displaystyleP_{irightarrow[n]}(t)=left.frac{2pi}{hbar},overline{|H_{ni}|^{,2}},rho(E_n),t,right|_{E_nsimeqE_i},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1920.png"
 ALT="$\displaystyle P_{i\rightarrow[n]} (t) = \left.\frac{2\pi}{\hbar}  \overline{\vert H_{ni}\vert^{ 2}}  \rho(E_n) t  \right\vert _{E_n\simeq E_i},$">|; 

$key = q/displaystylelangle({mitDelta}p)^2rangle=frac{hbar^2}{4,sigma^2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img511.png"
 ALT="$\displaystyle \langle ({\mit\Delta} p)^2\rangle = \frac{\hbar^2}{4 \sigma^2}.
$">|; 

$key = q/displaystyle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="9" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img639.png"
 ALT="$\displaystyle .$">|; 

$key = q/displaystylef({bfk}',{bfk})simeq-frac{2,m}{hbar^2,q}int_0^inftydr',r',V(r')sin(q,r').;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2195.png"
 ALT="$\displaystyle f({\bf k}', {\bf k}) \simeq - \frac{2 m}{\hbar^2 q} \int_0^\infty dr' r'  V(r') \sin(q  r').$">|; 

$key = q/displaystyle=[u,v_1],v_2+v_1,[u,v_2],[u,[v,w]]+[v,[w,u]]+[w,[u,v]]=0,nonumber;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="466" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img502.png"
 ALT="$\displaystyle = [u, v_1]  v_2 + v_1  [u, v_2], [u, [v, w] ]+ [v, [w, u] ] + [w, [u, v]] = 0,\nonumber$">|; 

$key = q/displaystyleP_{irightarrown}(t_0,t)=|langlen|,U_I(t_0,t),|irangle|^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1867.png"
 ALT="$\displaystyle P_{i\rightarrow n} (t_0, t) = \vert\langle n\vert  U_I(t_0, t)  \vert i\rangle\vert^{ 2}.$">|; 

$key = q/displaystylefrac{hbar}{2},exp(,{rmi},S_z,{mitDelta}varphislashhbar),(,|+ranglela|-ranglelangle+|,),exp(-{rmi},S_z,{mitDelta}varphislashhbar),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="405" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1083.png"
 ALT="$\displaystyle \frac{\hbar}{2}  \exp( {\rm i} S_z  {\mit\Delta}\varphi/\hbar...
...rt-\rangle \langle +\vert )  \exp(-{\rm i} S_z  {\mit\Delta}\varphi/\hbar),$">|; 

$key = q/|2,1,0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1576.png"
 ALT="$ \vert 2,1,0\rangle$">|; 

$key = q/P_2(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1828.png"
 ALT="$ P_2(t)$">|; 

$key = q/displaystyle=r,sintheta,cosvarphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img889.png"
 ALT="$\displaystyle = r  \sin\theta  \cos\varphi,$">|; 

$key = q/displaystyleL^+,Y_{l,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img918.png"
 ALT="$\displaystyle L^+  Y_{l m}$">|; 

$key = q/c_sslashc_{s-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2898.png"
 ALT="$ c_s/c_{s-1}$">|; 

$key = q/{bfx}_2';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2459.png"
 ALT="$ {\bf x}_2'$">|; 

$key = q/displaystylefrac{2,m}{hbar^2}frac{|V_0|}{mu,k}ll1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2216.png"
 ALT="$\displaystyle \frac{2 m}{\hbar^2} \frac{\vert V_0\vert}{\mu  k} \ll 1.$">|; 

$key = q/displaystyleX,YneqY,X.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img92.png"
 ALT="$\displaystyle X Y \neq Y X.$">|; 

$key = q/displaystylew_{irightarrown}=frac{pi}{hbar}frac{e^2}{epsilon_0,m_e^{,2},omega^2},U,|langlen|,exp[-{rmi},(omegaslashc),{bfn}cdot{bfx}],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="340" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2061.png"
 ALT="$\displaystyle w_{i\rightarrow n} = \frac{\pi}{\hbar} \frac{e^2}{\epsilon_0 m_e...
...}  U  \vert\langle n\vert  \exp[-{\rm i} (\omega/c) {\bf n}\cdot{\bf x}] $">|; 

$key = q/displaystyleepsilon=left({array}{rr}0&-{rmi}[0.5ex]{rmi}&0{array}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2858.png"
 ALT="$\displaystyle \epsilon = \left(\begin{array}{rr}0&amp;-{\rm i}\ [0.5ex]{\rm i}&amp;0\end{array}\right).$">|; 

$key = q/omega_B({bfx});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2487.png"
 ALT="$ \omega_B({\bf x})$">|; 

$key = q/sin^2(thetaslash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1292.png"
 ALT="$ \sin^2(\theta/2)$">|; 

$key = q/displaystyle=left({array}{c}0001{array}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="111" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1591.png"
 ALT="$\displaystyle = \left( \begin{array}{c} 0 \ 0\ 0 \ 1\end{array} \right).$">|; 

$key = q/v_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img279.png"
 ALT="$ v_1$">|; 

$key = q/displaystyledeltaHsimeq-e,phi+frac{1}{2,m_e},({bfp}+e,{bfA})^2+frac{e,hbar}{2,m_e},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2751.png"
 ALT="$\displaystyle \delta H\simeq -e \phi + \frac{1}{2 m_e} ({\bf p} + e {\bf A})^2 + \frac{e \hbar}{2 m_e} $">|; 

$key = q/p_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2568.png"
 ALT="$ p_y$">|; 

$key = q/displaystylelambda_{12}=lambda_{23}=lambda_{31}=+1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2443.png"
 ALT="$\displaystyle \lambda_{12} = \lambda_{23} = \lambda_{31}= +1,$">|; 

$key = q/L_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img736.png"
 ALT="$ L_y$">|; 

$key = q/langlek'|k''rangle=delta_{k',k''};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2396.png"
 ALT="$ \langle k'\vert k''\rangle = \delta_{k' k''}$">|; 

$key = q/displaystyleL^2=-hbar^2left(frac{1}{sintheta}frac{partial}{partialtheta},sintheta}+frac{1}{sin^2theta}frac{partial^2}{partialvarphi^2}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img899.png"
 ALT="$\displaystyle L^2 = - \hbar^2\left( \frac{1}{\sin\theta}\frac{\partial}{\partia...
...al\theta} + \frac{1}{\sin^2\theta}\frac{\partial^2} {\partial\varphi^2}\right).$">|; 

$key = q/displaystylefrac{{mitDelta}lambda}{lambda}simeq4times10^{-8}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2127.png"
 ALT="$\displaystyle \frac{{\mit\Delta}\lambda}{\lambda} \simeq 4\times 10^{-8}.
$">|; 

$key = q/displaystyleS_xleft[1-frac{({mitDelta}varphi)^2}{2!}+frac{({mitDelta}varphi)^4}{ta}varphi)^3}{3!}+frac{({mitDelta}varphi)^5}{5!}+cdotsright],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="481" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1091.png"
 ALT="$\displaystyle S_x\left[ 1- \frac{({\mit\Delta}\varphi)^2}{2!} + \frac{({\mit\De...
...{\mit\Delta}\varphi)^3}{3!}+ \frac{({\mit\Delta}\varphi)^5}{5!} +\cdots\right],$">|; 

$key = q/displaystylelim_{deltaxrightarrow0}frac{D,exp({rmi},gamma)-1}{deltax}=lim_{deltaxrightarrow0}frac{D-1+{rmi},gamma}{deltax}=d_x+{rmi},a_x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="386" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img467.png"
 ALT="$\displaystyle \lim_{\delta x\rightarrow 0 }\frac{D \exp({\rm i} \gamma)-1}{\d...
...elta x\rightarrow 0 }\frac{D-1+{\rm i}  \gamma}{\delta x}= d_x + {\rm i} a_x,$">|; 

$key = q/trightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1903.png"
 ALT="$ t\rightarrow \infty$">|; 

$key = q/displaystylelanglexi'|,xi=langlexi'|,xi'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img126.png"
 ALT="$\displaystyle \langle \xi'\vert \xi = \langle \xi'\vert \xi'.$">|; 

$key = q/m_1=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2475.png"
 ALT="$ m_1=1/2$">|; 

$key = q/displaystyle=intd^3x_1intd^3x_2,|psi_{100}({bfx}_1)|^{,2},|psi_{nlm}({bfx}_2)|^{,2},frac{e^2}{4pi,epsilon_0,r_{12}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="376" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2530.png"
 ALT="$\displaystyle = \int d^3 x_1\int d^3 x_2  \vert\psi_{100}({\bf x}_1)\vert^{ 2} \vert\psi_{nlm}({\bf x}_2)\vert^{ 2} \frac{e^2}{4\pi \epsilon_0 r_{12}},$">|; 

$key = q/displaystylesigma_l=frac{4pi}{k^2},(2,l+1),sin^2delta_l=frac{4pi}{k^2},(2,l+1),frac{1}{1+cot^2delta_l}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="384" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2348.png"
 ALT="$\displaystyle \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}.$">|; 

$key = q/displaystyle=langle+|,S_k,|+ranglelangle+|Arangle+langle+|,S_k,|-ranglelangle-|Arangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1158.png"
 ALT="$\displaystyle = \langle + \vert S_k \vert +\rangle \langle +\vert A\rangle + \langle +\vert S_k  \vert-\rangle \langle -\vert A\rangle,$">|; 

$key = q/displaystyle[{bfx},H_0]=frac{{rmi},hbar,{bfp}}{m_e},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2075.png"
 ALT="$\displaystyle [{\bf x}, H_0] = \frac{{\rm i}  \hbar  {\bf p}}{m_e},$">|; 

$key = q/displaystyle|n,lrangle'=|n,lrangle+sum_{n',l'neqn,l}frac{e_{n'l'nl}}{E_n-E_{n'}},|n',l'rangle+O(epsilon^2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1550.png"
 ALT="$\displaystyle \vert n, l\rangle' = \vert n,l\rangle + \sum_{n', l'\neq n, l} \frac{e_{n'l'nl}}{E_n-E_{n'}} \vert n',l'\rangle + O(\epsilon^2),$">|; 

$key = q/x=x';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img196.png"
 ALT="$ x=x'$">|; 

$key = q/displaystyle[u_1,u_2,v]_{cl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img263.png"
 ALT="$\displaystyle [u_1  u_2, v]_{cl}$">|; 

$key = q/displaystyleP(r)=f(y)exp(-y).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img949.png"
 ALT="$\displaystyle P(r) = f(y) \exp(-y).$">|; 

$key = q/{rmsinc}(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1899.png"
 ALT="$ {\rm sinc}(x)$">|; 

$key = q/D({mitDelta}x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img486.png"
 ALT="$ D({\mit\Delta} x)$">|; 

$key = q/exp(-{rmi},H);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img226.png"
 ALT="$ \exp(-{\rm i} H)$">|; 

$key = q/lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img902.png"
 ALT="$ \lambda$">|; 

$key = q/displaystyle=-nablaphi-frac{partial{bfA}}{partialt},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2029.png"
 ALT="$\displaystyle = - \nabla\phi - \frac{\partial {\bf A}}{\partial t},$">|; 

$key = q/displaystylecdot{bfB}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2745.png"
 ALT="$\displaystyle \cdot{\bf B}.$">|; 

$key = q/displaystyle[H,P_{123}]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2433.png"
 ALT="$\displaystyle [H,P_{123}] = 0.$">|; 

$key = q/displaystylea^0,b^0-a^1,b^1-a^2,a^2-a^3,b^3=a^{,mu},b_mu=a_mu,b^{,mu},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2554.png"
 ALT="$\displaystyle a^0 b^0-a^1 b^1-a^2 a^2-a^3 b^3 = a^{ \mu} b_\mu= a_\mu b^{ \mu},$">|; 

$key = q/displaystyle=frac{1}{sqrt{2}},(chi_+,chi_--chi_-,chi_+).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2473.png"
 ALT="$\displaystyle = \frac{1}{\sqrt{2}} (\chi_+ \chi_-\chi_- \chi_+).$">|; 

$key = q/displaystyle{bfJ}={bfL}+frac{hbar}{2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2786.png"
 ALT="$\displaystyle {\bf J} = {\bf L} + \frac{\hbar}{2} $">|; 

$key = q/|m_l,m_srangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1638.png"
 ALT="$ \vert m_l, m_s\rangle$">|; 

$key = q/displaystylelangleq_1'cdotsq_N'|,p_i=-{rmi},hbar,frac{partial}{partialq_i'},langleq_1'cdotsq_N'|.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img398.png"
 ALT="$\displaystyle \langle q_1'\cdots q_N'\vert  p_i = -{\rm i} \hbar \frac{\partial}{\partial q_i'}  \langle q_1'\cdots q_N'\vert.$">|; 

$key = q/displaystylephi({bfx}_1,{bfx}_2)=frac{1}{sqrt{2}}left[psi_{nlm}({bfx}_1),psi_{n'm'}({bfx}_2)pmpsi_{nlm}({bfx}_2),psi_{n'l'm'}({bfx}_1)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="442" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2497.png"
 ALT="$\displaystyle \phi({\bf x}_1,{\bf x}_2)=\frac{1}{\sqrt{2}}\left[\psi_{nlm}({\bf...
..._{n'l'm'}({\bf x}_2)\pm \psi_{nlm}({\bf x}_2) \psi_{n'l'm'}({\bf x}_1)\right],$">|; 

$key = q/S_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1289.png"
 ALT="$ S_y$">|; 

$key = q/displaystylew_{2rightarrow1}^{rmspn}simalpha^3,omega_{21},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2119.png"
 ALT="$\displaystyle w_{2\rightarrow 1}^{\rm spn} \sim \alpha^3 \omega_{21},$">|; 

$key = q/displaystylesimeqleft(frac{-{rmi}}{hbar}right)H_{ii}+lim_{etarightarrow0}left(frright)sum_{mneqi}frac{|H_{mi}|^{,2}}{E_i-E_m+{rmi},hbar,eta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1969.png"
 ALT="$\displaystyle \simeq \left(\frac{-{\rm i}}{\hbar}\right) H_{ii} + \lim_{\eta\ri...
...sum_{m\neq i} \frac{\vert H_{mi}\vert^{ 2}}{E_i - E_m+ {\rm i} \hbar  \eta}.$">|; 

$key = q/displaystyle=2,{rmi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1177.png"
 ALT="$\displaystyle = 2 {\rm i} $">|; 

$key = q/displaystyleH,|k'',k''',k'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2417.png"
 ALT="$\displaystyle H \vert k'' k''' k'\rangle$">|; 

$key = q/displaystylechi'=left(frac{hbar}{2}right)sigma_k,chi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1188.png"
 ALT="$\displaystyle \chi' = \left(\frac{\hbar}{2}\right) \sigma_k  \chi.$">|; 

$key = q/j_l(k,r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2226.png"
 ALT="$ j_l(k r)$">|; 

$key = q/displaystyleu_l(0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2279.png"
 ALT="$\displaystyle u_l(0) = 0$">|; 

$key = q/displaystyle=-1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2558.png"
 ALT="$\displaystyle =-1,$">|; 

$key = q/displaystyle=1+frac{1}{8},[gamma_mu,gamma_nu],{mitDelta}omega^{,mu,nu},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2695.png"
 ALT="$\displaystyle = 1 + \frac{1}{8} [\gamma_\mu,\gamma_\nu] {\mit\Delta}\omega^{ \mu \nu},$">|; 

$key = q/overline{|H_{ni}|^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="49" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1922.png"
 ALT="$ \overline{\vert H_{ni}\vert^{ 2}}$">|; 

$key = q/2,gammaslashhbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1831.png"
 ALT="$ 2 \gamma/\hbar$">|; 

$key = q/Sigma^{,2}=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2794.png"
 ALT="$ \Sigma^{ 2}=3$">|; 

$key = q/displaystyleP(xi')=frac{|langleA|xi'rangle|^{,2}}{sum_{xi'}|langleA|xi'rangle|^{,2}}=frac{|langleA|xi'rangle|^{,2}}{langleA|Arangle}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img140.png"
 ALT="$\displaystyle P(\xi') = \frac{ \vert\langle A\vert \xi'\rangle\vert^{ 2}}{\sum...
...}} =\frac{\vert\langle A\vert \xi'\rangle\vert^{ 2}}{\langle A\vert A\rangle}.$">|; 

$key = q/displaystyletan(k,a+delta_0)=frac{k}{kappa},tanh(kappa,a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2324.png"
 ALT="$\displaystyle \tan(k a+ \delta_0) = \frac{k}{\kappa}  \tanh (\kappa  a)$">|; 

$key = q/displaystyleS_z,|s,s_zrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1020.png"
 ALT="$\displaystyle S_z  \vert s, s_z\rangle$">|; 

$key = q/displaystylebeta_{l+}=beta_{l-}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2282.png"
 ALT="$\displaystyle \beta_{l+} = \beta_{l-}.$">|; 

$key = q/displaystylecdot{bfx}),({bfx}cdot{bfp})-({bfx}cdot{bfp}),(;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2845.png"
 ALT="$\displaystyle \cdot{\bf x}) ({\bf x}\cdot{\bf p}) - ({\bf x}\cdot{\bf p}) ($">|; 

$key = q/omega_{21}pm2,gammaslashhbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1832.png"
 ALT="$ \omega_{21} \pm 2 \gamma/\hbar$">|; 

$key = q/displaystyleT(t,t_0)=expleft[-{rmi},H,(t-t_0)slashhbarright].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img578.png"
 ALT="$\displaystyle T(t, t_0) = \exp\left[-{\rm i}  H (t-t_0)/\hbar \right].$">|; 

$key = q/f(l,m)=l,(l+1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img785.png"
 ALT="$ f(l,m) = l (l+1)$">|; 

$key = q/displaystylefrac{E}{m_e,c^2}=left{1+frac{alpha^2}{left(i+[(j+1slash2)^2-alpha^2]^{1slash2}right)^2}right}^{-1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="354" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2915.png"
 ALT="$\displaystyle \frac{E}{m_e c^2} =\left\{1 + \frac{\alpha^2}{\left(i+[(j+1/2)^2-\alpha^2]^{1/2}\right)^2}\right\}^{-1/2}.$">|; 

$key = q/displaystylec_i^{(0)}(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1958.png"
 ALT="$\displaystyle c_i^{(0)} (t)$">|; 

$key = q/displaystyle=-{rmi},hbar,(L_x,L_y+L_y,L_x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img763.png"
 ALT="$\displaystyle = -{\rm i} \hbar (L_x  L_y + L_y  L_x),$">|; 

$key = q/|l+1slash2,l+1slash2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1650.png"
 ALT="$ \vert l+1/2, l+1/2\rangle$">|; 

$key = q/displaystylec_n^{(2)}(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1878.png"
 ALT="$\displaystyle c_n^{(2)}(t)$">|; 

$key = q/chi';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1230.png"
 ALT="$ \chi'$">|; 

$key = q/displaystyle=langle-|,S_k,|+ranglelangle+|Arangle+langle-|,S_k,|-ranglelangle-|Arangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1160.png"
 ALT="$\displaystyle = \langle - \vert S_k \vert +\rangle \langle +\vert A\rangle + \langle -\vert S_k  \vert-\rangle \langle -\vert A\rangle,$">|; 

$key = q/displaystyleP_{irightarrown}(t)=|c_n^{(1)}|^{,2}=frac{|H_{ni}|^{,2}}{hbar^2}frac{exp(2,eta,t)}{eta^2+omega_{ni}^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1952.png"
 ALT="$\displaystyle P_{i\rightarrow n}(t) = \vert c_n^{(1)}\vert^{ 2} = \frac{\vert H_{ni}\vert^{ 2}}{\hbar^2} \frac{\exp(2  \eta  t)}{\eta^2 + \omega_{ni}^{ 2}}.$">|; 

$key = q/displaystylefrac{partial}{partialq_i},frac{partial}{partialq_j}=frac{partial}{partialq_j},frac{partial}{partialq_i}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img391.png"
 ALT="$\displaystyle \frac{\partial}{\partial q_i} \frac{\partial}{\partial q_j} = \frac{\partial }{\partial q_j}  \frac{\partial }{\partial q_i}.$">|; 

$key = q/displaystyle=1-frac{1}{8},[gamma_mu,gamma_nu],{mitDelta}omega^{,mu,nu}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2697.png"
 ALT="$\displaystyle = 1 - \frac{1}{8} [\gamma_\mu,\gamma_\nu] {\mit\Delta}\omega^{ \mu \nu}.$">|; 

$key = q/displaystylelangle+|-rangle=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1033.png"
 ALT="$\displaystyle \langle +\vert -\rangle = 0.$">|; 

$key = q/displaystyleH=frac{p^2}{2,m}+V(r).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1003.png"
 ALT="$\displaystyle H = \frac{p^2}{2 m} + V(r).
$">|; 

$key = q/3s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1682.png"
 ALT="$ 3s$">|; 

$key = q/x_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img506.png"
 ALT="$ x_0$">|; 

$key = q/tandelta_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2291.png"
 ALT="$ \tan\delta_l$">|; 

$key = q/displaystyleP_{123},|k',k'',k'''rangle=|k''',k',k''rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2431.png"
 ALT="$\displaystyle P_{123}  \vert k' k'' k'''\rangle =\vert k''' k' k''\rangle.$">|; 

$key = q/langlen,l,m|,x,|n',l',m'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2124.png"
 ALT="$ \langle n,l,m\vert x \vert n',l',m'\rangle$">|; 

$key = q/displaystylecdot{bfL}+hbar)]^{,2}=(;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2825.png"
 ALT="$\displaystyle \cdot{\bf L}+\hbar)]^{ 2} = ($">|; 

$key = q/displaystyle=a^nu_{~mu},x_{nu'}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2663.png"
 ALT="$\displaystyle = a^\nu_{&nbsp;\mu} x_{\nu'}.$">|; 

$key = q/m_j=m_l+m_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1639.png"
 ALT="$ m_j=m_l+m_s$">|; 

$key = q/displaystyle[L^2,L_z]=[L_x^{,2},L_z]+[L_y^{,2},L_z]+[L_z^{,2},L_z].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img761.png"
 ALT="$\displaystyle [L^2, L_z] = [L_x^{ 2}, L_z] + [L_y^{ 2}, L_z] + [L_z^{ 2}, L_z].$">|; 

$key = q/displaystylep_x={rmi},hbar,d_x.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img485.png"
 ALT="$\displaystyle p_x = {\rm i} \hbar  d_x.$">|; 

$key = q/displaystylelangleS_xranglerightarrowlangleA_R|,S_x,|A_Rrangle=langleA|,T_z^{dag},S_x,T_z,|Arangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1080.png"
 ALT="$\displaystyle \langle S_x\rangle \rightarrow \langle A_R\vert  S_x  \vert A_R \rangle = \langle A\vert  T_z^{\dag }  S_x  T_z  \vert A\rangle.$">|; 

$key = q/j_2,(j_2+1),hbar^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1310.png"
 ALT="$ j_2 (j_2+1)  \hbar^2$">|; 

$key = q/displaystyle=[(y,p_z-z,p_y),(z,p_x-x,p_z)]=y,[p_z,z],p_x+x,p_y,[z,p_z];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="425" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img743.png"
 ALT="$\displaystyle = [(y p_z-z p_y), (z p_x-x  p_z)] = y [p_z, z] p_x + x p_y [z, p_z]$">|; 

$key = q/sigma_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1271.png"
 ALT="$ \sigma_x$">|; 

$key = q/langleA|xi'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img130.png"
 ALT="$ \langle A\vert \xi'\rangle$">|; 

$key = q/n=n'=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2500.png"
 ALT="$ n=n'=1$">|; 

$key = q/displaystyletau^{dag}+tau=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img549.png"
 ALT="$\displaystyle \tau^{\dag } + \tau = 0.$">|; 

$key = q/displaystyleL^2,|l,mrangle=f(l,m),hbar^2,|l,mrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img783.png"
 ALT="$\displaystyle L^2  \vert l, m\rangle = f(l,m)  \hbar^2 \vert l, m\rangle,$">|; 

$key = q/displaystyle=E,|k''',k'',k'rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2426.png"
 ALT="$\displaystyle = E \vert k''' k'' k'\rangle,$">|; 

$key = q/p_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2569.png"
 ALT="$ p_z$">|; 

$key = q/j=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1703.png"
 ALT="$ j=1/2$">|; 

$key = q/displaystyleV(r)=left(frac{hbar^2}{2,m}right)gamma,delta(r-R),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2361.png"
 ALT="$\displaystyle V(r) = \left(\frac{\hbar^2}{2 m}\right)\gamma \delta(r-R),
$">|; 

$key = q/displaystylesimeqfrac{c_{s-1}}{a}+frac{c_{s-1}'}{a_2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2900.png"
 ALT="$\displaystyle \simeq \frac{c_{s-1}}{a}+\frac{c_{s-1}'}{a_2},$">|; 

$key = q/displaystyleE_1';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1457.png"
 ALT="$\displaystyle E_1'$">|; 

$key = q/{bfp}rightarrow-{rmi},hbar,nabla;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1237.png"
 ALT="$ {\bf p} \rightarrow -{\rm i} \hbar \nabla$">|; 

$key = q/displaystyle(E_n+e_{nn}-E)-sum_{kneqn}frac{|e_{nk}|^{,2}}{E_k-E_n}simeq0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1489.png"
 ALT="$\displaystyle (E_n + e_{nn} - E) - \sum_{k\neq n} \frac{\vert e_{nk}\vert^{ 2}} {E_k-E_n} \simeq 0.$">|; 

$key = q/displaystylex={rmi},hbar,frac{d}{dp_x}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img413.png"
 ALT="$\displaystyle x = {\rm i} \hbar  \frac{d}{dp_x}.$">|; 

$key = q/langlen|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1793.png"
 ALT="$ \langle n\vert$">|; 

$key = q/n_{rmmin}=-l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img959.png"
 ALT="$ n_{\rm min} = -l$">|; 

$key = q/displaystyleR_z({mitDelta}varphi_y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img872.png"
 ALT="$\displaystyle R_z({\mit\Delta}\varphi_y)$">|; 

$key = q/displaystyleg(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2876.png"
 ALT="$\displaystyle g(y)$">|; 

$key = q/c_1(0)=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1810.png"
 ALT="$ c_1(0) = 1$">|; 

$key = q/displaystyle=-frac{e}{2,m_e},({bfL}+2,{bfS});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1691.png"
 ALT="$\displaystyle = - \frac{e}{2 m_e}  ({\bf L} + 2  {\bf S})$">|; 

$key = q/displaystyle{mitDelta}E=-frac{1}{2},alpha,|{bfE}|^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1529.png"
 ALT="$\displaystyle {\mit\Delta} E = - \frac{1}{2}  \alpha  \vert{\bf E}\vert^{ 2}.$">|; 

$key = q/displaystyleH_1=e,|{bfE}|,z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1494.png"
 ALT="$\displaystyle H_1= e  \vert{\bf E}\vert  z.$">|; 

$key = q/displaystylehbar,cleft(frac{d}{dr}+frac{k+1}{r}right)psi_b+(E-m_e,c^2+e,phi),psi_a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="322" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2860.png"
 ALT="$\displaystyle \hbar c\left(\frac{d}{d r} + \frac{k+1}{r}\right)\psi_b + (E-m_e c^2+e \phi) \psi_a$">|; 

$key = q/displaystyle{rmi},hbar,dot{alpha}_1=2,alpha_1,H-2,c,p^1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2717.png"
 ALT="$\displaystyle {\rm i} \hbar \dot{\alpha}_1 = 2 \alpha_1 H-2 c p^1.$">|; 

$key = q/displaystyle=-{rmi},hbar,frac{partial}{partialvarphi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img896.png"
 ALT="$\displaystyle = -{\rm i} \hbar \frac{\partial}{\partial\varphi}.$">|; 

$key = q/{bfL}={bfx}times{bfp};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img727.png"
 ALT="$ {\bf L} = {\bf x}\times{\bf p}$">|; 

$key = q/displaystylealpha<frac{16}{3},4pi,epsilon_0,a_0sum_{n>1}|langle1,0,0|,z,|n,1,0rangle|^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1536.png"
 ALT="$\displaystyle \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}.$">|; 

$key = q/_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1427.png"
 ALT="$ _p$">|; 

$key = q/displaystyleH=frac{hbar^2}{2,M}left(-frac{1}{r^2}frac{partial}{partialr},r^2,frac{partial}{partialr}+frac{L^2}{hbar^2r^2}right)+V(r).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img928.png"
 ALT="$\displaystyle H= \frac{\hbar^2}{2 M}\left(- \frac{1}{r^2}\frac{\partial}{\partial r}  r^2 \frac{\partial}{\partial r} + \frac{L^2}{\hbar^2 r^2}\right) + V(r).$">|; 

$key = q/displaystyle{bfL}times{bfL}={rmi},hbar,{bfL}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img750.png"
 ALT="$\displaystyle {\bf L}\times {\bf L} = {\rm i} \hbar  {\bf L}.$">|; 

$key = q/displaystyleleft.-|k'''rangle,|k''rangle,|k'rangle-|k'rangle,|k'''rangle,|k''rangle-|k''rangle,|k'rangle,|k'''rangleright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2449.png"
 ALT="$\displaystyle \left.-\vert k'''\rangle \vert k''\rangle \vert k'\rangle - \ve...
...,\vert k''\rangle- \vert k''\rangle \vert k'\rangle \vert k'''\rangle\right),$">|; 

$key = q/|{bfx'}|rightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img651.png"
 ALT="$ \vert{\bf x'}\vert\rightarrow
\infty$">|; 

$key = q/displaystyleleftlanglefrac{p_x^{,2}}{2,m}rightrangle=-E+langleVrangle+leftlanglex,frac{dV}{dx}rightrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img723.png"
 ALT="$\displaystyle \left\langle \frac{p_x^{ 2}}{2 m}\right\rangle = -E + \langle V\rangle+\left\langle x \frac{dV}{dx}\right\rangle.
$">|; 

$key = q/displaystyleleft{{mitDelta}xi,{mitDelta}etaright};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img184.png"
 ALT="$\displaystyle \left\{ {\mit\Delta} \xi, {\mit\Delta} \eta\right\}$">|; 

$key = q/cotdelta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2366.png"
 ALT="$ \cot \delta_0$">|; 

$key = q/displaystyletimes;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1176.png"
 ALT="$\displaystyle \times$">|; 

$key = q/displaystyleintdxi',|xi'ranglelanglexi'|=1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img235.png"
 ALT="$\displaystyle \int d\xi'  \vert\xi'\rangle\langle \xi'\vert= 1,
$">|; 

$key = q/c_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="$ c_1$">|; 

$key = q/ineqn;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1886.png"
 ALT="$ i\neq n$">|; 

$key = q/j=l-1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1632.png"
 ALT="$ j=l-1/2$">|; 

$key = q/psi_{nlmpm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1673.png"
 ALT="$ \psi_{nlm\pm}$">|; 

$key = q/displaystyle=-[v,u],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img493.png"
 ALT="$\displaystyle = - [v, u],$">|; 

$key = q/displaystylealpha=frac{9}{2},4pi,epsilon_0,a_0^{,3}=4.5,4pi,epsilon_0,a_0^{,3}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1540.png"
 ALT="$\displaystyle \alpha = \frac{9}{2}  4\pi  \epsilon_0  a_0^{ 3} = 4.5 4\pi  \epsilon_0  a_0^{ 3}.$">|; 

$key = q/R_z({mitDelta}varphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1075.png"
 ALT="$ R_z({\mit\Delta}\varphi)$">|; 

$key = q/r_2=|{bfx}_2|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2491.png"
 ALT="$ r_2=\vert{\bf x}_2\vert$">|; 

$key = q/x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$ x$">|; 

$key = q/displaystylelanglei|(|jrangle)=delta_{i,j}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="$\displaystyle \langle i\vert(\vert j\rangle) = \delta_{i j}.$">|; 

$key = q/displaystylelanglen|,exp[,{rmi},(omegaslashc),{bfn}cdot{bfx}],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2072.png"
 ALT="$\displaystyle \langle n\vert  \exp[ {\rm i} (\omega/c) {\bf n}\cdot{\bf x}] $">|; 

$key = q/displaystyle{mitDelta}l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2085.png"
 ALT="$\displaystyle {\mit\Delta} l$">|; 

$key = q/gamma^5,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2812.png"
 ALT="$ \gamma^5 $">|; 

$key = q/m,hbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img841.png"
 ALT="$ m \hbar$">|; 

$key = q/displaystyleH=m_e,c^2+deltaH,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2746.png"
 ALT="$\displaystyle H = m_e c^2+ \delta H,$">|; 

$key = q/displaystylecdot{bfL}),(;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2788.png"
 ALT="$\displaystyle \cdot {\bf L}) ($">|; 

$key = q/displaystylecdot{bfp})=r,p_r+{rmi},(beta,zeta-hbar).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2853.png"
 ALT="$\displaystyle \cdot{\bf p}) = r p_r+{\rm i} (\beta \zeta-\hbar).$">|; 

$key = q/displaystyleT({mitDelta}phi)=expleft(-{rmi},{bfS}cdot{bfn},{mitDelta}varphislashhbarright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1213.png"
 ALT="$\displaystyle T ({\mit\Delta}\phi) = \exp\left(-{\rm i}  {\bf S}\cdot{\bf n} {\mit\Delta}\varphi/\hbar\right),$">|; 

$key = q/displaystyle=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img268.png"
 ALT="$\displaystyle = 0.$">|; 

$key = q/displaystyle{bfL}cdot{bfS}=frac{J^{,2}-L^2-S^{,2}}{2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1628.png"
 ALT="$\displaystyle {\bf L} \cdot{\bf S} = \frac{J^{ 2} - L^2 - S^{ 2}}{2}.$">|; 

$key = q/langlej_1,j_2;m_1,m_2|j_1,j_2;j,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1363.png"
 ALT="$ \langle j_1,j_2;m_1,m_2\vert j_1,j_2;j,m\rangle$">|; 

$key = q/[L^2,L_z]=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img990.png"
 ALT="$ [L^2, L_z] = 0$">|; 

$key = q/displaystyle{bfS}times{bfS}={rmi},hbar,{bfS}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1014.png"
 ALT="$\displaystyle {\bf S} \times {\bf S} = {\rm i} \hbar   {\bf S}.$">|; 

$key = q/displaystylefrac{dx_i}{dt}=frac{p_i}{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img611.png"
 ALT="$\displaystyle \frac{d x_i}{dt} = \frac{p_i}{m}$">|; 

$key = q/(1slash2)(x,p_x+p_x,x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img584.png"
 ALT="$ (1/2)(x p_x + p_x x)$">|; 

$key = q/m_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1707.png"
 ALT="$ m_s$">|; 

$key = q/displaystyleomega_{21};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1847.png"
 ALT="$\displaystyle \omega_{21}$">|; 

$key = q/displaystylelangleL^pmrangle=langleS^pmrangle=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1720.png"
 ALT="$\displaystyle \langle L^\pm \rangle = \langle S^\pm\rangle = 0$">|; 

$key = q/displaystyleJ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2531.png"
 ALT="$\displaystyle J$">|; 

$key = q/I=q,vslash2pi,r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1104.png"
 ALT="$ I=q v/2\pi  r$">|; 

$key = q/displaystyle{bfp}cdot{bfA}={bfA}cdot{bfp},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2036.png"
 ALT="$\displaystyle {\bf p}\cdot {\bf A} = {\bf A}\cdot {\bf p},$">|; 

$key = q/N_ntimesN_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1562.png"
 ALT="$ N_n\times N_n$">|; 

$key = q/r_1=|{bfx}_1|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2490.png"
 ALT="$ r_1=\vert{\bf x}_1\vert$">|; 

$key = q/displaystylelanglepsi_A^{,ast}(x)stackrel{rmDC}{longleftrightarrow}psi_A(x)rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img326.png"
 ALT="$\displaystyle \langle \psi_A^{ \ast}(x) \stackrel{\rm DC}{\longleftrightarrow} \psi_A(x)\rangle.$">|; 

$key = q/displaystyleA_0(r)=B,frac{sinh(kappa,r)}{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2320.png"
 ALT="$\displaystyle A_0(r) = B  \frac{\sinh(\kappa  r)}{r},$">|; 

$key = q/displaystyleE_0=-frac{mu,e^4}{32pi^2,epsilon_0^{,2},hbar^2}=-13.6,{rmeV};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img975.png"
 ALT="$\displaystyle E_0 = - \frac{\mu  e^4}{32\pi^2 \epsilon_0^{ 2}  \hbar^2} = - 13.6 {\rm eV}$">|; 

$key = q/P_{irightarrown};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1901.png"
 ALT="$ P_{i\rightarrow n}$">|; 

$key = q/displaystyle|2rangle-frac{e_{12}}{E_1-E_2},|1rangle+cdots.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1464.png"
 ALT="$\displaystyle \vert 2\rangle - \frac{e_{12}}{E_1-E_2}  \vert 1\rangle +\cdots.$">|; 

$key = q/displaystyledelta_l(E_0)=frac{pi}{2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2343.png"
 ALT="$\displaystyle \delta_l(E_0) = \frac{\pi}{2}.$">|; 

$key = q/displaystyleleft({array}{rr}cos({mitDelta}varphislash2)-{rmi},n_zsin({mitDelta}vphislash2)+{rmi},n_zsin({mitDelta}varphislash2){array}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="461" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1228.png"
 ALT="$\displaystyle \left(\begin{array}{rr} \cos({\mit\Delta}\varphi/2) - {\rm i} n_...
...lta}\varphi/2) + {\rm i}  n_z \sin({\mit\Delta}\varphi/2) \end{array} \right).$">|; 

$key = q/varphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img849.png"
 ALT="$ \varphi$">|; 

$key = q/displaystyle[q_i,q_j];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img293.png"
 ALT="$\displaystyle [q_i, q_j]$">|; 

$key = q/displaystylerightarrow(1,0)equivchi_+^{dag},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1147.png"
 ALT="$\displaystyle \rightarrow (1, 0) \equiv \chi_+^{\dag },$">|; 

$key = q/displaystylecdot{bfx})^2=frac{1}{2}sum_{i,j=1,3}{alpha_i,alpha_j},x^i,x^j=sum_{i=1,3}(x^i)^{,2}=r^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2841.png"
 ALT="$\displaystyle \cdot{\bf x})^2 = \frac{1}{2}\sum_{i,j=1,3}\{\alpha_i,\alpha_j\} x^i x^j = \sum_{i=1,3}(x^i)^{ 2}=r^2,$">|; 

$key = q/x,p_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img583.png"
 ALT="$ x p_x$">|; 

$key = q/j^{,mu}rightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2935.png"
 ALT="$ j^{ \mu}\rightarrow 0$">|; 

$key = q/mu=0,1,2,3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2542.png"
 ALT="$ \mu= 0,1,2,3$">|; 

$key = q/displaystyleE=frac{(E_1+E_2)pmsqrt{(E_1-E_2)^{,2}+4,|e_{12}|^{,2}}}{2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="314" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1454.png"
 ALT="$\displaystyle E = \frac{(E_1+E_2) \pm \sqrt{(E_1-E_2)^{ 2} + 4 \vert e_{12}\vert^{ 2}}}{2}.$">|; 

$key = q/partialslashpartialq_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img386.png"
 ALT="$ \partial/\partial q_i$">|; 

$key = q/displaystylew_{irightarrown}=frac{pi}{hbar^2}frac{e^2}{epsilon_0,m_e^{,2},omega_ni}),|langlen|,exp[,{rmi},(omega_{ni}slashc),{bfn}cdot{bfx}],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="383" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2065.png"
 ALT="$\displaystyle w_{i\rightarrow n} = \frac{\pi}{\hbar^2} \frac{e^2}{\epsilon_0 m...
..., \vert\langle n\vert  \exp[ {\rm i} (\omega_{ni}/c) {\bf n}\cdot{\bf x}] $">|; 

$key = q/displaystylembox{hspace{1.75cm}for{tgeq0{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1892.png"
 ALT="$\displaystyle \mbox{\hspace{1.75cm}for $t\geq 0$}$">|; 

$key = q/alpha_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2921.png"
 ALT="$ \alpha_2$">|; 

$key = q/displaystyleR_y({mitDelta}varphi_y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img870.png"
 ALT="$\displaystyle R_y({\mit\Delta}\varphi_y)$">|; 

$key = q/displaystyleleft(frac{H}{c}+frac{e}{c},phiright)^2=({bfp}+e,{bfA})^2+m_e^{,2},c^2+e,hbar,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2744.png"
 ALT="$\displaystyle \left(\frac{H}{c}+\frac{e}{c} \phi\right)^2 = ({\bf p}+e {\bf A})^2 + m_e^{ 2} c^2+e \hbar $">|; 

$key = q/displaystyle|f(E)|^{,2}proptofrac{1}{(E-[E_i+{rmRe}({mitDelta}_i)])^2+{mitGamma}_i^{,2}slash4}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1993.png"
 ALT="$\displaystyle \vert f(E)\vert^{ 2} \propto \frac{1}{(E - [E_i +{\rm Re}({\mit\Delta}_i)])^2 + {\mit\Gamma}_i^{ 2}/4}.$">|; 

$key = q/displaystyleE_{n00}-E_{100}geqE_{200}-E_{100}=frac{3}{4}frac{e^2}{8pi,epsilon_0,a_0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="301" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1535.png"
 ALT="$\displaystyle E_{n00}-E_{100} \geq E_{200} - E_{100} = \frac{3}{4} \frac{e^2}{8\pi \epsilon_0  a_0}.$">|; 

$key = q/e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img204.png"
 ALT="$ e$">|; 

$key = q/displaystyleA=sqrt{frac{m,omega}{2,hbar}},x+{rmi},frac{p_x}{sqrt{2,m,omega,hbar}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img665.png"
 ALT="$\displaystyle A = \sqrt{\frac{m \omega}{2 \hbar}} x + {\rm i} \frac{p_x}{\sqrt{2 m \omega \hbar}}.
$">|; 

$key = q/displaystyleH_B=-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1689.png"
 ALT="$\displaystyle H_B = -$">|; 

$key = q/2p(m=-1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1610.png"
 ALT="$ 2p(m=-1)$">|; 

$key = q/t+dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1910.png"
 ALT="$ t+dt$">|; 

$key = q/displaystylefrac{dsigma}{d{mitOmega}}simeqleft(frac{2,m,Z,Z',e^2}{4pi,epsilon_0,hbar^2}right)^2frac{1}{16,k^4,sin^4(thetaslash2)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2205.png"
 ALT="$\displaystyle \frac{d\sigma}{d{\mit\Omega}} \simeq \left(\frac{2 m  Z  Z'  e^2}{4\pi \epsilon_0 \hbar^2}\right)^2 \frac{1}{ 16  k^4  \sin^4( \theta/2)}.$">|; 

$key = q/displaystylefrac{partial}{partialq_i},frac{partial}{partialq_j},psirangle=frac{prac{partial}{partialq_j},frac{partial}{partialq_i},psirangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img390.png"
 ALT="$\displaystyle \frac{\partial}{\partial q_i} \frac{\partial}{\partial q_j}  \p...
... \frac{\partial }{\partial q_j}  \frac{\partial }{\partial q_i}  \psi\rangle,$">|; 

$key = q/displaystyle|n,l^{(1)}rangle=sum_{k=1,N_n}langlen,k|n,l^{(1)}rangle|n,krangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1555.png"
 ALT="$\displaystyle \vert n,l^{(1)}\rangle = \sum_{k=1,N_n} \langle n,k\vert n,l^{(1)}\rangle \vert n,k\rangle.$">|; 

$key = q/displaystylex_d=x-deltax.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img479.png"
 ALT="$\displaystyle x_d = x -\delta x.$">|; 

$key = q/displaystyledot{x}=frac{[x,H]}{{rmi},hbar}=c,alpha_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2706.png"
 ALT="$\displaystyle \dot{x} = \frac{[x,H]}{{\rm i} \hbar} = c \alpha_1,$">|; 

$key = q/omega+domega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2064.png"
 ALT="$ \omega+d\omega$">|; 

$key = q/displaystylew_{irightarrow[n]}=frac{dP_{irightarrow[n]}}{dt},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1925.png"
 ALT="$\displaystyle w_{i\rightarrow [n]} = \frac{d P_{i\rightarrow [n]}}{dt},$">|; 

$key = q/c_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img522.png"
 ALT="$ c_n$">|; 

$key = q/displaystyleV_{rmeff}(r)=-frac{e^2}{4pi,epsilon_0,r}+frac{l,(l+1),hbar^2}{2,mu,r^2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img946.png"
 ALT="$\displaystyle V_{\rm eff}(r) = -\frac{e^2}{4\pi  \epsilon_0  r} + \frac{l (l+1) \hbar^2}{2 \mu  r^2}.$">|; 

$key = q/{bfA}(bfr);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2026.png"
 ALT="$ {\bf A}(\bf r)$">|; 

$key = q/J_2^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1308.png"
 ALT="$ J_2^{ 2}$">|; 

$key = q/displaystyle(nabla^2+k^2),psi=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2221.png"
 ALT="$\displaystyle (\nabla^2 + k^2) \psi = 0.$">|; 

$key = q/displaystylelangle{bfx}'|,H_1,|psi^pmrangle=intd^3x'',langle{bfx}'|,H_1,|{bfx}''ranglelangle{bfx}''|psi^pmrangle,=V({bfx}'),psi^pm({bfx}').;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="439" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2153.png"
 ALT="$\displaystyle \langle {\bf x}'\vert H_1 \vert \psi^\pm\rangle = \int d^3 x''\...
...e \langle {\bf x}'' \vert\psi^\pm\rangle  = V({\bf x}')  \psi^\pm ({\bf x}').$">|; 

$key = q/sqrt{2,m,|V_0|,a^2slashhbar^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2336.png"
 ALT="$ \sqrt{2 m  \vert V_0\vert a^2/\hbar^2}$">|; 

$key = q/displaystylepsi_-(x',y',z');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1065.png"
 ALT="$\displaystyle \psi_-(x', y', z')$">|; 

$key = q/displaystyle{rmi},hbar,frac{dT}{dt},|At_0rangle=H(t),T,|At_0rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img559.png"
 ALT="$\displaystyle {\rm i} \hbar  \frac{d T}{dt} \vert At_0\rangle = H(t) T \vert At_0\rangle.$">|; 

$key = q/displaystylepsi^pm({bfx})=phi({bfx})-frac{2,m}{hbar^2}intd^3x',frac{exp(pm{rmi},{bfx}-{bfx}'|)}{4pi,|{bfx}-{bfx}'|},V({bfx}'),psi^pm({bfx}').;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="445" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2154.png"
 ALT="$\displaystyle \psi^\pm({\bf x}) = \phi({\bf x}) - \frac{2 m}{\hbar^2} \int d^3...
...\vert)}{4\pi \vert{\bf x} - {\bf x}'\vert}  V({\bf x}')  \psi^\pm({\bf x}').$">|; 

$key = q/1slashsqrt{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1417.png"
 ALT="$ 1/\sqrt{2}$">|; 

$key = q/g^{00}=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2563.png"
 ALT="$ g^{00}=1$">|; 

$key = q/p^{,mu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2664.png"
 ALT="$ p^{ \mu}$">|; 

$key = q/displaystyleH,|k'rangle,|k''rangle=E,|k'rangle,|k''rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2386.png"
 ALT="$\displaystyle H \vert k'\rangle \vert k''\rangle = E \vert k'\rangle \vert k''\rangle$">|; 

$key = q/m^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img800.png"
 ALT="$ m^2$">|; 

$key = q/{bfA}({bfx});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2123.png"
 ALT="$ {\bf A}({\bf x})$">|; 

$key = q/displaystylechi=psi({bfx}')left(!{array}{c}c_+c_-{array}!right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1250.png"
 ALT="$\displaystyle \chi = \psi({\bf x}') \left(\!\begin{array}{c} c_+\ c_-\end{array}\!\right).$">|; 

$key = q/displaystyle(xi,eta-eta,xi),|Arangle=|0rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img161.png"
 ALT="$\displaystyle (\xi \eta -\eta \xi)  \vert A\rangle = \vert\rangle,$">|; 

$key = q/displaystyleleft(frac{x'}{x_0}+x_0,frac{d}{dx'}right)psi_0(x')=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img693.png"
 ALT="$\displaystyle \left(\frac{x'}{x_0}+x_0 \frac{d}{d x'}\right)\psi_0(x')= 0,
$">|; 

$key = q/c,p^0-e,{mitPhi}^{,0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2927.png"
 ALT="$ c p^0-e {\mit\Phi}^{ 0}$">|; 

$key = q/displaystyleJ_{2z},|j_1,j_2;m_1,m_2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1348.png"
 ALT="$\displaystyle J_{2z}  \vert j_1,j_2; m_1,m_2\rangle$">|; 

$key = q/t^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1924.png"
 ALT="$ t^2$">|; 

$key = q/displaystyle{mitPsi}(p_x');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img431.png"
 ALT="$\displaystyle {\mit\Psi}(p_x')$">|; 

$key = q/displaystyle{bfS}_pcdot{bfS}_e=frac{1}{2},(S^2-S_e^{,2}-S_p^{,2}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1747.png"
 ALT="$\displaystyle {\bf S}_p\cdot{\bf S}_e = \frac{1}{2} (S^2-S_e^{ 2}-S_p^{ 2}).$">|; 

$key = q/displaystyle=frac{alpha}{1+{calE}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2867.png"
 ALT="$\displaystyle = \frac{\alpha}{1+{\cal E}},$">|; 

$key = q/|s,s_zrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1019.png"
 ALT="$ \vert s, s_z\rangle$">|; 

$key = q/displaystyle=-{rmi},hbarleft(y,frac{partial}{partialz}-z,frac{partial}{partialy}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img885.png"
 ALT="$\displaystyle = -{\rm i} \hbar\left(y \frac{\partial}{\partial z} - z \frac{\partial} {\partial y}\right),$">|; 

$key = q/displaystyle=pm1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2090.png"
 ALT="$\displaystyle = \pm 1.$">|; 

$key = q/displaystyleP_{123}=P_{12},P_{31}=P_{23},P_{12}=P_{31},P_{23}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2432.png"
 ALT="$\displaystyle P_{123} = P_{12} P_{31}=P_{23} P_{12}=P_{31} P_{23}.$">|; 

$key = q/epsilonll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1475.png"
 ALT="$ \epsilon\ll 1$">|; 

$key = q/rho({bfx'},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img641.png"
 ALT="$ \rho({\bf x'}, t)$">|; 

$key = q/displaystylechi(1,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2468.png"
 ALT="$\displaystyle \chi(1,0)$">|; 

$key = q/displaystyle{rmi},hbar,frac{partialU_I(t_0,t)}{partialt}=H_I(t_0,t),U_I(t_0,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1860.png"
 ALT="$\displaystyle {\rm i}  \hbar  \frac{\partial U_I(t_0, t)}{\partial t} = H_I(t_0, t)  U_I(t_0, t),$">|; 

$key = q/displaystylepartial_mu,j^{,mu}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2636.png"
 ALT="$\displaystyle \partial_\mu j^{ \mu} = 0,$">|; 

$key = q/alpha_1^{,2}=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2716.png"
 ALT="$ \alpha_1^{ 2}=1$">|; 

$key = q/displaystyle({bfp}+e,{bfA})times({bfp}+e,{bfA})=e,{bfp}times{bfA}+e,{bfA}times{batimes{bfA}-{rmi},e,hbar,{bfA}timesnabla=-{rmi},e,hbar,{bfB},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="619" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2742.png"
 ALT="$\displaystyle ({\bf p}+e {\bf A})\times ({\bf p}+e {\bf A}) = e {\bf p}\time...
...\bf A} -{ \rm i} e \hbar {\bf A}\times \nabla = -{\rm i} e \hbar {\bf B},$">|; 

$key = q/displaystyle|E_n-E_i|stackrel{_{normalsize<}}{_{normalsizesim}}frac{2pi,hbar}{t}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1902.png"
 ALT="$\displaystyle \vert E_n - E_i\vert \stackrel {_{\normalsize &lt;}}{_{\normalsize\sim}}\frac{2\pi  \hbar}{t}.$">|; 

$key = q/{bfB}=B,{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1117.png"
 ALT="$ {\bf B}=
B {\bf e}_z$">|; 

$key = q/r,R(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img943.png"
 ALT="$ r  R(r)$">|; 

$key = q/exp(-Eslashk_B,T);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2114.png"
 ALT="$ \exp(-E/k_B T)$">|; 

$key = q/m_{rmmax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img801.png"
 ALT="$ m_{\rm max}$">|; 

$key = q/psi';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2670.png"
 ALT="$ \psi'$">|; 

$key = q/displaystylelangleF|(|Arangle+|Brangle)=langleF|(|Arangle)+langleF|(|Brangle),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="$\displaystyle \langle F\vert(\vert A\rangle + \vert B\rangle) = \langle F\vert(\vert A\rangle) + \langle F\vert(\vert B \rangle),$">|; 

$key = q/E>V_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2318.png"
 ALT="$ E&gt;V_0$">|; 

$key = q/displaystylelangletilde{A}|tilde{A}rangle=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img75.png"
 ALT="$\displaystyle \langle \tilde{A}\vert\tilde{A} \rangle = 1.$">|; 

$key = q/displaystyleqequiv|{bfk}-{bfk}'|=2,k,sin(thetaslash2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2196.png"
 ALT="$\displaystyle q \equiv \vert{\bf k} - {\bf k}'\vert = 2  k  \sin (\theta/2),$">|; 

$key = q/n',l',m'=n,l,m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2527.png"
 ALT="$ n',l',m'=n,l,m$">|; 

$key = q/L;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1541.png"
 ALT="$ L$">|; 

$key = q/displaystyleL^2,Y_{l,m}=l,(l+1),hbar^2,Y_{l,m},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img913.png"
 ALT="$\displaystyle L^2  Y_{l m} = l (l+1) \hbar^2 Y_{l m},$">|; 

$key = q/displaystylelangleA|Arangle=intdxi',|langleA|xi'rangle|^{,2}=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img201.png"
 ALT="$\displaystyle \langle A\vert A\rangle =\int d\xi'  \vert\langle A\vert\xi'\rangle\vert^{ 2} = 1.$">|; 

$key = q/displaystylec_{l,m}^+=sqrt{l,(l+1)-m,(m+1)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img826.png"
 ALT="$\displaystyle c_{l m}^+ = \sqrt{l (l+1)- m  (m+1)}.$">|; 

$key = q/|f(E)|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1994.png"
 ALT="$ \vert f(E)\vert^{ 2}$">|; 

$key = q/displaystyle[J^{,2},J_{2,z}];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1335.png"
 ALT="$\displaystyle [J^{ 2}, J_{2 z}]$">|; 

$key = q/displaystyle=left(!{array}{rr}0&-{rmi}{rmi}&0{array}!right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1169.png"
 ALT="$\displaystyle = \left(\!\begin{array}{rr} 0 &amp;-{\rm i}\ {\rm i}&amp;0\end{array}\!\right),$">|; 

$key = q/(0,1slash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1724.png"
 ALT="$ (0,1/2)$">|; 

$key = q/displaystylek'asimeqsqrt{frac{2,m,|V_0|,a^2}{hbar^2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2332.png"
 ALT="$\displaystyle k' a \simeq \sqrt{\frac{2  m  \vert V_0\vert  a^2}{\hbar^2}}.$">|; 

$key = q/displaystylecdot({bfa}times{bfb}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2737.png"
 ALT="$\displaystyle \cdot ({\bf a} \times {\bf b}),$">|; 

$key = q/x=v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img480.png"
 ALT="$ x=v$">|; 

$key = q/langlexi''|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img122.png"
 ALT="$ \langle \xi''\vert$">|; 

$key = q/displaystyle=delta_{ij}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img254.png"
 ALT="$\displaystyle = \delta_{ij}.$">|; 

$key = q/displaystyleH_{LS}=frac{1}{2,m_e^{,2},c^2}frac{1}{r}frac{dV}{dr},{bfL}cdot{bfS}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1718.png"
 ALT="$\displaystyle H_{LS} = \frac{1}{2  m_e^{ 2} c^2} \frac{1}{r} \frac{dV}{dr}  {\bf L}\cdot {\bf S}.$">|; 

$key = q/displaystyle=sum_{m_1}sum_{m_2}|langlej_1,j_2;m_1,m_2|j_1,j_2;j,mrangle|^{,2}=1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1384.png"
 ALT="$\displaystyle =\sum_{m_1}\sum_{m_2} \vert\langle j_1,j_2;m_1,m_2\vert j_1,j_2;j,m\rangle\vert^{ 2} =1,$">|; 

$key = q/langleB|(|Arangle);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="$ \langle B\vert(\vert A \rangle )$">|; 

$key = q/displaystylelanglel,m|,L^-,L^+,|l,mrangle=hbar^2,[l,(l+1)-m,(m+1)].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img816.png"
 ALT="$\displaystyle \langle l,m\vert  L^-  L^+ \vert l,m\rangle =\hbar^2  [l (l+1) - m (m+1)].$">|; 

$key = q/displaystyle=m,hbar,|j_1,j_2;j,mrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1359.png"
 ALT="$\displaystyle = m \hbar \vert j_1,j_2; j,m\rangle.$">|; 

$key = q/+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2392.png"
 ALT="$ +1$">|; 

$key = q/displaystylefrac{{mitGamma}_i}{hbar}=-frac{2,{rmIm}({mitDelta}_i)}{hbar}=frac{2pi}{hbar}sum_{mneqi}|H_{mi}|^{,2},delta(E_i-E_m).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1983.png"
 ALT="$\displaystyle \frac{{\mit\Gamma}_i}{\hbar} = - \frac{2 {\rm Im}({\mit\Delta}_i...
...= \frac{2\pi}{\hbar} \sum_{m\neq i} \vert H_{mi}\vert^{ 2} \delta(E_i - E_m).$">|; 

$key = q/displaystyle=2,{rmi},hbarleft(L^2,L_y,x-L_x,y+{rmi},hbar,zright)=2,{rmi},hbarleft(L_y,[L^2,x]-L_x,[L^2,y]+{rmi},hbar,[L^2,z]right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="545" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1514.png"
 ALT="$\displaystyle = 2  {\rm i}   \hbar \left( L^2, L_y  x - L_x  y + {\rm i} \...
...hbar \left( L_y  [L^2, x] - L_x  [ L^2, y] + {\rm i} \hbar  [L^2, z]\right)$">|; 

$key = q/displaystyle{gamma^{,mu},gamma^nu}=2,g^{,mu,nu}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2601.png"
 ALT="$\displaystyle \{\gamma^{ \mu},\gamma^\nu\} = 2 g^{ \mu \nu}.$">|; 

$key = q/d_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img465.png"
 ALT="$ d_x$">|; 

$key = q/omega_{in}=(E_i-E_n)slashhbar>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2069.png"
 ALT="$ \omega_{in} = (E_i-E_n)/\hbar&gt;0$">|; 

$key = q/displaystylecdot{bfa}),(;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1205.png"
 ALT="$\displaystyle \cdot {\bf a} )  ($">|; 

$key = q/theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img850.png"
 ALT="$ \theta$">|; 

$key = q/displaystyletimeslanglej_1,j_2;m_1',m_2'|j_1,j_2;j,mrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1389.png"
 ALT="$\displaystyle \times\langle j_1, j_2; m_1', m_2'\vert j_1, j_2; j, m\rangle.$">|; 

$key = q/B_0=E_0slashc=2,A_0,omegaslashc;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2058.png"
 ALT="$ B_0=E_0/c= 2 A_0 \omega/c$">|; 

$key = q/displaystyle|c_i|^{,2}+sum_{mneqi}|c_m|^{,2}simeq(1-{mitGamma}_i,tslashhbar)+sum_{mneqi}w_{irightarrowm},t=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1987.png"
 ALT="$\displaystyle \vert c_i\vert^{ 2} + \sum_{m\neq i} \vert c_m\vert^{ 2} \simeq (1- {\mit\Gamma}_i t/ \hbar) + \sum_{m\neq i} w_{i\rightarrow m}  t = 1.$">|; 

$key = q/delta(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img193.png"
 ALT="$ \delta(x)$">|; 

$key = q/displaystylefrac{dsigma}{d{mitOmega}},d{mitOmega}=frac{r^2,d{mitOmega},|{bfj}_{rmsca}|}{|{bfj}_{rminc}|},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2182.png"
 ALT="$\displaystyle \frac{d\sigma}{d {\mit\Omega}}  d{\mit\Omega} = \frac{ r^2 d{\mit\Omega}   \vert{\bf j}_{\rm sca}\vert}{\vert{\bf j}_{\rm inc}\vert},$">|; 

$key = q/displaystyleH,|k''rangle,|k'rangle=E,|k''rangle,|k'rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2387.png"
 ALT="$\displaystyle H \vert k''\rangle \vert k'\rangle = E \vert k''\rangle \vert k'\rangle,$">|; 

$key = q/displaystylefrac{dot{c}_i}{c_i}=left(frac{-{rmi}}{hbar}right){mitDelta}_i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1970.png"
 ALT="$\displaystyle \frac{\dot{c}_i}{c_i} = \left( \frac{-{\rm i}}{\hbar}\right) {\mit\Delta}_i,$">|; 

$key = q/2p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1572.png"
 ALT="$ 2p$">|; 

$key = q/displaystylec_n^{(0)}(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1874.png"
 ALT="$\displaystyle c_n^{(0)}(t)$">|; 

$key = q/a^2=a_1,a_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2910.png"
 ALT="$ a^2=a_1 a_2$">|; 

$key = q/displaystyle[H,A^dag];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img672.png"
 ALT="$\displaystyle [H,A^\dag ]$">|; 

$key = q/t=-infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1943.png"
 ALT="$ t=-\infty$">|; 

$key = q/displaystyleL_z={rmi},hbar,lim_{deltavarphirightarrow0}frac{R(deltavarphi)-1}{deltavarphi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img855.png"
 ALT="$\displaystyle L_z = {\rm i} \hbar  \lim_{\delta \varphi\rightarrow 0}\frac{R(\delta \varphi)-1} {\delta \varphi}.$">|; 

$key = q/langleH^{,2}ranglegeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img233.png"
 ALT="$ \langle H^{ 2}\rangle \geq 0$">|; 

$key = q/displaystylecdot{bff}_{ni}|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2078.png"
 ALT="$\displaystyle \cdot{\bf f}_{ni}\vert^{ 2}$">|; 

$key = q/displaystylesimeq-frac{(2,l-1)!!}{(k,r)^{l+1}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2296.png"
 ALT="$\displaystyle \simeq -\frac{(2 l-1)!!}{(k r)^{l+1}},$">|; 

$key = q/displaystyleleft.-sum_mfrac{H_{nm},H_{mi}}{E_m-E_i}exp(,{rmi},omega_{im},tslash2),{rmsinc}(omega_{nm},tslash2)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1935.png"
 ALT="$\displaystyle \left. - \sum_m\frac{H_{nm} H_{mi}}{E_m - E_i} \exp( {\rm i} \omega_{im} t/2) {\rm sinc}(\omega_{nm} t/2)\right],$">|; 

$key = q/displaystyle=left({array}{rr}1&0[0.5ex]0&-1{array}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2604.png"
 ALT="$\displaystyle = \left(\begin{array}{rr} 1&amp; 0\ [0.5ex]0&amp;-1\end{array}\right),$">|; 

$key = q/chi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1231.png"
 ALT="$ \chi$">|; 

$key = q/displaystyle={rmi},hbar,sum_{i=1,N}{bfL}_i={rmi},hbar,{bfL}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img758.png"
 ALT="$\displaystyle = {\rm i} \hbar \sum_{i=1,N} {\bf L}_i = {\rm i} \hbar  {\bf L}.$">|; 

$key = q/displaystyleint_{-infty}^{+infty}cdotsint_{-infty}^{+infty}dq_1'cdotsdq_N',|q_1'cdotsq_N'ranglelangleq_1'cdotsq_N'|=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="389" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img374.png"
 ALT="$\displaystyle \int_{-\infty}^{+\infty} \cdots\int_{-\infty}^{+\infty} dq_1' \cdots dq_N'  \vert q_1'\cdots q_N'\rangle \langle q_1'\cdots q_N'\vert = 1.$">|; 

$key = q/displaystyle=0,pm1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2092.png"
 ALT="$\displaystyle = 0, \pm 1.$">|; 

$key = q/j_l'(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2271.png"
 ALT="$ j_l'(x)$">|; 

$key = q/{calO}(1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2685.png"
 ALT="$ {\cal O}(1)$">|; 

$key = q/displaystyle=sum_{m_1'}sum_{m_2'}left(sqrt{j_1,(j_1+1)-m_1',(m_1'pm1)},|j_1,j_2;m_1'pm1,m_2'rangleright.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="433" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1387.png"
 ALT="$\displaystyle = \sum_{m_1'}\sum_{m_2'} \left( \sqrt{j_1 (j_1+1)- m_1' (m_1'\pm 1)}  \vert j_1, j_2; m_1'\pm 1, m_2'\rangle\right.$">|; 

$key = q/x_1=Z,r_1slasha_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2520.png"
 ALT="$ x_1=Z r_1/a_0$">|; 

$key = q/displaystylealpha,c_{s_0}+(s_0+k),c_{s_0}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2894.png"
 ALT="$\displaystyle \alpha c_{s_0}+(s_0+k) c_{s_0}'$">|; 

$key = q/displaystyleleft[-e,phi(r)+c,epsilon,(p_r-{rmi},hbarslashr)+{rmi},c,hbar,k,epsilon,betaslashr+beta,m_e,c^2right]psi=E,psi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="448" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2856.png"
 ALT="$\displaystyle \left[- e \phi(r) + c \epsilon (p_r-{\rm i} \hbar/r) + {\rm i} c \hbar k \epsilon \beta/r + \beta m_e c^2\right]\psi = E \psi,$">|; 

$key = q/(2,j_1+1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1375.png"
 ALT="$ (2 j_1+1)$">|; 

$key = q/displaystyle{bfU}=e,|{bfE}|left({array}{cccc}0&langle2,0,0|,z,|2,1,0rangle&0&0langle2,1,0|,z,|2,0,0rangle&0&0&00&0&0&00&0&0&0{array}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="431" HEIGHT="111" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1574.png"
 ALT="$\displaystyle {\bf U} = e  \vert{\bf E}\vert \left( \begin{array}{cccc} 0&amp;\lan...
...2,1,0\vert z \vert 2,0,0\rangle&amp;0&amp;0&amp;0\ 0&amp;0&amp;0&amp;0\ 0&amp;0&amp;0&amp;0 \end{array}\right),$">|; 

$key = q/displaystyle[L_y,L_z];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img746.png"
 ALT="$\displaystyle [L_y, L_z]$">|; 

$key = q/displaystylefrac{partial}{partialq_i},psirangle=frac{partialpsi}{partialq_i}rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img387.png"
 ALT="$\displaystyle \frac{\partial}{\partial q_i}  \psi\rangle =\frac{\partial\psi}{\partial q_i} \rangle.$">|; 

$key = q/displaystyle{mitDelta}E=leftlanglefrac{e^2}{4pi,epsilon_0,r_{12}}rightrangle=intac{-2,Z,(r_1+r_2)}{a_0}right]frac{e^2}{4pi,epsilon_0,r_{12}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="552" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2507.png"
 ALT="$\displaystyle {\mit\Delta} E = \left\langle \frac{e^2}{4\pi \epsilon_0 r_{12}...
...p\left[\frac{-2 Z (r_1+r_2)}{a_0}\right]\frac{e^2}{4\pi \epsilon_0 r_{12}}.$">|; 

$key = q/k,agg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2303.png"
 ALT="$ k a\gg 1$">|; 

$key = q/n';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2494.png"
 ALT="$ n'$">|; 

$key = q/D(a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img524.png"
 ALT="$ D(a)$">|; 

$key = q/displaystylep_x={rmi},hbar,lim_{deltaxrightarrow0}frac{D(deltax)-1}{deltax},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img854.png"
 ALT="$\displaystyle p_x = {\rm i} \hbar  \lim_{\delta x\rightarrow 0}\frac{D(\delta x)-1} {\delta x},$">|; 

$key = q/displaystyletandelta_0=frac{sin(k,a)slashk,a}{-cos(k,a)slashka}=-tan(k,a),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2286.png"
 ALT="$\displaystyle \tan\delta_0 = \frac{\sin (k a)/k a}{-\cos (k a)/ka} = -\tan (k a),$">|; 

$key = q/displaystyle=left(frac{-{rmi}}{hbar}right)^2sum_m|H_{mi}|^{,2}frac{exp(2,eta,t)}{2,eta,(eta+{rmi},omega_{mi})}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1963.png"
 ALT="$\displaystyle = \left(\frac{-{\rm i}}{\hbar} \right)^2 \sum_m \vert H_{mi}\vert^{ 2} \frac{\exp(2  \eta  t)}{2 \eta (\eta + {\rm i} \omega_{mi})}.$">|; 

$key = q/r_{12}=|{bfx}_1-{bfx}_2|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2492.png"
 ALT="$ r_{12} = \vert{\bf x}_1-{\bf x}_2\vert$">|; 

$key = q/displaystyle=langlex'|,{rme}^{,{rmi},gamma'}{rme}^{-{rmi},gamma''}|x''rangle_{rmld}={rme}^{,{rmi},(gamma'-gamma'')}langlex'|x''rangle_{rmold};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="313" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img355.png"
 ALT="$\displaystyle = \langle x'\vert  {\rm e}^{ {\rm i} \gamma'} {\rm e}^{-{\rm i...
... = {\rm e}^{ {\rm i} (\gamma'-\gamma'')} \langle x'\vert x''\rangle_{\rm old}$">|; 

$key = q/displaystyleH,|k''',k',k''rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2419.png"
 ALT="$\displaystyle H \vert k''' k' k''\rangle$">|; 

$key = q/displaystyle=sum_{xi'}langleA|xi'ranglelanglexi'|Arangle=sum_{xi'}|langleA|xi'rangle|^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img137.png"
 ALT="$\displaystyle = \sum_{\xi'} \langle A\vert\xi'\rangle \langle\xi'\vert A\rangle = \sum_{\xi'}\vert\langle A\vert\xi' \rangle\vert^{ 2},$">|; 

$key = q/displaystyle=frac{l,hbar^2}{2},|l+1slash2,mrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1665.png"
 ALT="$\displaystyle = \frac{l  \hbar^2}{2}  \vert l+ 1/2, m\rangle,$">|; 

$key = q/pmhbarslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1032.png"
 ALT="$ \pm \hbar/2$">|; 

$key = q/f^{,2}_{21};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2112.png"
 ALT="$ f^{ 2}_{21}$">|; 

$key = q/(2p)_{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1773.png"
 ALT="$ (2p)_{1/2}$">|; 

$key = q/n=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img985.png"
 ALT="$ n=1$">|; 

$key = q/|irangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="$ \vert i\rangle$">|; 

$key = q/displaystyle|A'ranglerightarrowleft(!{array}{c}langle+|A'ranglelangle-|A'rangle{array}!right)equivchi'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1156.png"
 ALT="$\displaystyle \vert A'\rangle \rightarrow \left(\!\begin{array}{c}\langle +\vert A'\rangle\ \langle -\vert A'\rangle\end{array}\!\right)\equiv \chi'.$">|; 

$key = q/U_I;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1859.png"
 ALT="$ U_I$">|; 

$key = q/displaystyle+sqrt{j_2,(j_2+1)-m_2,(m_2mp1)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1393.png"
 ALT="$\displaystyle + \sqrt{j_2 (j_2+1) - m_2 (m_2\mp 1)}$">|; 

$key = q/displaystyleL^-,|l,m_{rmmin}rangle=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img832.png"
 ALT="$\displaystyle L^-  \vert l, m_{\rm min}\rangle = 0.$">|; 

$key = q/displaystylew_{irightarrow[n]};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2014.png"
 ALT="$\displaystyle w_{i\rightarrow [n]}$">|; 

$key = q/displaystyle|xi_1'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img148.png"
 ALT="$\displaystyle \vert\xi_1'\rangle$">|; 

$key = q/displaystyleP(x',dx')=|psi_A(x')|^{,2},dx'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img310.png"
 ALT="$\displaystyle P(x', dx') = \vert\psi_A(x')\vert^{ 2}  dx'.$">|; 

$key = q/1slash(E_n-E_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1552.png"
 ALT="$ 1/(E_n - E_n)$">|; 

$key = q/E_{n10}=E_{n00};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1532.png"
 ALT="$ E_{n10} = E_{n00}$">|; 

$key = q/displaystyle[u_1+u_2,v];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img495.png"
 ALT="$\displaystyle [u_1+ u_2, v]$">|; 

$key = q/displaystyle{bfj}({bfx}',t)=left(frac{hbar}{m}right){rmIm}(psi^ast,nabla'psi).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img662.png"
 ALT="$\displaystyle {\bf j}({\bf x}', t) = \left(\frac{\hbar}{m} \right){\rm Im} (\psi^\ast  \nabla' \psi).
$">|; 

$key = q/f(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img195.png"
 ALT="$ f(x)$">|; 

$key = q/displaystylep_krightarrow-{rmi},hbar,frac{partial}{partialx_k'},{bf1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1194.png"
 ALT="$\displaystyle p_k \rightarrow -{\rm i} \hbar \frac{\partial}{\partial x_k'}  {\bf 1}.$">|; 

$key = q/c'=h^{-1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img426.png"
 ALT="$ c' = h^{-1/2}$">|; 

$key = q/displaystyleleft(frac{H}{c}+frac{e}{c},phiright)^2=left[mbox{boldmathalpha}cdot(ox{boldmathalpha}cdot({bfp}+e,{bfA})right]^{,2}+m_e^{,2},c^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="503" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2729.png"
 ALT="$\displaystyle \left(\frac{H}{c}+\frac{e}{c} \phi\right)^2 = \left[\mbox{\boldm...
...mbox{\boldmath$\alpha$}\cdot({\bf p}+e {\bf A})\right]^{ 2} + m_e^{ 2} c^2,$">|; 

$key = q/|Rtrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img534.png"
 ALT="$ \vert Rt\rangle$">|; 

$key = q/S';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2653.png"
 ALT="$ S'$">|; 

$key = q/langlepm|,S_k,|pmrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1164.png"
 ALT="$ \langle \pm \vert S_k \vert\pm \rangle$">|; 

$key = q/displaystylelanglen,l,m|,[L_z,z],|n',l',m'rangle=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1501.png"
 ALT="$\displaystyle \langle n,l, m\vert  [L_z, z]  \vert n',l',m'\rangle = 0,$">|; 

$key = q/displaystyle(E_m-E_n),langlem|Erangle+e_{mn}simeq0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1485.png"
 ALT="$\displaystyle (E_m - E_n)  \langle m \vert E \rangle + e_{mn} \simeq 0,$">|; 

$key = q/tan(k,R);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2365.png"
 ALT="$ \tan(k R)$">|; 

$key = q/r<a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2275.png"
 ALT="$ r&lt;a$">|; 

$key = q/displaystyle{rmi},hbar,frac{partialpsi}{partialt}=left[-e,phi+c,mbox{boldmathalpha}cdot({bfp}+e,{bfA})+beta,m_e,c^2right]psi=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="373" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2625.png"
 ALT="$\displaystyle {\rm i} \hbar \frac{\partial\psi}{\partial t} =\left[-e \phi +...
...ox{\boldmath$\alpha$}\cdot({\bf p}+e {\bf A})+ \beta m_e c^2\right]\psi = 0.$">|; 

$key = q/{bfx}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1240.png"
 ALT="$ {\bf x}'$">|; 

$key = q/|xi''_irangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img231.png"
 ALT="$ \vert\xi''_i\rangle$">|; 

$key = q/displaystyle{mitDelta}xi,{mitDelta}eta=frac{1}{2}left[{mitDelta}xi,{mitDelta}etaright]+frac{1}{2}left{{mitDelta}xi,{mitDelta}etaright},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img179.png"
 ALT="$\displaystyle {\mit\Delta} \xi  {\mit\Delta} \eta= \frac{1}{2} \left[ {\mit\De...
...} \eta\right] +\frac{1}{2} \left\{ {\mit\Delta} \xi, {\mit\Delta} \eta\right\},$">|; 

$key = q/displaystylef(theta)=sum_{l=0,infty}(2,l+1),frac{exp(,{rmi},delta_l)}{k},sindelta_l,P_l(costheta).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2252.png"
 ALT="$\displaystyle f(\theta) = \sum_{l=0,\infty} (2 l+1) \frac{\exp( {\rm i} \delta_l)} {k}  \sin\delta_l P_l(\cos\theta).$">|; 

$key = q/displaystyle[zeta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2815.png"
 ALT="$\displaystyle [\zeta,$">|; 

$key = q/langlecdots|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="$ \langle\cdots\vert$">|; 

$key = q/psi({bfx})simeqphi({bfx});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2210.png"
 ALT="$ \psi({\bf x})
\simeq \phi({\bf x})$">|; 

$key = q/E=hbar^2,k^2slash2,m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2363.png"
 ALT="$ E=\hbar^2 k^2/2 m$">|; 

$key = q/displaystyle[r,p_r]={rmi},hbar,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2784.png"
 ALT="$\displaystyle [r,p_r] = {\rm i}  \hbar,$">|; 

$key = q/=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2813.png"
 ALT="$ =$">|; 

$key = q/1slashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img989.png"
 ALT="$ 1/r$">|; 

$key = q/g_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1732.png"
 ALT="$ g_p$">|; 

$key = q/x'+dx';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img308.png"
 ALT="$ x'+dx'$">|; 

$key = q/rmi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="10" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img411.png"
 ALT="$ \rm i$">|; 

$key = q/displaystyle{bff}_{ni}=langlen|,{bfx},|irangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2081.png"
 ALT="$\displaystyle {\bf f}_{ni} =\langle n\vert {\bf x} \vert i\rangle,$">|; 

$key = q/displaystyle=delta_{ij},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img490.png"
 ALT="$\displaystyle = \delta_{ij},$">|; 

$key = q/displaystylefrac{dT}{dt},v_t+T,frac{dv_t}{dt}=v,frac{dT}{dt}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img569.png"
 ALT="$\displaystyle \frac{d T}{dt}  v_t + T  \frac{dv_t}{dt} = v  \frac{dT}{dt}.$">|; 

$key = q/langlen|mrangle=delta_{n,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1866.png"
 ALT="$ \langle n\vert m \rangle = \delta_{n m}$">|; 

$key = q/langlen|,H_1,|irangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1887.png"
 ALT="$ \langle n\vert H_1 \vert i\rangle$">|; 

$key = q/s_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1025.png"
 ALT="$ s_z$">|; 

$key = q/displaystyleleft[A,gamma^{,mu},A^{-1}left(p_mu-frac{e}{c},{mitPhi}_muright)-m_e,cright]psi'=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2674.png"
 ALT="$\displaystyle \left[A \gamma^{ \mu} A^{-1}\left(p_\mu- \frac{e}{c} {\mit\Phi}_\mu\right)-m_e c\right]\psi' = 0.$">|; 

$key = q/displaystylesum_m{rmi},hbar,frac{dc_m}{dt}exp[-{rmi},E_m,(t-t_0)slashhbar],|mrangle=sum_mc_m(t)exp[-{rmi},E_m,(t-t_0)slashhbar],H_1,|mrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="575" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1792.png"
 ALT="$\displaystyle \sum_m {\rm i} \hbar  \frac{d c_m}{dt}\exp[-{\rm i} E_m  (t-t...
...gle = \sum_m c_m(t) \exp[-{\rm i} E_m  (t-t_0)/\hbar]  H_1  \vert m\rangle.$">|; 

$key = q/A^dag;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img713.png"
 ALT="$ A^\dag$">|; 

$key = q/displaystyle=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2384.png"
 ALT="$\displaystyle = 1.$">|; 

$key = q/displaystylelanglex',y',z'|langle-|A'ranglerangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1184.png"
 ALT="$\displaystyle \langle x', y', z'\vert \langle -\vert A'\rangle\rangle$">|; 

$key = q/displaystylesigma_{rmtot}=left(frac{16pi}{9}right)frac{m^2,V_0^{,2},R^{,6}}{hbar^4}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2358.png"
 ALT="$\displaystyle \sigma_{\rm tot} = \left(\frac{16\pi}{9}\right) \frac{m^2 V_0^{ 2} R^{ 6}}{\hbar^4}.
$">|; 

$key = q/displaystylecdot({bfp}+e,{bfA})+beta,m_e,c^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2728.png"
 ALT="$\displaystyle \cdot({\bf p}+e {\bf A})+ \beta m_e c^2.$">|; 

$key = q/displaystyleJ^{,2}=J_1^{,2}+J_2^{,2}+2,{bfJ}_1cdot{bfJ}_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1327.png"
 ALT="$\displaystyle J^{ 2} = J_1^{ 2} + J_2^{ 2} + 2 {\bf J}_1 \cdot {\bf J}_2.$">|; 

$key = q/X,|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img78.png"
 ALT="$ X \vert A\rangle$">|; 

$key = q/c_n,y^{,n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img973.png"
 ALT="$ c_n  y^{ n}$">|; 

$key = q/dj_l(x)slashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2272.png"
 ALT="$ dj_l(x)/dx$">|; 

$key = q/displaystyleA^dag=1-frac{1}{8},gamma^0,[gamma_mu,gamma_nu],gamma^0,{mitDelta}omega^{,mu,nu}=gamma^0,A^{-1},gamma^0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="330" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2699.png"
 ALT="$\displaystyle A^\dag = 1-\frac{1}{8} \gamma^0 [\gamma_\mu,\gamma_\nu] \gamma^0 {\mit\Delta}\omega^{ \mu \nu} = \gamma^0 A^{-1} \gamma^0.$">|; 

$key = q/displaystyleP(xi')propto|langleA|xi'rangle|^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img139.png"
 ALT="$\displaystyle P(\xi') \propto \vert\langle A\vert \xi'\rangle\vert^{ 2}.$">|; 

$key = q/displaystyleL_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img730.png"
 ALT="$\displaystyle L_y$">|; 

$key = q/cdot{bfx}=gamma^5,r,epsilon;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2850.png"
 ALT="$ \cdot{\bf x} = \gamma^5 r \epsilon$">|; 

$key = q/displaystyle=frac{[s,(s+1)-j,(j-1)]^{1slash2}}{2,{rmi},s},delta_{j,,l+1}-frac{[s,(s+1)-j,(j+1)]^{1slash2}}{2,{rmi},s},delta_{j,,l-1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="492" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1281.png"
 ALT="$\displaystyle = \frac{[ s (s+1) - j (j-1)]^{1/2} }{2 {\rm i} s} \delta_{j   l+1}- \frac{[s (s+1) - j (j+1)]^{1/2} }{2 {\rm i} s} \delta_{j   l-1}.$">|; 

$key = q/alpha_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2710.png"
 ALT="$ \alpha_1$">|; 

$key = q/displaystyler^2,epsilon,p_r-r,p_r,r,epsilon={rmi},hbar,r,epsilon.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2851.png"
 ALT="$\displaystyle r^2 \epsilon p_r - r p_r r \epsilon = {\rm i} \hbar r \epsilon.$">|; 

$key = q/(1,-1slash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1725.png"
 ALT="$ (1,-1/2)$">|; 

$key = q/c^2,p_x,H^{,-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2722.png"
 ALT="$ c^2 p_x H^{ -1}$">|; 

$key = q/2,Hslashh;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2723.png"
 ALT="$ 2 H/h$">|; 

$key = q/[n_{rmmin}(n_{rmmin}-1)-l,(l+1)]=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img957.png"
 ALT="$ [n_{\rm min} (n_{\rm min} -1) - l (l+1) ] =0$">|; 

$key = q/[cdots]_{cl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img577.png"
 ALT="$ [\cdots]_{cl}$">|; 

$key = q/langleA|,X;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img103.png"
 ALT="$ \langle A\vert  X$">|; 

$key = q/displaystyleH_0,|n,lrangle=E_n,|n,lrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1542.png"
 ALT="$\displaystyle H_0  \vert n, l\rangle = E_n  \vert n, l\rangle,$">|; 

$key = q/displaystylev_d=D,v,D^{dag}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img460.png"
 ALT="$\displaystyle v_d = D  v  D^{\dag }.$">|; 

$key = q/displaystylev_d,|Adrangle=D,|Brangle=D,v,|Arangle=D,v,D^{dag},|Adrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img458.png"
 ALT="$\displaystyle v_d  \vert Ad\rangle = D  \vert B\rangle = D  v  \vert A\rangle = D  v  D^{\dag }  \vert Ad\rangle.$">|; 

$key = q/displaystyleH_1=-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1737.png"
 ALT="$\displaystyle H_1 = -$">|; 

$key = q/displaystylelanglem-1slash2,1slash2|l+1slash2,mrangle=sqrt{frac{l+m+1slash2}{2,l+1}},langlel,1slash2|l+1slash2,l+1slash2rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="512" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1647.png"
 ALT="$\displaystyle \langle m-1/2, 1/2\vert l+1/2, m\rangle = \sqrt{\frac{l+m+1/2}{2 l+1}}  \langle l, 1/2\vert l+1/2, l+1/2\rangle.$">|; 

$key = q/displaystyleleft(c',|x',y',z'rangle+c'',|x'',y'',z''rangleright),|+rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1055.png"
 ALT="$\displaystyle \left(c'  \vert x', y', z'\rangle + c'' \vert x'', y'', z''\rangle\right)  \vert+\rangle$">|; 

$key = q/displaystylepsi_n(x')=frac{1}{pi^{1slash4},(2^n,n!)^{1slash2},x_0^{,n+1slash2}}l}right)^nexpleft[-frac{1}{2}left(frac{x'}{x_0}right)^2right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="463" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img695.png"
 ALT="$\displaystyle \psi_n(x') = \frac{1}{\pi^{1/4} (2^n n!)^{1/2} x_0^{ n+1/2}}\...
...c{d}{dx'}\right)^n\exp\left[-\frac{1}{2}\left(
\frac{x'}{x_0}\right)^2\right].
$">|; 

$key = q/displaystyle=sqrt{frac{l+m+1slash2}{2,l+1}},|m-1slash2,1slash2rangle+sqrt{frac{l-m+1slash2}{2,l+1}},|m+1slash2,-1slash2rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="508" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1654.png"
 ALT="$\displaystyle =\sqrt{\frac{l+m+1/2}{2 l+1}} \vert m-1/2, 1/2\rangle+\sqrt{\frac{l-m+1/2}{2 l+1}} \vert m+1/2, -1/2\rangle,$">|; 

$key = q/R_{n,l}(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1660.png"
 ALT="$ R_{n l}(r)$">|; 

$key = q/displaystyleint_{-infty}^{+infty}dx',|psi_A(x')|^{,2}=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img312.png"
 ALT="$\displaystyle \int_{-\infty}^{+\infty} dx'  \vert\psi_A(x')\vert^{ 2}= 1.$">|; 

$key = q/|Atrangle=T,|At_0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img558.png"
 ALT="$ \vert At\rangle = T  \vert At_0\rangle$">|; 

$key = q/cosalpha_2,chi_++sinalpha_2,{rme}^{,{rmi},beta_2},chi_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1432.png"
 ALT="$ \cos\alpha_2 \chi_+
+ \sin\alpha_2 {\rm e}^{ {\rm i} \beta_2} \chi_-$">|; 

$key = q/{bfL}=m_e,{bfx}times{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1616.png"
 ALT="$ {\bf L} = m_e  {\bf x}\times{\bf v}$">|; 

$key = q/cos(pmomega,Tslash2+delta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1137.png"
 ALT="$ \cos( \pm \omega  T/2 + \delta)$">|; 

$key = q/displaystylepsi^pm({bfx})=phi({bfx})-frac{2,m}{hbar^2}intd^3x',frac{exp(pm{rmi},fx}'|,)}{4pi,|{bfx}-{bfx}'|},langle{bfx}'|,H_1,|psi^pmrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="441" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2151.png"
 ALT="$\displaystyle \psi^\pm({\bf x}) = \phi({\bf x}) - \frac{2 m}{\hbar^2} \int d^3...
...rt{\bf x} - {\bf x}'\vert}  \langle {\bf x}' \vert H_1 \vert\psi^\pm\rangle.$">|; 

$key = q/A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="$ A$">|; 

$key = q/|Rrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="$ \vert R\rangle$">|; 

$key = q/displaystyle|i,t_0,trangle=sum_mc_m(t)exp[-{rmi},E_m,(t-t_0)slashhbar],|mrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="334" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1863.png"
 ALT="$\displaystyle \vert i, t_0, t\rangle = \sum_m c_m(t) \exp[ -{\rm i}   E_m (t-t_0)/\hbar]  \vert m\rangle.$">|; 

$key = q/displaystylefrac{dsigma}{d{mitOmega}}=frac{sin^2(k,a)}{k^2}simeqa^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2300.png"
 ALT="$\displaystyle \frac{d\sigma}{d{\mit\Omega}} = \frac{\sin^2 (k a)}{k^2} \simeq a^2$">|; 

$key = q/displaystyle(left[{mitDelta}xi,{mitDelta}etaright])^{dag}=({mitDelta}xi,{mitDeltelta}xi,{mitDelta}eta=-left[{mitDelta}xi,{mitDelta}etaright],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="476" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img186.png"
 ALT="$\displaystyle (\left[ {\mit\Delta} \xi, {\mit\Delta} \eta\right])^{\dag } = ({\...
...a}\xi  {\mit\Delta}\eta = - \left[ {\mit\Delta} \xi, {\mit\Delta} \eta\right],$">|; 

$key = q/displaystyleL^+,|l,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img843.png"
 ALT="$\displaystyle L^+  \vert l, m\rangle$">|; 

$key = q/displaystyle[epsilon,p_r]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2852.png"
 ALT="$\displaystyle [\epsilon,p_r]= 0.$">|; 

$key = q/partialslashpartialx_{mu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2576.png"
 ALT="$ \partial /\partial x_{\mu}$">|; 

$key = q/n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img220.png"
 ALT="$ n$">|; 

$key = q/displaystyle[Sigma_1,Sigma_1];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2768.png"
 ALT="$\displaystyle [\Sigma_1,\Sigma_1]$">|; 

$key = q/rho(E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1912.png"
 ALT="$ \rho(E)$">|; 

$key = q/displaystylelanglex',y',z'|langle+|A'ranglerangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1182.png"
 ALT="$\displaystyle \langle x', y', z'\vert \langle +\vert A'\rangle\rangle$">|; 

$key = q/displaystyle[L^+,L_z];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img774.png"
 ALT="$\displaystyle [L^+, L_z ]$">|; 

$key = q/displaystylecdot{bfp}=epsilon,(p_r-{rmi},hbarslashr)+{rmi},epsilon,beta,zetaslashr.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2854.png"
 ALT="$\displaystyle \cdot {\bf p} = \epsilon (p_r-{\rm i} \hbar/r) + {\rm i} \epsilon \beta \zeta/r.$">|; 

$key = q/c_i(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1948.png"
 ALT="$ c_i(t)$">|; 

$key = q/displaystyleleft(!{array}{c}langle+|A'rangle[0.5ex]langle-|A'rangle{array}!rightarray}{c}langle+|Arangle[0.5ex]langle-|Arangle{array}!right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1161.png"
 ALT="$\displaystyle \left(\!\begin{array}{c}\langle +\vert A'\rangle\ [0.5ex] \langl...
...}\langle +\vert A\rangle\ [0.5ex] \langle -\vert A\rangle\end{array}\!\right).$">|; 

$key = q/displaystylelangleB|Arangle=langleA|Brangle^ast.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="$\displaystyle \langle B\vert A \rangle = \langle A\vert B \rangle^\ast.$">|; 

$key = q/displaystyle(l+l'+2),(l+l'),(l-l'+1),(l-l'-1),langlen,l,m|,z,|n',l',m'rangle=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="482" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1522.png"
 ALT="$\displaystyle (l+l'+2) (l+l') (l-l'+1) (l-l'-1) \langle n,l,m\vert z \vert n',l',m' \rangle = 0.$">|; 

$key = q/displaystylefrac{1}{r_{12}}=frac{1}{(r_1^{,2}+r_2^{,2}-2,r_1,r_2,cosgamma)^{1slah2}}=sum_{l=0,infty}frac{r_<^{,l}}{r_>^{,l+1}},P_l(cosgamma),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="420" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2510.png"
 ALT="$\displaystyle \frac{1}{r_{12}} = \frac{1}{(r_1^{ 2}+r_2^{ 2}-2 r_1 r_2 \co...
...mma)^{1/2} }= \sum_{l=0,\infty} \frac{r_&lt;^{ l}}{r_&gt;^{ l+1}} P_l(\cos\gamma),$">|; 

$key = q/displaystyleR(deltavarphi)=1-{rmi},L_z,deltavarphislashhbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img856.png"
 ALT="$\displaystyle R(\delta \varphi) = 1 -{\rm i} L_z \delta\varphi/\hbar$">|; 

$key = q/|l,1slash2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1649.png"
 ALT="$ \vert l,1/2\rangle$">|; 

$key = q/displaystyleL^pm=pmhbar,exp(pm{rmi},varphi)left(frac{partial}{partialtheta}pm{rmi},cottheta,frac{partial}{partialvarphi}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img897.png"
 ALT="$\displaystyle L^\pm = \pm \hbar \exp(\pm{\rm i} \varphi)\left(\frac{\partial}{\partial\theta} \pm{\rm i}  \cot\theta \frac{\partial}{\partial\varphi}\right).$">|; 

$key = q/displaystyleintd{mitOmega},Y_{l,m}(theta,phi)=sqrt{4pi},delta_{l,0},delta_{m,0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2516.png"
 ALT="$\displaystyle \int d{\mit\Omega}  Y_{l m}(\theta,\phi) = \sqrt{4\pi} \delta_{l 0} \delta_{m 0},$">|; 

$key = q/H_0+H_{LS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1694.png"
 ALT="$ H_0 + H_{LS}$">|; 

$key = q/displaystylephi(r)=frac{e}{4pi,epsilon_0,r}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2862.png"
 ALT="$\displaystyle \phi(r) = \frac{e}{4\pi \epsilon_0 r}.$">|; 

$key = q/exp(2,yslasha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2905.png"
 ALT="$ \exp(2 y/a)$">|; 

$key = q/displaystylelangle{mitDelta}xi,{mitDelta}etarangle=frac{1}{2},langleleft[{mitDelfrac{1}{2},langleleft{{mitDelta}xi,{mitDelta}etaright}rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="294" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img187.png"
 ALT="$\displaystyle \langle{\mit\Delta} \xi  {\mit\Delta} \eta\rangle= \frac{1}{2} \...
...frac{1}{2}  \langle\left\{ {\mit\Delta} \xi, {\mit\Delta} \eta\right\}\rangle,$">|; 

$key = q/displaystyle=-2,{rmi},Sigma_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2772.png"
 ALT="$\displaystyle =-2 {\rm i} \Sigma_2,$">|; 

$key = q/omega_{21}=(E_2-E_1)slashhbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1807.png"
 ALT="$ \omega_{21} = (E_2 - E_1)/\hbar$">|; 

$key = q/displaystyleP_{13};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2411.png"
 ALT="$\displaystyle P_{13}$">|; 

$key = q/L^-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img769.png"
 ALT="$ L^-$">|; 

$key = q/displaystylefrac{c_s}{c_{s-1}}simeqfrac{2}{a,s}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2904.png"
 ALT="$\displaystyle \frac{c_s}{c_{s-1}}\simeq \frac{2}{a s}.$">|; 

$key = q/displaystyle{bfB}=-frac{{bfv}times{bfE}}{c^2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1613.png"
 ALT="$\displaystyle {\bf B} = - \frac{{\bf v} \times {\bf E}}{c^2}.$">|; 

$key = q/hbarslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1165.png"
 ALT="$ \hbar/2$">|; 

$key = q/|j_1,j_2;j,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1351.png"
 ALT="$ \vert j_1, j_2; j, m\rangle$">|; 

$key = q/displaystyle=-sqrt{frac{l-m+1slash2}{2,l+1}},|m-1slash2,1slash2rangle+sqrt{frac{l+m+1slash2}{2,l+1}},|m+1slash2,-1slash2rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="520" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1655.png"
 ALT="$\displaystyle = - \sqrt{\frac{l-m+1/2}{2 l+1}}  \vert m-1/2,1/2\rangle + \sqrt{\frac{l+m+1/2}{2 l+1}}  \vert m+1/2, -1/2\rangle.$">|; 

$key = q/displaystylechi'=left(frac{hbar}{2}right)sigma_k,chi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1162.png"
 ALT="$\displaystyle \chi' =\left( \frac{\hbar}{2}\right)\sigma_k  \chi,$">|; 

$key = q/j_1=l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1629.png"
 ALT="$ j_1=l$">|; 

$key = q/displaystyle=cos({mitDelta}varphislash2),{bf1}-{rmi},sin({mitDelta}varphislash2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1226.png"
 ALT="$\displaystyle = \cos({\mit\Delta}\varphi/2) {\bf 1}- {\rm i} \sin ({\mit\Delta}\varphi/2) $">|; 

$key = q/displaystyleX,(c,|Arangle)=c,X,|Arangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img80.png"
 ALT="$\displaystyle X (c  \vert A\rangle) = c  X  \vert A\rangle,$">|; 

$key = q/|x|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1898.png"
 ALT="$ \vert x\vert$">|; 

$key = q/displaystyle|A,t_0,trangle=sum_nc_nexp[-{rmi},E_n,(t-t_0)slashhbar],|nrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="313" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1780.png"
 ALT="$\displaystyle \vert A, t_0, t\rangle = \sum_n c_n \exp[-{\rm i} E_n  (t-t_0)/\hbar] \vert n\rangle.$">|; 

$key = q/displaystyle+left(frac{{rmi}^3lambda^3}{3!}right)[G,[G,[G,A]]]+cdots,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1088.png"
 ALT="$\displaystyle + \left(\frac{{\rm i}^3\lambda^3}{3!}\right)[G, [G, [G,A]]]+\cdots,$">|; 

$key = q/displaystylefrac{dp_i}{dt}=frac{1}{{rmi},hbar},[p_i,H]=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img599.png"
 ALT="$\displaystyle \frac{dp_i}{dt} = \frac{1}{{\rm i} \hbar} [p_i, H] = 0,$">|; 

$key = q/displaystyletimeslanglej_1,j_2;m_1mp1,m_2|j_1,j_2;j,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1392.png"
 ALT="$\displaystyle \times\langle j_1, j_2; m_1\mp 1, m_2\vert j_1,j_2;j, m\rangle$">|; 

$key = q/langleA|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.png"
 ALT="$ \langle A\vert A\rangle$">|; 

$key = q/|x'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img300.png"
 ALT="$ \vert x'\rangle$">|; 

$key = q/langles,j|,S_k,|s,lrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1266.png"
 ALT="$ \langle s, j\vert S_k  \vert s, l\rangle$">|; 

$key = q/displaystyleS_z,|pmrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1027.png"
 ALT="$\displaystyle S_z  \vert\pm \rangle$">|; 

$key = q/1slashgamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2367.png"
 ALT="$ 1/\gamma$">|; 

$key = q/displaystyle|k',k''rangle_-=frac{1}{sqrt{2}},left(|k'rangle,|k''rangle-|k''rangle,|k'rangleright).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2395.png"
 ALT="$\displaystyle \vert k' k''\rangle_- = \frac{1}{\sqrt{2}} \left(\vert k'\rangle \vert k''\rangle - \vert k''\rangle \vert k'\rangle\right).$">|; 

$key = q/displaystyle={rmi},hbar,L_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img745.png"
 ALT="$\displaystyle = {\rm i} \hbar  L_z,$">|; 

$key = q/L_y,L_xneqL_x,L_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img884.png"
 ALT="$ L_y  L_x \neq L_x  L_y$">|; 

$key = q/displaystyleP_{12},H,P_{12}^{,2},|k'rangle,|k''rangle=E,P_{12},|k'rangle,|k''rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="268" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2388.png"
 ALT="$\displaystyle P_{12} H P_{12}^{ 2} \vert k'\rangle \vert k''\rangle = E P_{12} \vert k'\rangle \vert k''\rangle,$">|; 

$key = q/displaystylelangle{bfS}_pcdot{bfS}_erangle=frac{1}{4},hbar^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1753.png"
 ALT="$\displaystyle \langle {\bf S}_p\cdot{\bf S}_e\rangle = \frac{1}{4} \hbar^2$">|; 

$key = q/displaystyle{mitDelta}E_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1601.png"
 ALT="$\displaystyle {\mit\Delta} E_1$">|; 

$key = q/displaystyle(sigma_1)_{j,l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1278.png"
 ALT="$\displaystyle (\sigma_1)_{j l}$">|; 

$key = q/E_n=E_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1904.png"
 ALT="$ E_n=E_i$">|; 

$key = q/displaystyle|AranglerightarrowT_z(2pi),|Arangle=-|Arangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1102.png"
 ALT="$\displaystyle \vert A\rangle \rightarrow T_z(2\pi) \vert A\rangle = -\vert A\rangle.$">|; 

$key = q/{bfp}=(p_x,p_y,p_z)=(p^1,p^2,p^3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2618.png"
 ALT="$ {\bf p}=(p_x,p_y,p_z)=(p^1,p^2,p^3)$">|; 

$key = q/displaystyle=|k',k''',k''rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2406.png"
 ALT="$\displaystyle = \vert k' k''' k''\rangle,$">|; 

$key = q/displaystyle|c^+_{l,m}|^{,2}=[l,(l+1)-m,(m+1)].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img825.png"
 ALT="$\displaystyle \vert c^+_{l m}\vert^{ 2} = [l (l+1) - m  (m+1)].$">|; 

$key = q/displaystyle{bfL}cdot{bfS},|j=lpm1slash2,m_j=mrangle=frac{hbar^2}{2}left[j,(j+1)-l,(l+1)-3slash4right]|j,mrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="499" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1663.png"
 ALT="$\displaystyle {\bf L} \cdot{\bf S}  \vert j=l\pm 1/2,m_j= m\rangle = \frac{\hbar^2}{2} \left[ j (j+1) - l (l+1) - 3/4\right]\vert j,m\rangle,$">|; 

$key = q/(deltax)^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img472.png"
 ALT="$ (\delta x)^{ 2}$">|; 

$key = q/displaystyleJ_2^{,2},|j_1,j_2;m_1,m_2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1344.png"
 ALT="$\displaystyle J_2^{ 2}  \vert j_1,j_2; m_1,m_2\rangle$">|; 

$key = q/displaystylembox{hspace{2.0cm}for{t<0{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1890.png"
 ALT="$\displaystyle \mbox{\hspace{2.0cm}for $t&lt;0$}$">|; 

$key = q/displaystylembox{hspace{1.88cm}for{k{even};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1218.png"
 ALT="$\displaystyle \mbox{\hspace{1.88cm}for $k$ even}$">|; 

$key = q/displaystyle|lpm1slash2,mrangle=pmsqrt{frac{lpmm+1slash2}{2,l+1}},|m-1slash2,1slle+sqrt{frac{lmpm+1slash2}{2,l+1}},|m+1slash2,-1slash2rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="609" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1699.png"
 ALT="$\displaystyle \vert l\pm 1/2, m\rangle = \pm \sqrt{\frac{l\pm m +1/2}{2 l+1}}\...
... m-1/2, 1/2\rangle+\sqrt{\frac{l\mp m+1/2}{2 l+1}}  \vert m+1/2, -1/2\rangle.$">|; 

$key = q/displaystyleleft(frac{d}{dx}right)_{rmnew}={rme}^{-{rmi},gamma}frac{d}{dx},{rme}^{,{rmi},gamma}=frac{d}{dx}+{rmi},frac{dgamma}{dx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img363.png"
 ALT="$\displaystyle \left(\frac{d}{dx}\right)_{\rm new} = {\rm e}^{-{\rm i} \gamma}\...
...dx} {\rm e}^{ {\rm i} \gamma} = \frac{d}{dx} + {\rm i}  \frac{d\gamma}{dx},$">|; 

$key = q/displaystylecdot{bfn});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1221.png"
 ALT="$\displaystyle \cdot {\bf n})$">|; 

$key = q/displaystylelambda_{123}=+1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2442.png"
 ALT="$\displaystyle \lambda_{123} = +1,$">|; 

$key = q/displaystyleg_{22};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2559.png"
 ALT="$\displaystyle g_{22}$">|; 

$key = q/displaystyleh=6.6261times10^{-34}~~{rmJ,s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img289.png"
 ALT="$\displaystyle h = 6.6261 \times 10^{-34}&nbsp;&nbsp;{\rm J s}$">|; 

$key = q/displaystyleB_{PB}simalpha^2frac{e,m_e}{epsilon_0,h,a_0}simeq25,,,{rmtesla}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1729.png"
 ALT="$\displaystyle B_{PB} \sim \alpha^2 \frac{e  m_e}{\epsilon_0  h  a_0} \simeq 25   {\rm tesla}.$">|; 

$key = q/displaystyle=|+rangle|x',y',z'rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1054.png"
 ALT="$\displaystyle = \vert+\rangle \vert x', y', z'\rangle,$">|; 

$key = q/displaystyleP(xi')=|langleA|xi'rangle|^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img141.png"
 ALT="$\displaystyle P(\xi')= \vert\langle A\vert \xi'\rangle\vert^{ 2}.$">|; 

$key = q/-2,-1,0,1,2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img837.png"
 ALT="$ -2, -1, 0, 1, 2$">|; 

$key = q/(deltax)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img474.png"
 ALT="$ (\delta x)^2$">|; 

$key = q/displaystylelanglel,m|,L^2-L_z^{,2},|l,mrangle=langlel,m|,L_x^{,2}+L_y^{,2},|l,manglel,m|,L_x^{,2},|l,mrangle+langlel,m|,L_y^{,2},|l,mrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="562" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img787.png"
 ALT="$\displaystyle \langle l, m \vert L^2 - L_z^{ 2} \vert l, m\rangle =\langle l...
...,L_x^{ 2} \vert l, m\rangle+ \langle l, m\vert L_y^{ 2} \vert l, m\rangle.$">|; 

$key = q/displaystyle~~~{rmor}~~~E_nsimeqE_i-hbar,omega,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2010.png"
 ALT="$\displaystyle&nbsp;&nbsp;&nbsp;{\rm or}&nbsp;&nbsp;&nbsp; E_n \simeq E_i - \hbar  \omega,$">|; 

$key = q/j=lpm1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1695.png"
 ALT="$ j=l\pm 1/2$">|; 

$key = q/displaystyle|Erangle=langle1|Erangle|1rangle+langle2|Erangle|2rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1443.png"
 ALT="$\displaystyle \vert E\rangle = \langle 1\vert E\rangle \vert 1\rangle + \langle 2\vert E\rangle \vert 2\rangle.$">|; 

$key = q/displaystyle|k',k',k'''rangle=|k',k'',k''rangle=|k''',k'',k'''rangle=|0rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2451.png"
 ALT="$\displaystyle \vert k' k' k'''\rangle = \vert k' k'' k''\rangle = \vert k''' k'' k'''\rangle = \vert\rangle.$">|; 

$key = q/p^0+alpha_1,p^1+alpha_2,p^2+alpha_3,p^3+beta,m_e,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2588.png"
 ALT="$ p^0 +\alpha_1 p^1+\alpha_2 p^2+\alpha_3 p^3+\beta m_e c$">|; 

$key = q/displaystyleL_z,Y_{l,m}=m,hbar,Y_{l,m},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img914.png"
 ALT="$\displaystyle L_z  Y_{l m} = m \hbar Y_{l m},$">|; 

$key = q/displaystyleleftlanglefrac{a_0}{r}rightrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1765.png"
 ALT="$\displaystyle \left\langle \frac{a_0}{r}\right\rangle$">|; 

$key = q/dV=r^2,d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1675.png"
 ALT="$ dV = r^2 d{\mit\Omega}$">|; 

$key = q/hslash2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img288.png"
 ALT="$ h/2\pi$">|; 

$key = q/displaystyleeta_l(k,r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2295.png"
 ALT="$\displaystyle \eta_l(k r)$">|; 

$key = q/q=|{bfk}-{bfk}'|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2371.png"
 ALT="$ q=\vert{\bf k}-{\bf k}'\vert$">|; 

$key = q/x';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img114.png"
 ALT="$ x'$">|; 

$key = q/displaystyleleft(frac{d}{dx}right)^{dag}=-frac{d}{dx}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img345.png"
 ALT="$\displaystyle \left(\frac{d}{dx}\right)^{\dag }= - \frac{d}{dx}.$">|; 

$key = q/displaystyleD(a),V(x),D(a)^{,dag}=V(x-a),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img516.png"
 ALT="$\displaystyle D(a) V(x) D(a)^{ \dag } = V(x-a),
$">|; 

$key = q/displaystylelanglexrangle=x_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img508.png"
 ALT="$\displaystyle \langle x\rangle = x_0,$">|; 

$key = q/displaystyle|j_1,j_2;j,mrangle=sum_{m_1}sum_{m_2}langlej_1,j_2;m_1,m_2|j_1,j_2;j,mrangle|j_1,j_2;m_1,m_2rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="471" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1362.png"
 ALT="$\displaystyle \vert j_1,j_2;j,m\rangle = \sum_{m_1}\sum_{m_2} \langle j_1,j_2;m_1,m_2\vert j_1,j_2;j,m\rangle \vert j_1,j_2;m_1,m_2\rangle.$">|; 

$key = q/O(epsilon^2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1484.png"
 ALT="$ O(\epsilon^2)$">|; 

$key = q/m_e,c^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2748.png"
 ALT="$ m_e c^2$">|; 

$key = q/displaystylelangleq_1'cdotsq_N'|,frac{partial}{partialq_i},psirangle=langleq_1'ci'}=frac{partial}{partialq_i'},langleq_1'cdotsq_N'|psirangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="543" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img396.png"
 ALT="$\displaystyle \langle q_1'\cdots q_N'\vert \frac{\partial}{\partial q_i}  \ps...
...'} = \frac{\partial } {\partial q_i'} \langle q_1'\cdots q_N'\vert\psi\rangle.$">|; 

$key = q/langleB|,X;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img94.png"
 ALT="$ \langle B\vert X$">|; 

$key = q/langlex'|p_x'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img418.png"
 ALT="$ \langle x'\vert p_x'\rangle$">|; 

$key = q/displaystyle[{bfL},H]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img929.png"
 ALT="$\displaystyle [{\bf L}, H] = 0.$">|; 

$key = q/f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img484.png"
 ALT="$ f$">|; 

$key = q/displaystylecdot{bfp}]=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2816.png"
 ALT="$\displaystyle \cdot {\bf p}] = 0,$">|; 

$key = q/displaystylembox{boldmath{{sigma{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1175.png"
 ALT="$\displaystyle \mbox{\boldmath$\sigma$}$">|; 

$key = q/displaystylesum_nc_nleft[n,(n-1),y^{,n-2}-2,n,y^{,n-1}-l,(l+1),y^{,n-2}+frac{2,mu,e^2,a}{4pi,epsilon_0,hbar^2},y^{,n-1}right]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="514" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img952.png"
 ALT="$\displaystyle \sum_n c_n \left[ n (n-1) y^{ n-2} - 2 n  y^{ n-1} - l (l+...
...+ \frac{2 \mu  e^2  a}{4\pi  \epsilon_0  \hbar^2}  y^{ n-1} \right] = 0.$">|; 

$key = q/|+ranglerightleftharpoons|-rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1851.png"
 ALT="$ \vert+\rangle \rightleftharpoons \vert-\rangle$">|; 

$key = q/{bfa};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2738.png"
 ALT="$ {\bf a}$">|; 

$key = q/displaystyle=~frac{partialH}{partialp_i},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img243.png"
 ALT="$\displaystyle =&nbsp;\frac{\partial H}{\partial p_i},$">|; 

$key = q/displaystylelanglex'|Arangle=psi_A(x').;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img305.png"
 ALT="$\displaystyle \langle x'\vert A\rangle = \psi_A(x').$">|; 

$key = q/displaystyleS^+,|s,jrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1274.png"
 ALT="$\displaystyle S^+  \vert s, j\rangle$">|; 

$key = q/displaystylep_krightarrow-{rmi},hbar,frac{partial}{partialx_k'},{bf1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1260.png"
 ALT="$\displaystyle p_k \rightarrow -{\rm i} \hbar  \frac{\partial}{\partial x_k'}  {\bf 1},$">|; 

$key = q/+s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1265.png"
 ALT="$ +s$">|; 

$key = q/displaystyle[L_x^{,2},L_z];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img762.png"
 ALT="$\displaystyle [L_x^{ 2}, L_z]$">|; 

$key = q/displaystyle=-a^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2549.png"
 ALT="$\displaystyle = -a^2,$">|; 

$key = q/J_1^{,2},J_2^{,2},J^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1338.png"
 ALT="$ J_1^{ 2}, J_2^{ 2}, J^{ 2},$">|; 

$key = q/displaystyle|Arangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="$\displaystyle \vert A\rangle.$">|; 

$key = q/displaystylephi({bfx})=frac{exp(,{rmi},k,z)}{(2pi)^{3slash2}}=frac{exp(,{rmi},k,rcostheta)}{(2pi)^{3slash2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2219.png"
 ALT="$\displaystyle \phi({\bf x}) = \frac{\exp( {\rm i} k z)}{(2\pi)^{3/2}}= \frac{\exp( {\rm i} k r\cos\theta)}{(2\pi)^{3/2}},$">|; 

$key = q/psi_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img505.png"
 ALT="$ \psi_0$">|; 

$key = q/displaystyleP_{21};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2381.png"
 ALT="$\displaystyle P_{21}$">|; 

$key = q/displaystyle{bfJ}_2times{bfJ}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1303.png"
 ALT="$\displaystyle {\bf J}_2\times {\bf J}_2$">|; 

$key = q/{mitDelta}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1980.png"
 ALT="$ {\mit\Delta}_i$">|; 

$key = q/4times4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2602.png"
 ALT="$ 4\times 4$">|; 

$key = q/displaystylelanglen,l'|,H_1,|n,lrangle=lambda_{n,l},delta_{l,l'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1554.png"
 ALT="$\displaystyle \langle n, l' \vert H_1 \vert n, l\rangle = \lambda_{n l}  \delta_{l l'},$">|; 

$key = q/B_1llB_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1835.png"
 ALT="$ B_1\ll B_0$">|; 

$key = q/4.5times10^{-5},{rmeV};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1775.png"
 ALT="$ 4.5\times 10^{-5} {\rm eV}$">|; 

$key = q/displaystylelangleS_yrangle_t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1125.png"
 ALT="$\displaystyle \langle S_y\rangle_t$">|; 

$key = q/displaystylelim_{deltaxrightarrow0}frac{v_d-v}{deltax}=d_x,v-v,d_x.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img476.png"
 ALT="$\displaystyle \lim_{\delta x\rightarrow 0} \frac{v_d -v}{\delta x} = d_x  v -v  d_x.$">|; 

$key = q/displaystylelangleS_xrangle_t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1123.png"
 ALT="$\displaystyle \langle S_x\rangle_t$">|; 

$key = q/S^pm=S_xpm{rmi},S_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1842.png"
 ALT="$ S^\pm=S_x\pm {\rm i} S_y$">|; 

$key = q/displaystyle[L_x,p_y];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2760.png"
 ALT="$\displaystyle [L_x,p_y]$">|; 

$key = q/{bfx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img588.png"
 ALT="$ {\bf x}$">|; 

$key = q/{bfU};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1563.png"
 ALT="$ {\bf U}$">|; 

$key = q/|n,m_{rmmax}-2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img811.png"
 ALT="$ \vert n, m_{\rm max}-2\rangle$">|; 

$key = q/langlen,l,m|,y,|n',l',m'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2125.png"
 ALT="$ \langle n,l,m\vert y \vert n',l',m'\rangle$">|; 

$key = q/displaystylepsi(r,theta,varphi)=R_{n,l}(r),Y_{l,m}(theta,varphi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img976.png"
 ALT="$\displaystyle \psi(r, \theta, \varphi) = R_{n l}(r)  Y_{l m} (\theta, \varphi),$">|; 

$key = q/{a,b}equiva,b+b,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1180.png"
 ALT="$ \{a,b\}\equiv a b+b a$">|; 

$key = q/k=pm(j+1slash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2831.png"
 ALT="$ k=\pm(j+1/2)$">|; 

$key = q/displaystyle{mitDelta}E_{nlm}=-frac{1}{2,m_e,c^2}left(E_n^{,2}-2,E_n,langleVrangle+langleV^{,2}rangleright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1763.png"
 ALT="$\displaystyle {\mit\Delta}E_{nlm} = -\frac{1}{2 m_e c^2}\left(E_n^{ 2} - 2 E_n \langle V\rangle + \langle V^{ 2}\rangle\right),
$">|; 

$key = q/displaystylealpha_1,H+H,alpha_1=alpha_1,c,alpha_1,p^1+c,alpha_1,p^1,alpha_1=2,c,p^1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2715.png"
 ALT="$\displaystyle \alpha_1 H+H \alpha_1 = \alpha_1 c \alpha_1 p^1 + c \alpha_1 p^1 \alpha_1 = 2 c p^1.$">|; 

$key = q/displaystylepsi=R_{2,1}(r),left[sqrt{1slash3},Y_{1,0}(theta,varphi),chi_++sqrt{2slash3},Y_{1,1}(theta,varphi),chi_-right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="409" HEIGHT="49" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1423.png"
 ALT="$\displaystyle \psi = R_{2 1}(r) \left[\sqrt{1/3} Y_{1 0}(\theta,\varphi) \chi_+ + \sqrt{2/3} Y_{1 1}(\theta,\varphi) \chi_-\right].
$">|; 

$key = q/displaystyle[u,v_1,v_2]_{cl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img265.png"
 ALT="$\displaystyle [u, v_1  v_2]_{cl}$">|; 

$key = q/{bfJ}times{bfJ}={rmi},hbar,{bfJ};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1099.png"
 ALT="$ {\bf J}\times {\bf J} = {\rm i} \hbar  {\bf J}$">|; 

$key = q/(1s)_{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1754.png"
 ALT="$ (1s)_{1/2}$">|; 

$key = q/displaystyle=eta',|xi'eta'rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img159.png"
 ALT="$\displaystyle = \eta' \vert\xi' \eta'\rangle.$">|; 

$key = q/displaystylex,p_x-p_x,x={rmi},hbar.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img349.png"
 ALT="$\displaystyle x p_x - p_x x = {\rm i}  \hbar.$">|; 

$key = q/L^2equivL_x^{,2}+L_y^{,2}+L_z^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img759.png"
 ALT="$ L^2 \equiv
L_x^{ 2} + L_y^{ 2}+L_z^{ 2}$">|; 

$key = q/j_1-j_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1379.png"
 ALT="$ j_1-j_2$">|; 

$key = q/displaystylelangleA_t|=langleA|,T.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img565.png"
 ALT="$\displaystyle \langle A_t\vert =\langle A\vert  T.$">|; 

$key = q/displaystyle{bfB};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2030.png"
 ALT="$\displaystyle {\bf B}$">|; 

$key = q/dot{c}_iequivdc_islashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1966.png"
 ALT="$ \dot{c}_i \equiv
d c_i/dt$">|; 

$key = q/displaystylecdot{bfx}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2838.png"
 ALT="$\displaystyle \cdot{\bf x}.$">|; 

$key = q/displaystylepsi_+(x',y',z');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1063.png"
 ALT="$\displaystyle \psi_+(x', y', z')$">|; 

$key = q/rho(E),dE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1914.png"
 ALT="$ \rho(E) dE$">|; 

$key = q/{bfB}=nablatimes{bfA};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2743.png"
 ALT="$ {\bf B} = \nabla\times {\bf A}$">|; 

$key = q/{mitDelta}t=t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1907.png"
 ALT="$ {\mit\Delta} t = t$">|; 

$key = q/m_s=pm1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1626.png"
 ALT="$ m_s = \pm 1/2$">|; 

$key = q/displaystyleneq0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1336.png"
 ALT="$\displaystyle \neq 0.$">|; 

$key = q/|E_nrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img683.png"
 ALT="$ \vert E_n\rangle$">|; 

$key = q/displaystylelanglen|,{bfp},|irangle=-{rmi},frac{m_e}{hbar}langlen|,[{bfx},H_0],|irangle={rmi},m_e,omega_{ni},langlen|,{bfx},|irangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2076.png"
 ALT="$\displaystyle \langle n\vert  {\bf p} \vert i\rangle = -{\rm i}  \frac{m_e}{...
...i\rangle = {\rm i} m_e \omega_{ni}  \langle n\vert {\bf x} \vert i\rangle.$">|; 

$key = q/displaystylew_{irightarrown}(t)=frac{dP_{irightarrown}}{dt}=frac{2,|H_{ni}|^{,2}}{hbar^2}frac{etaexp(2,eta,t)}{eta^2+omega_{ni}^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1953.png"
 ALT="$\displaystyle w_{i\rightarrow n}(t) = \frac{dP_{i\rightarrow n}}{dt} = \frac{2 ...
...ert^{ 2}}{\hbar^2} \frac{\eta \exp(2  \eta  t)}{\eta^2 + \omega_{ni}^{ 2}}.$">|; 

$key = q/displaystylecdot{bfp},|irangle|^{,2},delta(E_n-E_i+hbar,omega),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2055.png"
 ALT="$\displaystyle \cdot{\bf p}  \vert i\rangle\vert^{ 2}  \delta(E_n-E_i +\hbar \omega),$">|; 

$key = q/langleB||Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="$ \langle B\vert\vert A \rangle$">|; 

$key = q/displaystyle=frac{hbar}{2}left(,|+ranglelangle+|-|-ranglelangle-|,right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1041.png"
 ALT="$\displaystyle = \frac{\hbar}{2}\left(  \vert+\rangle \langle +\vert - \vert-\rangle \langle -\vert \right),$">|; 

$key = q/omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img664.png"
 ALT="$ \omega$">|; 

$key = q/chi(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1298.png"
 ALT="$ \chi(t)$">|; 

$key = q/displaystyle{mitDelta}omega^{alpha,beta},(g^nu_{~alpha},gamma_beta-g^nu_{~beta},a},(gamma^nu,sigma_{alpha,beta}-sigma_{alpha,beta},gamma^nu),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="402" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2689.png"
 ALT="$\displaystyle {\mit\Delta}\omega^{\alpha \beta} (g^\nu_{&nbsp;\alpha} \gamma_\bet...
...beta} (\gamma^\nu \sigma_{\alpha \beta}-\sigma_{\alpha \beta} \gamma^\nu),$">|; 

$key = q/l+1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1648.png"
 ALT="$ l+1/2$">|; 

$key = q/mneqk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1474.png"
 ALT="$ m\neq k$">|; 

$key = q/intdE_n,rho(E_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1929.png"
 ALT="$ \int dE_n \rho(E_n)$">|; 

$key = q/displaystyleU_I(t_0,t)=1-frac{rmi}{hbar}int_{t_0}^tdt',H_I(t_0,t'),U_I(t_0,t').;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="322" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1868.png"
 ALT="$\displaystyle U_I(t_0, t) = 1 - \frac{\rm i}{\hbar} \int_{t_0}^t dt'  H_I(t_0, t')  U_I(t_0, t') .$">|; 

$key = q/displaystyle(X^dag)^dag=X.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img216.png"
 ALT="$\displaystyle (X^\dag )^\dag = X.
$">|; 

$key = q/displaystyleH_0,|pmrangle=pmfrac{e,hbar,B_0}{2,m_e},|pmrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1840.png"
 ALT="$\displaystyle H_0  \vert\pm \rangle = \pm \frac{e  \hbar  B_0}{2  m_e}  \vert\pm \rangle.$">|; 

$key = q/-hbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img999.png"
 ALT="$ -\hbar$">|; 

$key = q/(x,y,z)equiv{bfx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img725.png"
 ALT="$ (x, y, z)\equiv {\bf x}$">|; 

$key = q/kllmu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2212.png"
 ALT="$ k\ll \mu$">|; 

$key = q/displaystyle=L_x+{rmi},L_y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img771.png"
 ALT="$\displaystyle = L_x + {\rm i}  L_y,$">|; 

$key = q/displaystyle={rmi},hbar,p_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2761.png"
 ALT="$\displaystyle = {\rm i} \hbar p_z,$">|; 

$key = q/displaystyle|n,l^{(1)}rangle'=|n,l^{(1)}rangle+sum_{n'neqn,l'}frac{e_{n'l'nl}}{E_n-E_{n'}},|n',l'rangle+O(epsilon^2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="374" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1569.png"
 ALT="$\displaystyle \vert n, l^{(1)}\rangle' = \vert n,l^{(1)}\rangle + \sum_{n'\neq n, l'} \frac{e_{n'l'nl}}{E_n-E_{n'}} \vert n',l'\rangle + O(\epsilon^2).$">|; 

$key = q/hslash2,m_e,c^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2725.png"
 ALT="$ h/2 m_e c^2$">|; 

$key = q/partial_mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2634.png"
 ALT="$ \partial_\mu$">|; 

$key = q/(0,-1slash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1727.png"
 ALT="$ (0,-1/2)$">|; 

$key = q/displaystyle[L^2,[L^2,z]];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1513.png"
 ALT="$\displaystyle [L^2, [L^2, z]]$">|; 

$key = q/J_{2z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1340.png"
 ALT="$ J_{2z}$">|; 

$key = q/|psi_+(x',y',z')|^{,2},dx'dy'dz';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1070.png"
 ALT="$ \vert\psi_+ (x', y', z')\vert^{ 2} dx' dy' dz'$">|; 

$key = q/displaystyleP_{12},H,P_{12}=H,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2390.png"
 ALT="$\displaystyle P_{12} H P_{12} = H,$">|; 

$key = q/displaystylelanglem-1slash2,1slash2|l+1slash2,mrangle=sqrt{frac{l+m+1slash2}{l+m+3slash2}},langlem+1slash2,1slash2|l+1slash2,m+1rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="558" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1645.png"
 ALT="$\displaystyle \langle m-1/2, 1/2\vert l+1/2, m\rangle = \sqrt{\frac{l+m+1/2}{l+m+3/2}}  \langle m+1/2, 1/2\vert l+1/2, m+1\rangle.$">|; 

$key = q/displaystyle|Arangle=langle+|Arangle|+rangle+langle-|Arangle|-rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1074.png"
 ALT="$\displaystyle \vert A\rangle = \langle +\vert A\rangle \vert+\rangle + \langle -\vert A\rangle \vert-\rangle$">|; 

$key = q/displaystyle{bfk}'=k,{bfe}_r.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2167.png"
 ALT="$\displaystyle {\bf k}' = k {\bf e}_r.$">|; 

$key = q/displaystylelanglexirangle=langleA|,xi,|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img144.png"
 ALT="$\displaystyle \langle\xi\rangle = \langle A\vert \xi \vert A\rangle$">|; 

$key = q/displaystyle=c_+,|x',y',z'rangle|+rangle+c_-,|x',y',z'rangle|-rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1058.png"
 ALT="$\displaystyle = c_+   \vert x', y', z'\rangle\vert+\rangle+ c_- \vert x', y', z'\rangle\vert-\rangle,$">|; 

$key = q/displaystyle{rmi},hbar,dot{Sigma}_1=[Sigma_1,H].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2766.png"
 ALT="$\displaystyle {\rm i} \hbar \dot{\Sigma}_1= [\Sigma_1,H].$">|; 

$key = q/l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="10" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img780.png"
 ALT="$ l$">|; 

$key = q/displaystylej_l(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2228.png"
 ALT="$\displaystyle j_l(y)$">|; 

$key = q/X,Y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img104.png"
 ALT="$ X Y$">|; 

$key = q/displaystyle=y,p_z-z,p_y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img729.png"
 ALT="$\displaystyle = y  p_z - z  p_y,$">|; 

$key = q/{mitDelta}omega^{alpha,beta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2691.png"
 ALT="$ {\mit\Delta}\omega^{\alpha \beta}$">|; 

$key = q/displaystylecdot{bfL}times{bfp},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2799.png"
 ALT="$\displaystyle \cdot{\bf L}\times{\bf p},$">|; 

$key = q/{bfA};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2033.png"
 ALT="$ {\bf A}$">|; 

$key = q/displaystyle{rmi},hbar,frac{dc_1}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1803.png"
 ALT="$\displaystyle {\rm i}  \hbar  \frac{d c_1}{dt}$">|; 

$key = q/i,j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1306.png"
 ALT="$ i, j$">|; 

$key = q/displaystyle={bfL}cdot{bfp}+{rmi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2797.png"
 ALT="$\displaystyle = {\bf L}\cdot {\bf p} + {\rm i} $">|; 

$key = q/t-t_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img531.png"
 ALT="$ t-t_0$">|; 

$key = q/displaystyle=[u_1,v_1]_{qm},u_2,v_2+u_1,[u_2,v_1]_{qm},v_2+v_1,[u_1,v_2]_{qm},u_2+v_1,u_1,[u_2,v_2]_{qm}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="513" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img276.png"
 ALT="$\displaystyle = [u_1, v_1]_{qm}  u_2  v_2 + u_1 [u_2, v_1]_{qm}  v_2 + v_1 [u_1, v_2]_{qm} u_2 + v_1  u_1 [u_2, v_2]_{qm}.$">|; 

$key = q/omegaslashc=2pislashlambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2071.png"
 ALT="$ \omega/c =2\pi/\lambda$">|; 

$key = q/displaystyleH_1(t)=V,exp(,{rmi},omega,t)+V^dagger,exp(-{rmi},omega,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1999.png"
 ALT="$\displaystyle H_1(t) = V \exp( {\rm i} \omega  t) + V^\dagger  \exp(-{\rm i} \omega  t),$">|; 

$key = q/displaystylefrac{d}{dt}left(frac{r}{a_0}right)^3=-frac{1}{tau},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img208.png"
 ALT="$\displaystyle \frac{d}{dt}\left(\frac{r}{a_0}\right)^3 =- \frac{1}{\tau},
$">|; 

$key = q/displaystyleH,A^dag,|Erangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img679.png"
 ALT="$\displaystyle H A^\dag \vert E\rangle$">|; 

$key = q/displaystyleH=-e,phi(r)+c,gamma^5,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2755.png"
 ALT="$\displaystyle H = - e \phi(r) + c \gamma^5 $">|; 

$key = q/c_2(0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1811.png"
 ALT="$ c_2(0) = 0$">|; 

$key = q/nablacdot{bfA}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2037.png"
 ALT="$ \nabla\cdot{\bf A} = 0$">|; 

$key = q/p_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img241.png"
 ALT="$ p_i$">|; 

$key = q/displaystylefrac{d^2x_i}{dt^2}=frac{1}{{rmi},hbar}left[frac{dx_i}{dt},Hright]=fr}frac{dp_i}{dt}=-frac{1}{m}frac{partialV({bfx})}{partialx_i}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="431" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img612.png"
 ALT="$\displaystyle \frac{d^2 x_i}{dt^2} = \frac{1}{{\rm i} \hbar} \left[\frac{dx_i}...
...1}{m}\frac{d p_i}{dt} = - \frac{1}{m} \frac{\partial V({\bf x})}{\partial x_i}.$">|; 

$key = q/langleE_n|E_nrangle=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img684.png"
 ALT="$ \langle E_n\vert E_n\rangle = 1$">|; 

$key = q/m_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1318.png"
 ALT="$ m_2$">|; 

$key = q/phi({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2027.png"
 ALT="$ \phi({\bf r})$">|; 

$key = q/displaystyleE_{nl}'=E_n+lambda_{n,l}+sum_{n'neqn,l'}frac{|e_{n'l'nl}|^{,2}}{E_n-E_{n'}}+O(epsilon^3),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="310" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1568.png"
 ALT="$\displaystyle E_{nl}' = E_n + \lambda_{n l} + \sum_{n'\neq n, l'} \frac{\vert e_{n'l'nl}\vert^{ 2}}{E_n - E_{n'}} + O(\epsilon^3),$">|; 

$key = q/psi_{s_z}({bfx'});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1288.png"
 ALT="$ \psi_{s_z}({\bf x'})$">|; 

$key = q/displaystyle=m_1,hbar,|j_1,j_2;m_1,m_2rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1347.png"
 ALT="$\displaystyle = m_1 \hbar \vert j_1,j_2; m_1,m_2\rangle,$">|; 

$key = q/displaystyle=intdxi',langleA|xi'ranglelanglexi'|,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img199.png"
 ALT="$\displaystyle = \int d\xi' \langle A\vert\xi'\rangle \langle \xi'\vert,$">|; 

$key = q/gamma^{,mu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2595.png"
 ALT="$ \gamma^{ \mu}$">|; 

$key = q/mu=0,3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2596.png"
 ALT="$ \mu=0,3$">|; 

$key = q/c,alpha_1,p^1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2714.png"
 ALT="$ c \alpha_1 p^1$">|; 

$key = q/displaystyleH,chi=E,chi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1248.png"
 ALT="$\displaystyle H  \chi = E  \chi,$">|; 

$key = q/displaystyle|Rdrangle=|Adrangle+|Bdrangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img445.png"
 ALT="$\displaystyle \vert Rd\rangle = \vert Ad\rangle + \vert Bd\rangle.$">|; 

$key = q/V=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2311.png"
 ALT="$ V=0$">|; 

$key = q/displaystylelim_{etarightarrow0}frac{eta}{eta^2+omega_{ni}^{~2}}=pi,delta(omega_{ni})=pi,hbar,delta(E_n-E_i).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="313" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1956.png"
 ALT="$\displaystyle \lim_{\eta\rightarrow 0} \frac{\eta}{\eta^2+ \omega_{ni}^{&nbsp;2}} =\pi \delta(\omega_{ni}) = \pi \hbar  \delta(E_n - E_i).$">|; 

$key = q/{mitDelta}xi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img164.png"
 ALT="$ {\mit\Delta} \xi$">|; 

$key = q/displaystyle|2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1594.png"
 ALT="$\displaystyle \vert 2\rangle$">|; 

$key = q/displaystyle=langleS_yrangle_{t=0}cos(omega,t)+langleS_xrangle_{t=0}sin(omega,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1126.png"
 ALT="$\displaystyle = \langle S_y\rangle_{t=0} \cos(\omega  t) + \langle S_x\rangle_{t=0} \sin(\omega  t),$">|; 

$key = q/lambda_{123};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2437.png"
 ALT="$ \lambda_{123}$">|; 

$key = q/displaystyle{mitDelta}E_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1603.png"
 ALT="$\displaystyle {\mit\Delta} E_2$">|; 

$key = q/displaystyledot{L}_x=c,gamma^5,(Sigma_2,p_z-Sigma_3,p_y).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2765.png"
 ALT="$\displaystyle \dot{L}_x = c \gamma^5 (\Sigma_2 p_z-\Sigma_3 p_y).$">|; 

$key = q/j_2=s=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1630.png"
 ALT="$ j_2 = s = 1/2$">|; 

$key = q/displaystyle|Rtrangle=T,|Rt_0rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img536.png"
 ALT="$\displaystyle \vert Rt\rangle = T  \vert Rt_0\rangle,$">|; 

$key = q/displaystyled_x,x-x,d_x=-1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img481.png"
 ALT="$\displaystyle d_x  x - x d_x = -1.$">|; 

$key = q/displaystylev_d,|Adrangle=|Bdrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img456.png"
 ALT="$\displaystyle v_d  \vert Ad\rangle = \vert Bd\rangle,$">|; 

$key = q/displaystyle|Rt_0rangle=|At_0rangle+|Bt_0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img532.png"
 ALT="$\displaystyle \vert Rt_0\rangle = \vert At_0\rangle + \vert B t_0\rangle$">|; 

$key = q/displaystylefrac{{mitDeltaE}}{|E_0|}=frac{2,Z^{,6}}{pi^2}intintfrac{d^3x_1}{a_0^{,3}},expleft(-2,Z,frac{r_1+r_2}{a_0}right)frac{a_0}{r_{12}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="403" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2508.png"
 ALT="$\displaystyle \frac{{\mit\Delta E}}{\vert E_0\vert} = \frac{2 Z^{ 6}}{\pi^2}\...
..._2}{a_0^{ 3}} \exp\left(-2 Z \frac{r_1+r_2}{a_0}\right) \frac{a_0}{r_{12}},$">|; 

$key = q/displaystylechiequivleft(!{array}{c}langle+|Aranglelangle-|Arangle{array}!right)=left(!{array}{c}c_+c_-{array}!right)=c_+,chi_++c_-,chi_-,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1152.png"
 ALT="$\displaystyle \chi \equiv \left(\!\begin{array}{c}\langle +\vert A\rangle\ \la...
...!\begin{array}{c} c_+\ c_- \end{array}\!\right) = c_+  \chi_+ + c_-  \chi_-,$">|; 

$key = q/J;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2534.png"
 ALT="$ J$">|; 

$key = q/mu=m_e,m_pslash(m_e+m_p)simeqm_e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img940.png"
 ALT="$ \mu = m_e  m_p/(m_e+ m_p)\simeq m_e$">|; 

$key = q/gamma>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2362.png"
 ALT="$ \gamma&gt;0$">|; 

$key = q/displaystylefrac{d^2P}{dr^2}-frac{2,mu}{hbar^2}left[frac{l,(l+1),hbar^2}{2,mu,r^2}-frac{e^2}{4pi,epsilon_0,r}-Eright]P=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img945.png"
 ALT="$\displaystyle \frac{d^2 P}{d r^2} - \frac{2 \mu}{\hbar^2}\left[ \frac{l (l+1) \hbar^2}{2 \mu  r^2} - \frac{e^2}{4\pi  \epsilon_0  r}-E\right] P =0,$">|; 

$key = q/displaystyleD=1+deltax,d_x.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img470.png"
 ALT="$\displaystyle D = 1 + \delta x d_x.$">|; 

$key = q/displaystyle|4rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1598.png"
 ALT="$\displaystyle \vert 4\rangle$">|; 

$key = q/displaystylelangleF|=sum_{i=1,N}f_i,langlei|.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="$\displaystyle \langle F\vert = \sum_{i=1,N} f_i  \langle i\vert.$">|; 

$key = q/alpha=e^2slash(2,epsilon_0,h,c)simeq1slash137;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2082.png"
 ALT="$ \alpha = e^2/(2 \epsilon_0  h  c) \simeq 1/137$">|; 

$key = q/[xi,eta];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img297.png"
 ALT="$ [\xi, \eta]$">|; 

$key = q/displaystyle=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img251.png"
 ALT="$\displaystyle = 0,$">|; 

$key = q/displaystylelangleA|AranglelangleB|Brangle-|langleA|Brangle|^{,2}geq0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img172.png"
 ALT="$\displaystyle \langle A\vert A\rangle \langle B\vert B\rangle - \vert\langle A\vert B\rangle\vert^{ 2} \geq 0,$">|; 

$key = q/displaystyleV(r)=-frac{e^2}{4pi,epsilon_0,r}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img938.png"
 ALT="$\displaystyle V(r) = -\frac{e^2}{4\pi \epsilon_0 r}.$">|; 

$key = q/displaystylefrac{{mitDeltaE}}{|E_0|};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2517.png"
 ALT="$\displaystyle \frac{{\mit\Delta E}}{\vert E_0\vert}$">|; 

$key = q/langlen|,x,|irangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2088.png"
 ALT="$ \langle n\vert x \vert i \rangle $">|; 

$key = q/displaystyle=langlex',y',z'|,p_klangle+||Aranglerangle=-{rmi},hbarfrac{partial}{partialx_k'}langlex',y',z'|langle+||Aranglerangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="373" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1190.png"
 ALT="$\displaystyle = \langle x', y', z'\vert p_k \langle +\vert\vert A\rangle\rangl...
...l} {\partial x_k'} \langle x', y', z'\vert \langle +\vert\vert A\rangle\rangle,$">|; 

$key = q/displaystylelanglen,l,m|,z,|n',l',m'rangle=int!int!intdV',psi^ast_{nlm}(r',theta',varphi'),r'costheta',psi_{n'm'l'}(r',theta',varphi'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="553" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1525.png"
 ALT="$\displaystyle \langle n,l,m\vert z \vert n',l',m' \rangle = \int\!\int\!\int ...
...nlm}(r',\theta',\varphi')  r'\cos\theta'  \psi_{n'm'l'}(r',\theta',\varphi'),$">|; 

$key = q/|Branglerightarrow|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.png"
 ALT="$ \vert B\rangle \rightarrow \vert A \rangle$">|; 

$key = q/|k''rangle,|k'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2377.png"
 ALT="$ \vert k''\rangle \vert k'\rangle$">|; 

$key = q/psi_+({bfx'});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1242.png"
 ALT="$ \psi_+({\bf x'})$">|; 

$key = q/displaystyle=[l,(l+1)-(m-1slash2),(m+1slash2)]^{1slash2},langlem+1slash2,1slash2|l+1slash2,m+1rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="512" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1644.png"
 ALT="$\displaystyle = [l (l+1)-(m-1/2) (m+1/2)]^{1/2}  \langle m+1/2, 1/2\vert l+1/2, m+1\rangle,$">|; 

$key = q/displaystyleintd^3x',|psi|^{,2}=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1241.png"
 ALT="$\displaystyle \int d^3x'  \vert\psi\vert^{ 2}= 1.$">|; 

$key = q/displaystyle{mitDelta}E=frac{mu_0,g_p,e^2}{8pi,m_p,m_e}leftlanglefrac{3,({bfS}_pe^2}{3,m_p,m_e},langle{bfS}_pcdot{bfS}_erangle,|psi(0)|^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="558" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1743.png"
 ALT="$\displaystyle {\mit\Delta} E = \frac{\mu_0 g_p e^2}{8\pi m_p m_e}\left\lang...
...}{3 m_p m_e} \langle{\bf S}_p\cdot{\bf S}_e\rangle \vert\psi(0)\vert^{ 2}.$">|; 

$key = q/2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img864.png"
 ALT="$ 2\pi$">|; 

$key = q/displaystyle{alpha_i,beta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2592.png"
 ALT="$\displaystyle \{\alpha_i,\beta\}$">|; 

$key = q/displaystyleV(x-a)=V(x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img709.png"
 ALT="$\displaystyle V(x-a)= V(x),
$">|; 

$key = q/H_1rightarrowexp(eta,t),H_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1940.png"
 ALT="$ H_1\rightarrow \exp(\eta t) 
H_1$">|; 

$key = q/displaystyleleft[p^0,p^0-frac{1}{2}sum_{i,j=1,3}{alpha_i,alpha_j},p^i,p^j-sum_{i=1,3}{alpha_i,beta},p^i,m_e,c-beta^{,2},m_e^2,c^2right]psi=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="506" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2589.png"
 ALT="$\displaystyle \left[p^0 p^0-\frac{1}{2}\sum_{i,j=1,3}\{\alpha_i,\alpha_j\} p^...
...m_{i=1,3}\{\alpha_i,\beta\} p^i m_e c-\beta^{ 2} m_e^2 c^2\right]\psi= 0,$">|; 

$key = q/displaystylealpha=frac{e^2}{2,epsilon_0,h,c}=frac{1}{137};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1688.png"
 ALT="$\displaystyle \alpha = \frac{e^2}{2 \epsilon_0  h  c} = \frac{1}{137}$">|; 

$key = q/displaystyleH=frac{p_x^{,2}}{2,m}+frac{1}{2},m,omega^2,x^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1759.png"
 ALT="$\displaystyle H = \frac{p_x^{ 2}}{2 m} + \frac{1}{2} m \omega^2 x^2.
$">|; 

$key = q/displaystylelangle{bfL}cdot{bfS}rangle=langle,L_z,S_z+(L^+,S^-+L^-,S^+)slash2,rangle=hbar^2,m_l,m_s,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1719.png"
 ALT="$\displaystyle \langle {\bf L}\cdot {\bf S}\rangle = \langle  L_z  S_z + (L^+  S^- + L^-  S^+)/2  \rangle= \hbar^2  m_l  m_s,$">|; 

$key = q/+e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2929.png"
 ALT="$ +e$">|; 

$key = q/eta_l(k,r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2227.png"
 ALT="$ \eta_l(k r)$">|; 

$key = q/displaystyleomega_{ni}+omegasimeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2009.png"
 ALT="$\displaystyle \omega_{ni} + \omega \simeq 0$">|; 

$key = q/displaystyleleftlanglefrac{1}{r}frac{dV}{dr}rightrangle=int_0^inftydr,r^2,(R_{n,l})^ast,frac{1}{r}frac{dV}{dr},R_{n,l}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1680.png"
 ALT="$\displaystyle \left\langle \frac{1}{r}\frac{dV}{dr} \right\rangle = \int_0^\infty dr  r^2 (R_{n l})^\ast  \frac{1}{r}\frac{dV}{dr}  R_{n l}.$">|; 

$key = q/displaystyle(xi'-xi''^{,ast}),langlexi''|xi'rangle=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img123.png"
 ALT="$\displaystyle (\xi' - \xi''^{ \ast})  \langle \xi''\vert\xi'\rangle = 0.$">|; 

$key = q/displaystyle=langleS_zrangle_{t=0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1128.png"
 ALT="$\displaystyle = \langle S_z\rangle_{t=0}.$">|; 

$key = q/s=s_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2892.png"
 ALT="$ s=s_0$">|; 

$key = q/displaystyle=-frac{rmi}{hbar}int_{t_0}^tdt',exp[,{rmi},omega_{ni},(t'-t_0)],H_{ni}(t'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1881.png"
 ALT="$\displaystyle = -\frac{\rm i}{\hbar} \int_{t_0}^t dt'  \exp[ {\rm i}  \omega_{ni}  (t'-t_0)]  H_{ni}(t') ,$">|; 

$key = q/displaystyle=left({array}{c}1slashsqrt{2}-1slashsqrt{2}00{array}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="115" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1587.png"
 ALT="$\displaystyle = \left( \begin{array}{c} 1/\sqrt{2} \ - 1/\sqrt{2} \ 0 \ 0 \end{array} \right),$">|; 

$key = q/displaystyleH=frac{L_x^{,2}+L_y^{,2}}{2,I_1}+frac{L_z^{,2}}{2,I_2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1000.png"
 ALT="$\displaystyle H = \frac{L_x^{ 2}+L_y^{ 2}}{2 I_1} + \frac{L_z^{ 2}}{2 I_2}.
$">|; 

$key = q/|xi_i'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img228.png"
 ALT="$ \vert\xi_i'\rangle$">|; 

$key = q/n=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img986.png"
 ALT="$ n=2$">|; 

$key = q/lambda_{12}=pm1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2438.png"
 ALT="$ \lambda_{12}=\pm 1$">|; 

$key = q/displaystylefrac{dv_t}{dt}=[v_t,H_t]_{qm},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img574.png"
 ALT="$\displaystyle \frac{dv_t}{dt} = [v_t, H_t]_{qm},$">|; 

$key = q/displaystylefrac{mu,e^2,a}{4pi,epsilon_0,hbar^2}=n,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img972.png"
 ALT="$\displaystyle \frac{\mu  e^2  a}{4\pi  \epsilon_0  \hbar^2} = n,$">|; 

$key = q/displaystylelim_{deltaxrightarrow0}frac{|Adrangle-|Arangle}{deltax}=lim_{deltaxrightarrow0}frac{D-1}{deltax},|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img463.png"
 ALT="$\displaystyle \lim_{\delta x\rightarrow 0 } \frac{\vert A d\rangle - \vert A\ra...
...}{\delta x} = \lim_{\delta x\rightarrow 0 }\frac{D-1}{\delta x} \vert A\rangle$">|; 

$key = q/displaystylelanglex'|p_x'rangle=c'exp(,{rmi},p_x',x'slashhbar),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img421.png"
 ALT="$\displaystyle \langle x' \vert p_x'\rangle = c' \exp( {\rm i}  p_x'  x'/\hbar),$">|; 

$key = q/omega_{ni}=(E_n-E_i)slashhbar>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2068.png"
 ALT="$ \omega_{ni} = (E_n-E_i)/\hbar&gt;0$">|; 

$key = q/displaystylephi(p)rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img404.png"
 ALT="$\displaystyle \phi(p)\rangle$">|; 

$key = q/exp(,{rmi},H)equivsum_{n=0,infty}({rmi},H)^nslashn!;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img225.png"
 ALT="$ \exp( {\rm i} H)\equiv\sum_{n=0,\infty}
({\rm i} H)^n/n!$">|; 

$key = q/S_p^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1749.png"
 ALT="$ S_p^{ 2}$">|; 

$key = q/E_{200};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1600.png"
 ALT="$ E_{200}$">|; 

$key = q/displaystylecdot{bfp});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2796.png"
 ALT="$\displaystyle \cdot{\bf p})$">|; 

$key = q/V(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1611.png"
 ALT="$ V(r)$">|; 

$key = q/m_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1317.png"
 ALT="$ m_1$">|; 

$key = q/displaystyle|Arangle+|0rangle=|Arangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="$\displaystyle \vert A\rangle + \vert\rangle = \vert A\rangle$">|; 

$key = q/displaystylesum_{l''=1,N_n}langlen,l'|H_1|n,l''ranglelanglen,l''|n,l^{(1)}rangle=lambda_{n,l},langlen,l'|n,l^{(1)}rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="378" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1560.png"
 ALT="$\displaystyle \sum_{l''=1,N_n}\langle n, l'\vert H_1\vert n, l''\rangle \langle...
...vert n, l^{(1)}\rangle = \lambda_{n l}  \langle n, l'\vert n, l^{(1)}\rangle.$">|; 

$key = q/displaystylelambda_{123}=lambda_{12},lambda_{31}=lambda_{23},lambda_{12}=lambda_{31},lambda_{23}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2441.png"
 ALT="$\displaystyle \lambda_{123} = \lambda_{12} \lambda_{31} = \lambda_{23} \lambda_{12}=\lambda_{31} \lambda_{23}.$">|; 

$key = q/k'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2375.png"
 ALT="$ k''$">|; 

$key = q/displaystyle{mitDelta}E_{nlmpm}=intdV,(psi_{nlmpm})^dagger,H_{LS},psi_{nlmpm},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1674.png"
 ALT="$\displaystyle {\mit\Delta} E_{nlm\pm} = \int dV (\psi_{nlm\pm})^\dagger  H_{LS} \psi_{nlm\pm},$">|; 

$key = q/displaystyle=j_2,(j_2+1),hbar^2,|j_1,j_2;j,mrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1355.png"
 ALT="$\displaystyle = j_2 (j_2+1) \hbar^2 \vert j_1,j_2; j,m\rangle,$">|; 

$key = q/displaystyleR({mitDelta}varphi_1),R({mitDelta}varphi_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img862.png"
 ALT="$\displaystyle R({\mit\Delta}\varphi_1) R({\mit\Delta}\varphi_2)$">|; 

$key = q/1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1290.png"
 ALT="$ 1/2$">|; 

$key = q/displaystyle[u,v];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img492.png"
 ALT="$\displaystyle [u, v]$">|; 

$key = q/displaystyleP_n(t)=|c_nexp[-{rmi},E_n(t-t_0)slashhbar]|^{,2}=|c_n|^{,2}=P_n(t_0).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="378" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1781.png"
 ALT="$\displaystyle P_n(t) = \vert c_n \exp[-{\rm i} E_n (t-t_0)/\hbar]\vert^{ 2} = \vert c_n\vert^{ 2} = P_n(t_0).$">|; 

$key = q/displaystylecosalpha,|Brangle+sinalpha,|Crangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="$\displaystyle \cos\alpha \vert B\rangle + \sin\alpha \vert C\rangle$">|; 

$key = q/displaystyle=-{rmi},hbarleft(x,frac{partial}{partialy}-y,frac{partial}{partialx}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img887.png"
 ALT="$\displaystyle = -{\rm i} \hbar\left(x \frac{\partial}{\partial y} - y \frac{\partial} {\partial x}\right),$">|; 

$key = q/displaystyle=R({mitDelta}varphi_1+{mitDelta}varphi_2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img863.png"
 ALT="$\displaystyle = R({\mit\Delta}\varphi_1+ {\mit\Delta}\varphi_2).$">|; 

$key = q/|langlex'|Arangle|^{,2},dx';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img309.png"
 ALT="$ \vert\langle x'\vert A\rangle\vert^{ 2} dx'$">|; 

$key = q/displaystylepsi^dag=left(psi_0^{,ast},psi_1^{,ast},psi_2^{,ast},psi_3^{,ast}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2628.png"
 ALT="$\displaystyle \psi^\dag = \left(\psi_0^{ \ast}, \psi_1^{ \ast},\psi_2^{ \ast},\psi_3^{ \ast}\right),$">|; 

$key = q/displaystylefrac{partialrho}{partialt}+nablacdot{bfj}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2643.png"
 ALT="$\displaystyle \frac{\partial\rho}{\partial t} + \nabla\cdot{\bf j} = 0.$">|; 

$key = q/displaystyle=(X,Y),|Arangle=X,Y,|Arangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img89.png"
 ALT="$\displaystyle = (X Y)  \vert A\rangle = X Y \vert A \rangle,$">|; 

$key = q/langlen|,y,|irangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2089.png"
 ALT="$ \langle n\vert y \vert i\rangle $">|; 

$key = q/displaystyle(H_0+H_1),|psirangle=E,|psirangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2141.png"
 ALT="$\displaystyle (H_0 + H_1)  \vert\psi\rangle = E \vert\psi\rangle,$">|; 

$key = q/displaystyleJ_1^{,2},|j_1,j_2;j,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1352.png"
 ALT="$\displaystyle J_1^{ 2}  \vert j_1,j_2; j,m\rangle$">|; 

$key = q/4times1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2619.png"
 ALT="$ 4\times 1$">|; 

$key = q/displaystyle={rmi},hbar,L_x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img747.png"
 ALT="$\displaystyle = {\rm i} \hbar  L_x,$">|; 

$key = q/{bfx}={bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2160.png"
 ALT="$ {\bf x} = {\bf0}$">|; 

$key = q/psi^dag,gamma^0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2631.png"
 ALT="$ \psi^\dag \gamma^0$">|; 

$key = q/(3p);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1686.png"
 ALT="$ (3p)$">|; 

$key = q/displaystyleleft(p^0,p^0-p^1,p^1-p^2,p^2-p^3,p^3+m_e^2,c^2right)psi=left(p^{,mu},p_mu+m_e^2,c^2right)psi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="472" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2585.png"
 ALT="$\displaystyle \left(p^0 p^0- p^1 p^1-p^2 p^2-p^3 p^3+m_e^2 c^2\right)\psi = \left(p^{ \mu} p_\mu+m_e^2 c^2\right)\psi.$">|; 

$key = q/hbarslashm_e,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2726.png"
 ALT="$ \hbar/m_e c$">|; 

$key = q/displaystyleP_{irightarrow}(t)=frac{2,t}{hbar}intdx,|H_{ni}|^{,2},rho(E_n),{rmsinc}^2(x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1916.png"
 ALT="$\displaystyle P_{i\rightarrow} (t) = \frac{2  t}{\hbar} \int dx  \vert H_{ni}\vert^{ 2}  \rho(E_n)  {\rm sinc}^2(x),$">|; 

$key = q/beta_{l+};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2274.png"
 ALT="$ \beta_{l+}$">|; 

$key = q/displaystyle=cos^2(gamma,tslashhbar),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1824.png"
 ALT="$\displaystyle =\cos^2 (\gamma  t / \hbar),$">|; 

$key = q/displaystylefrac{1}{2}left{|omega_A({bfx}_1)|^{,2},|omega_B({bfx}_2)|^{,2}+|omegt}({bfx}_2),omega_B^{,ast}({bfx}_1)right]right}d^3x_1,d^3x_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="697" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2485.png"
 ALT="$\displaystyle \frac{1}{2}\left\{\vert\omega_A({\bf x}_1)\vert^{ 2} \vert\omeg...
... \ast}({\bf x}_2) \omega_B^{ \ast}({\bf x}_1)\right]\right\}d^3 x_1 d^3x_2.$">|; 

$key = q/displaystyleomega_{nm}=frac{E_n-E_m}{hbar}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1796.png"
 ALT="$\displaystyle \omega_{nm} = \frac{E_n - E_m}{\hbar}.$">|; 

$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="269" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img919.png"
 ALT="$\displaystyle = [l (l+1)- m (m+1)]^{1/2} \hbar Y_{l  m+1},$">|; 

$key = q/Vneq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2268.png"
 ALT="$ V\neq 0$">|; 

$key = q/-78.8,{rmeV};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2523.png"
 ALT="$ -78.8 {\rm eV}$">|; 

$key = q/displaystylerightarrowH-q,phi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2025.png"
 ALT="$\displaystyle \rightarrow H-q \phi,$">|; 

$key = q/displaystyle~~~{rmor}~~~E_nsimeqE_i+hbar,omega.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2012.png"
 ALT="$\displaystyle&nbsp;&nbsp;&nbsp;{\rm or}&nbsp;&nbsp;&nbsp; E_n \simeq E_i + \hbar  \omega.$">|; 

$key = q/displaystyle(L^2-L_z^{,2}-hbar,L_z),|l,m_{rmmax}rangle=|0rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img806.png"
 ALT="$\displaystyle (L^2 - L_z^{ 2} - \hbar  L_z)  \vert l, m_{\rm max}\rangle = \vert\rangle.$">|; 

$key = q/displaystyleP_{32};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2409.png"
 ALT="$\displaystyle P_{32}$">|; 

$key = q/displaystylelim_{epsilonrightarrow0}frac{1}{x+{rmi},epsilon}={calP},frac{1}{x}-{rmi},pi,delta(x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1972.png"
 ALT="$\displaystyle \lim_{\epsilon\rightarrow 0} \frac{1}{x+{\rm i} \epsilon} = {\cal P} \frac{1}{x} - {\rm i} \pi \delta(x),$">|; 

$key = q/1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img227.png"
 ALT="$ 1$">|; 

$key = q/displaystylef({bfk}',{bfk})simeq-frac{m}{2pi,hbar^2}int_0^infty!int_0^pi!int_0^{eta',dphi',r'^{,2},sintheta',exp(,{rmi},q,r'costheta'),V(r'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="566" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2194.png"
 ALT="$\displaystyle f({\bf k}', {\bf k}) \simeq - \frac{m}{2\pi  \hbar^2} \int_0^\in...
...,d\phi' r'^{ 2} \sin\theta'  \exp( {\rm i}   q  r'\cos\theta')   V(r'),$">|; 

$key = q/displaystyle|1rangle';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1461.png"
 ALT="$\displaystyle \vert 1\rangle'$">|; 

$key = q/P_{123};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2430.png"
 ALT="$ P_{123}$">|; 

$key = q/displaystyle=intd^3x_1intd^3x_2,psi_{100}({bfx}_1),psi_{nlm}({bfx}_2),frac{e^2}{,r_{12}},psi_{100}^{,ast}({bfx}_2),psi_{nlm}^{,ast}({bfx}_1).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="471" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2532.png"
 ALT="$\displaystyle = \int d^3 x_1\int d^3 x_2  \psi_{100}({\bf x}_1) \psi_{nlm}({\...
...lon_0 r_{12}} \psi_{100}^{ \ast}({\bf x}_2) \psi_{nlm}^{ \ast}({\bf x}_1).$">|; 

$key = q/langlem|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1470.png"
 ALT="$ \langle m\vert$">|; 

$key = q/displaystyle=chi_+,chi_+,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2467.png"
 ALT="$\displaystyle = \chi_+ \chi_+,$">|; 

$key = q/displaystylelangle2,0,0|,z,|2,1,0rangle=langle2,1,0|,z,|2,0,0rangle=3,a_0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="333" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1579.png"
 ALT="$\displaystyle \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.$">|; 

$key = q/displaystyle=|2,1,1rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1597.png"
 ALT="$\displaystyle = \vert 2,1,1\rangle,$">|; 

$key = q/displaystylelangle+|+rangle=langle-|-rangle=1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1034.png"
 ALT="$\displaystyle \langle +\vert + \rangle=\langle -\vert - \rangle = 1,$">|; 

$key = q/displaystylelanglep_x'|p_x''rangle=delta(p_x'-p_x''),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img402.png"
 ALT="$\displaystyle \langle p_x'\vert p_x''\rangle = \delta(p_x'-p_x''),$">|; 

$key = q/displaystyle[zeta,epsilon]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2844.png"
 ALT="$\displaystyle [\zeta,\epsilon] = 0.$">|; 

$key = q/displaystylesigma_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1170.png"
 ALT="$\displaystyle \sigma_3$">|; 

$key = q/ErightarrowE+e,phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2620.png"
 ALT="$ E\rightarrow E+e \phi$">|; 

$key = q/displaystyleXleft(|Arangle+|Brangleright)=X,|Arangle+X,|Brangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img79.png"
 ALT="$\displaystyle X \left(\vert A\rangle + \vert B \rangle\right) = X \vert A\rangle + X \vert B \rangle,$">|; 

$key = q/m=m'=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2501.png"
 ALT="$ m=m'=0$">|; 

$key = q/deltaH;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2747.png"
 ALT="$ \delta H$">|; 

$key = q/|k',k''rangle_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2399.png"
 ALT="$ \vert k' k''\rangle_-$">|; 

$key = q/displaystylew_{irightarrow[n]}=left.frac{2pi}{hbar},overline{|H_{ni}|^{,2}},rho(E_n)right|_{E_nsimeqE_i}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1926.png"
 ALT="$\displaystyle w_{i\rightarrow [n]} = \left.\frac{2\pi}{\hbar}  \overline{\vert H_{ni}\vert^{ 2}}  \rho(E_n) \right\vert _{E_n\simeq E_i}.$">|; 

$key = q/|cdotsrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="$ \vert\cdots \rangle$">|; 

$key = q/|psirangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2142.png"
 ALT="$ \vert\psi\rangle$">|; 

$key = q/displaystyle=pmfrac{2^7}{3^5},a_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2130.png"
 ALT="$\displaystyle = \pm\frac{2^7}{3^5} a_0,$">|; 

$key = q/displaystylesigma_{rmtotal};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2254.png"
 ALT="$\displaystyle \sigma_{\rm total}$">|; 

$key = q/displaystyle=psi^dag,psi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2640.png"
 ALT="$\displaystyle = \psi^\dag \psi,$">|; 

$key = q/E_i-E_m+{rmi},hbar,eta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1942.png"
 ALT="$ E_i -
E_m +{\rm i} \hbar \eta$">|; 

$key = q/displaystyle|Aranglestackrel{rmDC}{longleftrightarrow}langleA|,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="$\displaystyle \vert A \rangle \stackrel{\rm DC}{\longleftrightarrow}\langle A\vert,$">|; 

$key = q/|j,mrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1419.png"
 ALT="$ \vert j,m\rangle$">|; 

$key = q/|m_1|leq1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1399.png"
 ALT="$ \vert m_1\vert \leq 1/2$">|; 

$key = q/displaystyle{mitDelta}m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2087.png"
 ALT="$\displaystyle {\mit\Delta} m$">|; 

$key = q/displaystylelangleJ_kranglerightarrowsum_lR_{k,l},langleJ_lrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1097.png"
 ALT="$\displaystyle \langle J_k \rangle \rightarrow \sum_l R_{k l}  \langle J_l\rangle,$">|; 

$key = q/displaystyleR({mitDelta}varphi)=lim_{Nrightarrowinfty}left(1-{rmi},frac{{mitDelta}varphi}{N}frac{L_z}{hbar}right)^N,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img858.png"
 ALT="$\displaystyle R({\mit\Delta}\varphi) = \lim_{N\rightarrow\infty} \left(1-{\rm i}  \frac{{\mit\Delta} \varphi}{N} \frac{L_z}{\hbar}\right)^N,$">|; 

$key = q/lambda_3=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1582.png"
 ALT="$ \lambda_3=0$">|; 

$key = q/yrightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img956.png"
 ALT="$ y\rightarrow 0$">|; 

$key = q/|psi|^{,2}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img963.png"
 ALT="$ \vert\psi\vert^{ 2} =0$">|; 

$key = q/displaystyleP_{irightarrown}(t)=frac{|H_{ni}|^{,2},t^2}{hbar^2},{rmsinc}^2left[frac{(E_n-E_i),t}{2,hbar}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1895.png"
 ALT="$\displaystyle P_{i\rightarrow n}(t) = \frac{\vert H_{ni}\vert^{ 2}  t^2}{\hbar^2}  {\rm sinc}^2\left[ \frac{(E_n-E_i) t}{2 \hbar}\right],$">|; 

$key = q/k''';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2402.png"
 ALT="$ k'''$">|; 

$key = q/displaystyle=-frac{{rmi}}{hbar},H_{ni}int_0^tdt',exp[,{rmi},omega_{ni},(t'-t)]=frac{H_{ni}}{E_n-E_i},[1-exp(,{rmi},omega_{ni},t)],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="483" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1893.png"
 ALT="$\displaystyle = -\frac{{\rm i}}{\hbar}  H_{ni} \int_0^t dt' \exp[ {\rm i}  ...
...{ni} (t'-t)]= \frac{H_{ni}}{E_n - E_i}  [1- \exp( {\rm i} \omega_{ni} t)],$">|; 

$key = q/displaystyleH,|k''',k'',k'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2425.png"
 ALT="$\displaystyle H \vert k''' k'' k'\rangle$">|; 

$key = q/H_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1435.png"
 ALT="$ H_1$">|; 

$key = q/displaystyleS_krightarrowleft(frac{hbar}{2}right)sigma_k.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1172.png"
 ALT="$\displaystyle S_k \rightarrow \left(\frac{\hbar}{2} \right)\sigma_k.$">|; 

$key = q/displaystylej^{,mu'}=a^{,mu}_{~nu},j^{,nu},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2703.png"
 ALT="$\displaystyle j^{ \mu'} = a^{ \mu}_{&nbsp;\nu} j^{ \nu},$">|; 

$key = q/displaystyle{rmi},hbar,ddot{alpha}_1=2,dot{alpha}_1,H,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2718.png"
 ALT="$\displaystyle {\rm i} \hbar \ddot{\alpha}_1 = 2 \dot{\alpha}_1 H,$">|; 

$key = q/2,m_e,c^2slashh;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2724.png"
 ALT="$ 2 m_e c^2/h$">|; 

$key = q/displaystylecosalpha=langlem-1slash2,1slash2|l+1slash2,mrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1640.png"
 ALT="$\displaystyle \cos\alpha = \langle m-1/2,1/2\vert l+1/2, m\rangle,$">|; 

$key = q/{bfS}={bfS}_1+{bfS}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2456.png"
 ALT="$ {\bf S}={\bf S}_1+{\bf S}_2$">|; 

$key = q/P(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img944.png"
 ALT="$ P(r)$">|; 

$key = q/displaystylepsi({bfx})=frac{1}{(2pi)^{3slash2}}left[exp(,{rmi},k,rcostheta)+frac{exp(,{rmi},k,r),f(theta)}{r}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="385" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2220.png"
 ALT="$\displaystyle \psi({\bf x}) = \frac{1}{(2\pi)^{3/2}} \left[ \exp( {\rm i} k r\cos\theta) + \frac{\exp( {\rm i} k r)  f(\theta)} {r} \right],$">|; 

$key = q/displaystyle=left[1-frac{(mbox{boldmathsigma}cdot{bfn})^2}{2!}left(frac{{mitDeltfn})^4}{4!}left(frac{{mitDelta}varphi}{2}right)^4+cdotsright];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1224.png"
 ALT="$\displaystyle = \left[ 1 - \frac{(\mbox{\boldmath$\sigma$}\cdot {\bf n})^2}{2!}...
...t {\bf n})^4}{4!} \left(\frac{{\mit\Delta}\varphi}{2} \right)^4 + \cdots\right]$">|; 

$key = q/l>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img962.png"
 ALT="$ l&gt;0$">|; 

$key = q/tan(k',a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2327.png"
 ALT="$ \tan (k' a)$">|; 

$key = q/displaystyle=langlex',y',z'|langle-||Aranglerangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1066.png"
 ALT="$\displaystyle = \langle x', y', z' \vert\langle -\vert\vert A\rangle\rangle.$">|; 

$key = q/displaystylepsi_a(y)=frac{{rme}^{-yslasha}}{y},f(y),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2869.png"
 ALT="$\displaystyle \psi_a(y) = \frac{{\rm e}^{-y/a}}{y} f(y),$">|; 

$key = q/c_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2890.png"
 ALT="$ c_s$">|; 

$key = q/displaystylelangle1,0,0|,z^2,|1,0,0rangle=a_0^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1538.png"
 ALT="$\displaystyle \langle 1,0,0\vert z^2 \vert 1,0,0\rangle = a_0^{ 2}.$">|; 

$key = q/displaystylelangleA|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img134.png"
 ALT="$\displaystyle \langle A\vert$">|; 

$key = q/displaystyleH_t=T^{dag},H,T=H.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img579.png"
 ALT="$\displaystyle H_t = T^{\dag }  H  T = H.$">|; 

$key = q/displaystyle=frac{rmi}{hbar}sum_mfrac{H_{nm},H_{mi}}{E_m-E_i}int_0^tdt'left[exp(,{rmi},omega_{ni},t',)-exp(,{rmi},omega_{nm},t'],right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="413" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1931.png"
 ALT="$\displaystyle =\frac{\rm i}{\hbar} \sum_m \frac{H_{nm}  H_{mi}}{E_m - E_i} \in...
...[\exp( {\rm i} \omega_{ni} t' ) - \exp( {\rm i} \omega_{nm} t'] \right)$">|; 

$key = q/displaystylenablacdot{bfE}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2930.png"
 ALT="$\displaystyle \nabla\cdot {\bf E} = 0$">|; 

$key = q/2pi,a^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2308.png"
 ALT="$ 2\pi  a^2$">|; 

$key = q/sigma_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2350.png"
 ALT="$ \sigma_l$">|; 

$key = q/displaystylew_{irightarrown}=4pi^2,alpha,frac{c}{hbar},u(omega_{ni}),|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2077.png"
 ALT="$\displaystyle w_{i\rightarrow n} = 4\pi^2 \alpha \frac{c}{\hbar} u(\omega_{ni}) \vert$">|; 

$key = q/|-rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1839.png"
 ALT="$ \vert-\rangle$">|; 

$key = q/displaystyleD^{dag},D=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img453.png"
 ALT="$\displaystyle D^{\dag }  D = 1.$">|; 

$key = q/{mitDelta}p_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img440.png"
 ALT="$ {\mit\Delta} p_x$">|; 

$key = q/displaystyle[J^{,2},J_1^{,2}]=[J^{,2},J_2^{,2}]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1332.png"
 ALT="$\displaystyle [J^{ 2}, J_1^{ 2}] = [J^{ 2}, J_2^{ 2}] = 0.$">|; 

$key = q/S_{1,z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2477.png"
 ALT="$ S_{1 z}$">|; 

$key = q/displaystyleH=frac{p^2}{2,m_e}+V(r)-frac{p^4}{8,m_e^{,3},c^2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1762.png"
 ALT="$\displaystyle H = \frac{p^2}{2 m_e} + V(r) - \frac{p^4}{8 m_e^{ 3} c^2}.
$">|; 

$key = q/_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1426.png"
 ALT="$ _n$">|; 

$key = q/c^ast,langleA|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="$ c^\ast  \langle A\vert$">|; 

$key = q/displaystylembox{hspace{1cm}as{|{{bf{x}'|rightarrowinfty{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img638.png"
 ALT="$\displaystyle \mbox{\hspace{1cm}as $\vert{\bf x}'\vert\rightarrow \infty$}$">|; 

$key = q/c_+slashc_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1252.png"
 ALT="$ c_+/c_-$">|; 

$key = q/displaystyleH=frac{p_x^{,2}}{2,m}+frac{1}{2},m,omega^2,x^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img663.png"
 ALT="$\displaystyle H = \frac{p_x^{ 2}}{2 m}+ \frac{1}{2} m \omega^2 x^2,
$">|; 

$key = q/displaystyle=exp(0),|0,0rangle=|0,0rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img878.png"
 ALT="$\displaystyle =\exp(0) \vert,0\rangle = \vert,0\rangle,$">|; 

$key = q/intd^3x',|psi|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img971.png"
 ALT="$ \int d^3x'  \vert\psi\vert^{ 2}$">|; 

$key = q/{bfL}+(hbarslash2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2776.png"
 ALT="$ {\bf L} + (\hbar/2) $">|; 

$key = q/displaystyle=frac{l,hbar^2}{2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1669.png"
 ALT="$\displaystyle = \frac{l  \hbar^2}{2},$">|; 

$key = q/displaystyle[u,v]_{qm}=frac{u,v-v,u}{{rmi},hbar},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img287.png"
 ALT="$\displaystyle [u, v]_{qm} = \frac{u v - v u}{{\rm i}  \hbar},$">|; 

$key = q/E<V_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2319.png"
 ALT="$ E&lt;V_0$">|; 

$key = q/displaystyleL^-,L^+|l,m_{rmmax}rangle=|0rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img804.png"
 ALT="$\displaystyle L^-  L^+ \vert l, m_{\rm max}\rangle = \vert\rangle.$">|; 

$key = q/displaystyleleft[gamma^{,mu}left(p_mu+frac{e}{c},{mitPhi}_muright)-m_e,cright]psbar,partial_mu+frac{e}{c},{mitPhi}_muright)-m_e,cright]psi=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="474" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2624.png"
 ALT="$\displaystyle \left[\gamma^{ \mu}\left(p_\mu+\frac{e}{c} {\mit\Phi}_\mu\right...
...} \hbar \partial_\mu+\frac{e}{c} {\mit\Phi}_\mu\right)-m_e c\right]\psi=0 ,$">|; 

$key = q/displaystyle{calY}_{l,m}^{j=lpm1slash2}equiv{calY}_{l,m}^{pm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1656.png"
 ALT="$\displaystyle {\cal Y}_{l m}^{j=l\pm 1/2}\equiv {\cal Y}_{l m}^{\pm}$">|; 

$key = q/m>m_{rmmax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img802.png"
 ALT="$ m&gt; m_{\rm max}$">|; 

$key = q/r'slashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2164.png"
 ALT="$ r'/r$">|; 

$key = q/langle1slash2,1slash2;1,1|1slash2,1slash2;1slash2,1slash2rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1406.png"
 ALT="$ \langle 1/2, 1/2; 1, 1\vert
1/2, 1/2; 1/2, 1/2
\rangle$">|; 

$key = q/displaystyleL^2,|l,m_{rmmax}rangle=(L_z^{,2}+hbar,L_z),|l,m_{rmmax}rangle=m_{rmmax}(m_{rmmax}+1),hbar^2,|l,m_{rmmax}rangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="484" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img807.png"
 ALT="$\displaystyle L^2  \vert l, m_{\rm max}\rangle = (L_z^{ 2} + \hbar  L_z)  \...
...\rangle = m_{\rm max}(m_{\rm max} + 1)  \hbar^2  \vert l, m_{\rm max}\rangle.$">|; 

$key = q/displaystylelangle1|,H_1,|2rangle=langle2|,H_1,|1rangle^ast=gammaexp(,{rmi},omega,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1802.png"
 ALT="$\displaystyle \langle 1\vert H_1 \vert 2\rangle = \langle 2\vert H_1 \vert 1\rangle^\ast = \gamma \exp( {\rm i}  \omega t),$">|; 

$key = q/j_z,hbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1365.png"
 ALT="$ j_z \hbar$">|; 

$key = q/psi_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1246.png"
 ALT="$ \psi_+$">|; 

$key = q/c^{,-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2750.png"
 ALT="$ c^{ -2}$">|; 

$key = q/{bfx'}equiv(x_1',x_2',x_3');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img618.png"
 ALT="$ {\bf x'} \equiv (x_1', x_2', x_3')$">|; 

$key = q/displaystyle{rmi},hbar,frac{partialU(t_0,t)}{partialt}=(H_0+H_1),U(t_0,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1855.png"
 ALT="$\displaystyle {\rm i}  \hbar  \frac{\partial U(t_0, t)}{\partial t} = (H_0 + H_1)  U(t_0, t),$">|; 

$key = q/displaystyle{mitDelta}xi=xi-langlexirangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img163.png"
 ALT="$\displaystyle {\mit\Delta} \xi = \xi - \langle \xi \rangle,$">|; 

$key = q/displaystyle[J_1^{,2},J_{1,i}];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1328.png"
 ALT="$\displaystyle [J_1^{ 2}, J_{1 i} ]$">|; 

$key = q/v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img248.png"
 ALT="$ v$">|; 

$key = q/displaystylelangleA|,xi^{,2},|Arangle=langleA|,xi^{dag},xi,|Arangle=langleB|Brangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img789.png"
 ALT="$\displaystyle \langle A\vert \xi^{ 2} \vert A\rangle = \langle A\vert \xi^{\dag }  \xi \vert A\rangle = \langle B\vert B\rangle,$">|; 

$key = q/displaystyle=(langlel,m+1|,L_x,|l,mrangle+{rmi},langlel,m+1|,L_y,|l,mrangle)^ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img822.png"
 ALT="$\displaystyle =( \langle l, m+1\vert  L_x \vert l,m \rangle + {\rm i}  \langle l, m+1\vert  L_y \vert l,m \rangle)^\ast$">|; 

$key = q/alpha_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="$ \alpha_i$">|; 

$key = q/X^{dag};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img97.png"
 ALT="$ X^{\dag }$">|; 

$key = q/f(x');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img319.png"
 ALT="$ f(x')$">|; 

$key = q/p_i(t)=p_i(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img600.png"
 ALT="$ p_i(t) = p_i(0)$">|; 

$key = q/displaystyleintd^{,3}x',rho({bfx'},t)=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img646.png"
 ALT="$\displaystyle \int d^{ 3} x' \rho({\bf x'}, t) = 1.$">|; 

$key = q/|langlexi'|Arangle|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img307.png"
 ALT="$ \vert\langle \xi'\vert A\rangle\vert^{ 2}$">|; 

$key = q/displaystylepsi({bfx})=phi({bfx})+frac{2,m}{hbar^2}intd^3x',G({bfx},{bfx}'),langle{bfx}'|,H_1,|psirangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="350" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2148.png"
 ALT="$\displaystyle \psi({\bf x}) = \phi({\bf x}) + \frac{2 m}{\hbar^2} \int d^3 x' G({\bf x}, {\bf x}')  \langle {\bf x}' \vert H_1 \vert\psi\rangle,$">|; 

$key = q/rho(E_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1918.png"
 ALT="$ \rho(E_n)$">|; 

$key = q/displaystylee_{mk}=langlem|,H_1,|krangle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1472.png"
 ALT="$\displaystyle e_{mk} = \langle m \vert H_1 \vert k\rangle.$">|; 

$key = q/displaystylelangleF|(|Arangle)=sum_{i=1,N}f_i,alpha_i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="$\displaystyle \langle F\vert(\vert A\rangle) = \sum_{i=1,N} f_i  \alpha_i,$">|; 

$key = q/|x|stackrel{_{normalsize<}}{_{normalsizesim}}pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1900.png"
 ALT="$ \vert x\vert \stackrel {_{\normalsize &lt;}}{_{\normalsize\sim}}\pi$">|; 

$key = q/displaystyle=frac{1}{sqrt{2}}left(!{array}{rrr}0&1&01&0&10&1&0{array}!right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="88" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1285.png"
 ALT="$\displaystyle =\frac{1}{\sqrt{2}}\left(\! \begin{array}{rrr} 0 &amp;1&amp;0\ 1&amp;0&amp;1\ 0&amp;1&amp;0\end{array}\!\right),$">|; 

$key = q/I;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2533.png"
 ALT="$ I$">|; 

$key = q/displaystyle=c,psi^dag,gamma^0,gamma^i,psi=psi^dag,c,alpha_i,psi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2642.png"
 ALT="$\displaystyle = c \psi^\dag \gamma^0 \gamma^i \psi = \psi^\dag c \alpha_i \psi,$">|; 

$key = q/displaystylesqrt{j_1,(j_1+1)-m_1,(m_1mp1)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1391.png"
 ALT="$\displaystyle \sqrt{j_1 (j_1+1) - m_1 (m_1\mp 1)}$">|; 

$key = q/alpha=e^2slash(4pi,epsilon_0,hbar,c);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img212.png"
 ALT="$ \alpha=e^2/(4\pi \epsilon_0 \hbar c)$">|; 

$key = q/displaystyle=-frac{rmi}{hbar},H_{ni}int_{-infty}^tdt',exp[(eta+{rmi},omega_{ni})i},frac{exp[(eta+{rmi},omega_{ni}),t]}{eta+{rmi},omega_{ni}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="483" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1950.png"
 ALT="$\displaystyle = - \frac{\rm i}{\hbar}  H_{ni} \int_{-\infty}^t dt'  \exp[(\et...
..., \frac{ \exp[(\eta + {\rm i} \omega_{ni} )  t]}{\eta +{\rm i} \omega_{ni}},$">|; 

$key = q/y'+dy';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1067.png"
 ALT="$ y'+dy'$">|; 

$key = q/displaystyle[x_i,F({bfx},{bfp})];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img591.png"
 ALT="$\displaystyle [x_i, F({\bf x},{\bf p})]$">|; 

$key = q/displaystyle=delta_{in},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1880.png"
 ALT="$\displaystyle = \delta_{in},$">|; 

$key = q/psi_-({bfx'});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1244.png"
 ALT="$ \psi_-({\bf x'})$">|; 

$key = q/displaystyle[L_x,H]={rmi},hbar,c,gamma^5,(Sigma_2,p_z-Sigma_3,p_y),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2764.png"
 ALT="$\displaystyle [L_x,H] = {\rm i} \hbar c \gamma^5 (\Sigma_2 p_z-\Sigma_3 p_y),$">|; 

$key = q/displaystyleH_0=frac{p^2}{2,m_e}+V(r),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2046.png"
 ALT="$\displaystyle H_0 = \frac{p^2}{2 m_e} + V(r),$">|; 

$key = q/j=l+1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1631.png"
 ALT="$ j=l+1/2$">|; 

$key = q/M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img923.png"
 ALT="$ M$">|; 

$key = q/displaystylechi^{dag}equiv(langleA|+rangle,langleA|-rangle)=(c_+^{~ast},c_-^{~ast})=c_+^{~ast},chi_+^{dag}+c_-^{~ast},chi_-^{dag}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="367" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1154.png"
 ALT="$\displaystyle \chi^{\dag } \equiv (\langle A\vert+\rangle, \langle A\vert-\rang...
...ast}, c_-^{&nbsp;\ast}) =c_+^{&nbsp;\ast}  \chi_+^{\dag } + c_-^{&nbsp;\ast} \chi_-^{\dag }.$">|; 

$key = q/displaystyle=langlel,m+1|,L^+,|l,mrangle^ast,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img823.png"
 ALT="$\displaystyle = \langle l, m+1\vert L^+ \vert l, m\rangle^\ast,$">|; 

$key = q/{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1108.png"
 ALT="$ {\bf v}$">|; 

$key = q/t=t_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1778.png"
 ALT="$ t=t_0$">|; 

$key = q/u_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img278.png"
 ALT="$ u_1$">|; 

$key = q/r=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img936.png"
 ALT="$ r=0$">|; 

$key = q/displaystyleY_{l,,-m}=(-1)^m,Y_{l,m}^{,ast},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img910.png"
 ALT="$\displaystyle Y_{l  -m} = (-1)^m  Y_{l m}^{ \ast},$">|; 

$key = q/displaystylephi({bfx}_1',{bfx}_2')=frac{1}{sqrt{2}}left[omega_A({bfx}_1'),omega_B({bfx}_2')+omega_B({bfx}_1'),omega_A({bfx}_2')right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="379" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2481.png"
 ALT="$\displaystyle \phi({\bf x}_1',{\bf x}_2') = \frac{1}{\sqrt{2}}\left[\omega_A({\bf x}_1') \omega_B({\bf x}_2')+\omega_B({\bf x}_1') \omega_A({\bf x}_2')\right].$">|; 

$key = q/displaystyle=[u,v_1]+[u,v_2],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img498.png"
 ALT="$\displaystyle = [u, v_1] + [u, v_2],$">|; 

$key = q/displaystyle|Arangle=int_{-infty}^{+infty}dx',langlex'|Arangle|x'rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img303.png"
 ALT="$\displaystyle \vert A\rangle = \int_{-\infty}^{+\infty} dx'  \langle x'\vert A\rangle \vert x'\rangle$">|; 

$key = q/displaystylesum_{l=1,N_n}|n,lranglelanglen,l|=1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1559.png"
 ALT="$\displaystyle \sum_{l=1,N_n} \vert n,l\rangle \langle n,l\vert = 1,$">|; 

$key = q/displaystyleEsimeqfrac{1}{2},(E_1+E_2)pmfrac{1}{2},(E_1-E_2),(1+2,epsilon^2+cdots).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="364" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1456.png"
 ALT="$\displaystyle E\simeq \frac{1}{2}  (E_1+E_2) \pm \frac{1}{2} (E_1-E_2) (1+2 \epsilon^2 + \cdots).$">|; 

$key = q/displaystylepsi_A(x')={rme}^{,{rmi},k',x'},u(x'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img526.png"
 ALT="$\displaystyle \psi_A(x') = {\rm e}^{ {\rm i} k' x'} u(x'),
$">|; 

$key = q/displaystylepsi'=A,psi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2671.png"
 ALT="$\displaystyle \psi' = A \psi,$">|; 

$key = q/lambda_{23};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2435.png"
 ALT="$ \lambda_{23}$">|; 

$key = q/displaystyletilde{A}_0(r)=frac{sin(k,r)}{k,r}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2289.png"
 ALT="$\displaystyle \tilde{A}_0(r) = \frac{ \sin (k r)}{k r}.$">|; 

$key = q/displaystylew_{irightarrown}=frac{2pi}{hbar},|H_{ni}|^{,2},delta(E_n-E),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1928.png"
 ALT="$\displaystyle w_{i\rightarrow n} = \frac{2\pi}{\hbar}  \vert H_{ni}\vert^{ 2}  \delta(E_n - E),$">|; 

$key = q/45^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$ 45^\circ$">|; 

$key = q/{mitDelta}omega_mu^{~nu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2681.png"
 ALT="$ {\mit\Delta}\omega_\mu^{&nbsp;\nu}$">|; 

$key = q/z';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1048.png"
 ALT="$ z'$">|; 

$key = q/tausimeq1.6times10^{-11},{rms};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img213.png"
 ALT="$ \tau\simeq 1.6\times 10^{-11} {\rm s}$">|; 

$key = q/displaystyle[u,[v,w]_{cl}]_{cl}+[v,[w,u]_{cl}]_{cl}+[w,[u,v]_{cl}]_{cl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="317" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img267.png"
 ALT="$\displaystyle [u, [v, w]_{cl} ]_{cl}+ [v, [w, u]_{cl} ]_{cl} + [w, [u, v]_{cl}]_{cl}$">|; 

$key = q/displaystyleH=-frac{hbar^2}{2,M},nabla^2+V(r).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img926.png"
 ALT="$\displaystyle H = -\frac{\hbar^2}{2 M}  \nabla^2 + V(r).$">|; 

$key = q/displaystyleP_{12},|k',k'',k'''rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2403.png"
 ALT="$\displaystyle P_{12} \vert k' k'' k'''\rangle$">|; 

$key = q/J_1^{,2},J_2^{,2},J_{1,z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1337.png"
 ALT="$ J_1^{ 2}, J_2^{ 2}, J_{1 z}$">|; 

$key = q/displaystyle=langle-|,S_k,|+ranglelanglex',y',z'|langle+||Aranglerangle+langle-|,S_k,|-ranglelanglex',y',z'|langle-||Aranglerangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="432" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1185.png"
 ALT="$\displaystyle = \langle - \vert S_k  \vert+\rangle \langle x',y',z'\vert\lang...
... S_k  \vert-\rangle \langle x',y',z'\vert\langle -\vert\vert A\rangle\rangle,$">|; 

$key = q/(-1,-1slash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1728.png"
 ALT="$ (-1,-1/2)$">|; 

$key = q/displaystyle=intdxi',|xi'ranglelanglexi'|Arangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img198.png"
 ALT="$\displaystyle = \int d\xi' \vert\xi'\rangle\langle \xi'\vert A\rangle,$">|; 

$key = q/{bfx}cdot{bfp}=r,p_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2849.png"
 ALT="$ {\bf x}\cdot{\bf p}=r p_r$">|; 

$key = q/displaystylelangle1|,H_1,|1rangle=langle2|,H_1,|2rangle=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1801.png"
 ALT="$\displaystyle \langle 1\vert H_1 \vert 1\rangle = \langle 2\vert H_1 \vert 2\rangle = 0.$">|; 

$key = q/Texp(,{rmi},gamma);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img550.png"
 ALT="$ T\exp( {\rm i} \gamma)$">|; 

$key = q/|H_{ni}|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1919.png"
 ALT="$ \vert H_{ni}\vert^{ 2}$">|; 

$key = q/(-1,1slash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1726.png"
 ALT="$ (-1, 1/2)$">|; 

$key = q/c_{s+1}=c_{s+1}'=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2906.png"
 ALT="$ c_{s+1}=c_{s+1}'=0$">|; 

$key = q/displaystylecdot{bfp}+beta,m_e,c^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2705.png"
 ALT="$\displaystyle \cdot{\bf p} + \beta m_e c^2.$">|; 

$key = q/m_e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img206.png"
 ALT="$ m_e$">|; 

$key = q/displaystyle{bfe}_r=frac{bfx}{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2165.png"
 ALT="$\displaystyle {\bf e}_r = \frac{\bf x}{r}$">|; 

$key = q/{rmRe}({mitDelta}_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1996.png"
 ALT="$ {\rm Re}({\mit\Delta}_i)$">|; 

$key = q/psi(r,theta,varphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img901.png"
 ALT="$ \psi(r, \theta, \varphi)$">|; 

$key = q/{bfx}_{n,l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1566.png"
 ALT="$ {\bf x}_{n l}$">|; 

$key = q/j,(j+1),hbar^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1010.png"
 ALT="$ j (j+1) \hbar^2$">|; 

$key = q/displaystyleg_{00};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2556.png"
 ALT="$\displaystyle g_{00}$">|; 

$key = q/displaystylew_{irightarrown};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2017.png"
 ALT="$\displaystyle w_{i\rightarrow n}$">|; 

$key = q/displaystylelanglem|nrangle'=delta_{mn}+frac{e_{nm}^{,ast}}{E_m-E_n}+frac{e_{mn}}{E_n-E_m}+O(epsilon^2)=delta_{mn}+O(epsilon^2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="452" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1492.png"
 ALT="$\displaystyle \langle m\vert n\rangle' = \delta_{mn} + \frac{e_{nm}^{ \ast}}{E_m-E_n} + \frac{e_{mn}} {E_n-E_m} + O(\epsilon^2) = \delta_{mn} + O(\epsilon^2).$">|; 

$key = q/N_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2107.png"
 ALT="$ N_1$">|; 

$key = q/2,n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2933.png"
 ALT="$ 2 n$">|; 

$key = q/sqrt{langleA|Arangle};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img76.png"
 ALT="$ \sqrt{\langle A\vert A \rangle }$">|; 

$key = q/displaystyle=[u_1,v_1]_{qm},v_2,u_2+v_1,[u_1,v_2]_{qm},u_2+u_1,[u_2,v_1]_{qm},v_2+u_1,v_1,[u_2,v_2]_{qm},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="513" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img274.png"
 ALT="$\displaystyle = [u_1, v_1]_{qm}  v_2  u_2 + v_1 [u_1, v_2]_{qm}  u_2 + u_1 [u_2, v_1]_{qm} v_2 + u_1  v_1 [u_2, v_2]_{qm},$">|; 

$key = q/c_{s-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2881.png"
 ALT="$ c_{s-1}$">|; 

$key = q/displaystylepsi(r,theta,varphi)=R(r),Y(theta,varphi).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img903.png"
 ALT="$\displaystyle \psi(r, \theta, \varphi) = R(r)  Y(\theta, \varphi).$">|; 

$key = q/displaystylecdot{bfL}+hbar)^2=J^{,2}+frac{1}{4},hbar^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2826.png"
 ALT="$\displaystyle \cdot{\bf L}+\hbar)^2 = J^{ 2}+\frac{1}{4} \hbar^2,$">|; 

$key = q/displaystyle=-{rmi},hbarleft(z,frac{partial}{partialx}-x,frac{partial}{partialz}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img886.png"
 ALT="$\displaystyle = -{\rm i} \hbar\left(z \frac{\partial}{\partial x} - x \frac{\partial} {\partial z}\right),$">|; 

$key = q/{bfJ};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1322.png"
 ALT="$ {\bf J}$">|; 

$key = q/1sleftrightarrow2p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2128.png"
 ALT="$ 1s\leftrightarrow 2p$">|; 

$key = q/displaystylerightarrowleft(!{array}{c}01{array}!right)equivchi_-.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1145.png"
 ALT="$\displaystyle \rightarrow \left(\!\begin{array}{c}0\ 1\end{array}\!\right) \equiv \chi_-.$">|; 

$key = q/Sigma_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2936.png"
 ALT="$ \Sigma_i$">|; 

$key = q/V_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2198.png"
 ALT="$ V_0$">|; 

$key = q/k'asimeq4.49;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2334.png"
 ALT="$ k'a\simeq 4.49$">|; 

$key = q/displaystyle[L_x,r];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2758.png"
 ALT="$\displaystyle [L_x,r]$">|; 

$key = q/displaystylefrac{-{rmi},gammaslashhbar}{[gamma^2slashhbar^2+(omega-omega_{21})^{slashhbar^2+(omega-omega_{21})^{,2}slash4]^{1slash2},tright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="626" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1813.png"
 ALT="$\displaystyle \frac{-{\rm i}  \gamma/\hbar} {[\gamma^2/\hbar^2 + (\omega-\omeg...
.../2] \sin\left([\gamma^2/\hbar^2+(\omega-\omega_{21})^{ 2}/4]^{1/2} t\right),$">|; 

$key = q/displaystyle=|k'',k',k'''rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2404.png"
 ALT="$\displaystyle = \vert k'' k' k'''\rangle,$">|; 

$key = q/gamma^0,gamma^0=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2698.png"
 ALT="$ \gamma^0 \gamma^0=1$">|; 

$key = q/displaystyleV_{ni};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2003.png"
 ALT="$\displaystyle V_{ni}$">|; 

$key = q/displaystyledot{L}_x+frac{hbar}{2},dot{Sigma}_1=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2775.png"
 ALT="$\displaystyle \dot{L}_x +\frac{\hbar}{2} \dot{\Sigma}_1 = 0.$">|; 

$key = q/10^5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2094.png"
 ALT="$ 10^5$">|; 

$key = q/displaystyle[u,c];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img494.png"
 ALT="$\displaystyle [u, c]$">|; 

$key = q/gamma'equivgamma(x');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img353.png"
 ALT="$ \gamma'\equiv \gamma(x')$">|; 

$key = q/|At_0rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img546.png"
 ALT="$ \vert A t_0\rangle$">|; 

$key = q/displaystylecdot{bfn})^k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1216.png"
 ALT="$\displaystyle \cdot {\bf n})^k$">|; 

$key = q/|xi''rangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img127.png"
 ALT="$ \vert\xi''\rangle$">|; 

$key = q/displaystyle=j_1,(j_1+1),hbar^2,|j_1,j_2;j,mrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1353.png"
 ALT="$\displaystyle = j_1 (j_1+1) \hbar^2 \vert j_1,j_2; j,m\rangle,$">|; 

$key = q/displaystylelanglep_x'|p_x''rangle=int_{-infty}^{+infty}dx',langlep_x'|x'rangle,int_{-infty}^{infty}dx',exp[-{rmi},(p_x'-p_x''),x'slashhbar].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="568" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img423.png"
 ALT="$\displaystyle \langle p_x'\vert p_x''\rangle = \int_{-\infty}^{+\infty} dx' \l...
...ast}  c'' \int_{-\infty}^{\infty}dx' \exp[-{\rm i}  (p_x'-p_x'') x'/\hbar].$">|; 

$key = q/displaystyleD(a),x^m,D(a)^{,dag}=(x-a)^m.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img515.png"
 ALT="$\displaystyle D(a) x^m D(a)^{ \dag } = (x-a)^m.
$">|; 

$key = q/displaystyleleftlanglefrac{a_0^{,3}}{r^3}rightrangle;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1769.png"
 ALT="$\displaystyle \left\langle \frac{a_0^{ 3}}{r^3}\right\rangle$">|; 

$key = q/displaystyle{mitDelta}omega^nu_{~beta},gamma^{,beta}=-frac{rmi}{4},{mitDelta}omeamma^nu,sigma_{alpha,beta}-sigma_{alpha,beta},gamma^nuright).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2688.png"
 ALT="$\displaystyle {\mit\Delta}\omega^\nu_{&nbsp;\beta} \gamma^{ \beta} = -\frac{\rm i}...
...(\gamma^\nu \sigma_{\alpha \beta}- \sigma_{\alpha \beta} \gamma^\nu\right).$">|; 

1;

