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Spherical harmonics

The simultaneous eigenstates, $Y_{l,m}(\theta,\phi)$, of $L^2$ and $L_z$ are known as the spherical harmonics. Let us investigate their functional form.

Now, we know that

\begin{displaymath}
L_+\,Y_{l,l}(\theta,\phi) = 0,
\end{displaymath} (570)

since there is no state for which $m$ has a larger value than $+l$. Writing
\begin{displaymath}
Y_{l,l}(\theta,\phi) = \Theta_{l,l}(\theta)\,{\rm e}^{\,{\rm i}\,l\,\phi}
\end{displaymath} (571)

[see Eqs. (549) and (553)], and making use of Eq. (534), we obtain
\begin{displaymath}
\hbar\,{\rm e}^{\,{\rm i}\,\phi}\left(\frac{\partial}{\parti...
...ial\phi}\right)\Theta_{l,l}(\theta)\,{\rm e}^{\,i\,l\,\phi}=0.
\end{displaymath} (572)

This equation yields
\begin{displaymath}
\frac{d\Theta_{l,l}}{d\theta} - l\,\cot\theta\,\Theta_{l,l} = 0.
\end{displaymath} (573)

which can easily be solved to give
\begin{displaymath}
\Theta_{l,l}\sim (\sin\theta)^l.
\end{displaymath} (574)

Hence, we conclude that
\begin{displaymath}
Y_{l,l}(\theta,\phi) \sim (\sin\theta)^l\,{\rm e}^{\,{\rm i}\,l\,\phi}.
\end{displaymath} (575)

Likewise, it is easy to demonstrate that
\begin{displaymath}
Y_{l,-l}(\theta,\phi) \sim (\sin\theta)^l\,{\rm e}^{-{\rm i}\,l\,\phi}.
\end{displaymath} (576)

Once we know $Y_{l,l}$, we can obtain $Y_{l,l-1}$ by operating on $Y_{l,l}$ with the lowering operator $L_-$. Thus,

\begin{displaymath}
Y_{l,l-1} \sim L_-\,Y_{l,l} \sim {\rm e}^{-{\rm i}\,\phi}\le...
...tial\phi}\right) (\sin\theta)^l\,{\rm e}^{\,{\rm i}\,l\,\phi},
\end{displaymath} (577)

where use has been made of Eq. (534). The above equation yields
\begin{displaymath}
Y_{l,l-1}\sim {\rm e}^{\,{\rm i}\,(l-1)\,\phi}\left(\frac{d}{d\theta} +l\,\cot\theta\right)(\sin\theta)^l.
\end{displaymath} (578)

Now,
\begin{displaymath}
\left(\frac{d}{d\theta}+l\,\cot\theta\right)f(\theta)\equiv
...
...a)^l}\frac{d}{d\theta}\left[
(\sin\theta)^l\,f(\theta)\right],
\end{displaymath} (579)

where $f(\theta)$ is a general function. Hence, we can write
\begin{displaymath}
Y_{l,l-1}(\theta,\phi)\sim \frac{{\rm e}^{\,{\rm i}\,(l-1)\,...
...ac{1}{\sin\theta}\frac{d}{d\theta}\right)
(\sin\theta)^{2\,l}.
\end{displaymath} (580)

Likewise, we can show that
\begin{displaymath}
Y_{l,-l+1}(\theta,\phi)\sim L_+\,Y_{l,-l}\sim \frac{{\rm e}^...
...ac{1}{\sin\theta}\frac{d}{d\theta}\right)
(\sin\theta)^{2\,l}.
\end{displaymath} (581)

We can now obtain $Y_{l,l-2}$ by operating on $Y_{l,l-1}$ with the lowering operator. We get

\begin{displaymath}
Y_{l,l-2}\sim L_-\,Y_{l,l-1}\sim {\rm e}^{-{\rm i}\,\phi}\le...
...ac{1}{\sin\theta}\frac{d}{d\theta}\right)
(\sin\theta)^{2\,l},
\end{displaymath} (582)

which reduces to
\begin{displaymath}
Y_{l,l-2}\sim {\rm e}^{-{\rm i}\,(l-2)\,\phi}\left[\frac{d}{...
...ac{1}{\sin\theta}\frac{d}{d\theta}\right)
(\sin\theta)^{2\,l}.
\end{displaymath} (583)

Finally, making use of Eq. (579), we obtain
\begin{displaymath}
Y_{l,l-2}(\theta,\phi) \sim \frac{{\rm e}^{\,{\rm i}\,(l-2)\...
...{1}{\sin\theta}\frac{d}{d\theta}\right)^2
(\sin\theta)^{2\,l}.
\end{displaymath} (584)

Likewise, we can show that
\begin{displaymath}
Y_{l,-l+2}(\theta,\phi) \sim L_+\,Y_{l,-l+1}\sim \frac{{\rm ...
...{1}{\sin\theta}\frac{d}{d\theta}\right)^2
(\sin\theta)^{2\,l}.
\end{displaymath} (585)

A comparison of Eqs. (575), (580), and (584) reveals the general functional form of the spherical harmonics:

\begin{displaymath}
Y_{l,m}(\theta,\phi)\sim \frac{{\rm e}^{\,{\rm i}\,m\,\phi}}...
...\sin\theta}\frac{d}{d\theta}\right)^{l-m}
(\sin\theta)^{2\,l}.
\end{displaymath} (586)

Here, $m$ is assumed to be non-negative. Making the substitution $u=\cos\theta$, we can also write
\begin{displaymath}
Y_{l,m}(u,\phi)\sim {\rm e}^{\,{\rm i}\,m\,\phi}\,(1-u^2)^{-m/2}\left(\frac{d}{d u}\right)^{l-m}
(1-u^2)^l.
\end{displaymath} (587)

Finally, it is clear from Eqs. (576), (581), and (585) that
\begin{displaymath}
Y_{l,-m} \sim Y^{\,\ast}_{l,m}.
\end{displaymath} (588)

We now need to normalize our spherical harmonic functions so as to ensure that

\begin{displaymath}
\oint \vert Y_{l,m}(\theta,\phi)\vert^2\,d\Omega = 1.
\end{displaymath} (589)

After a great deal of tedious analysis, the normalized spherical harmonic functions are found to take the form
\begin{displaymath}
Y_{l,m}(\theta,\phi) =(-1)^m\, \left[\frac{2\,l+1}{4\,\pi}\,...
...right]^{1/2} P_{l,m}(\cos\theta)\,{\rm e}^{\,{\rm i}\,m\,\phi}
\end{displaymath} (590)

for $m\geq 0$, where the $P_{l,m}$ are known as associated Legendre polynomials, and are written
\begin{displaymath}
P_{l,m}(u) = (-1)^{l+m}\,\frac{(l+m)!}{(l-m)!}\,\frac{(1-u^2)^{-m/2}}{2^l\,l!}\left(\frac{d}{du}\right)^{l-m} (1-u^2)^l
\end{displaymath} (591)

for $m\geq 0$. Alternatively,
\begin{displaymath}
P_{l,m}(u) = (-1)^{l}\,\frac{(1-u^2)^{m/2}}{2^l\,l!}\left(\frac{d}{du}\right)^{l+m} (1-u^2)^l,
\end{displaymath} (592)

for $m\geq 0$. The spherical harmonics characterized by $m<0$ can be calculated from those characterized by $m>0$ via the identity
\begin{displaymath}
Y_{l,-m} = (-1)^m\,Y^{\,\ast}_{l,m}.
\end{displaymath} (593)

The spherical harmonics are orthonormal: i.e.,
\begin{displaymath}
\oint Y_{l',m'}^{\,\ast}\,Y_{l,m}\,d\Omega = \delta_{ll'}\,\delta_{mm'},
\end{displaymath} (594)

and also form a complete set. In other words, any function of $\theta $ and $\phi$ can be represented as a superposition of spherical harmonics. Finally, and most importantly, the spherical harmonics are the simultaneous eigenstates of $L_z$ and $L^2$ corresponding to the eigenvalues $m\,\hbar$ and $l\,(l+1)\,\hbar^2$, respectively.

Figure 15: The $\vert Y_{l,m}(\theta ,\phi )\vert^{\,2}$ plotted as a functions of $\theta $. The solid, short-dashed, and long-dashed curves correspond to $l,m=0,0$, and $1,0$, and $1,\pm 1$, respectively.
\begin{figure}
\epsfysize =4in
\centerline{\epsffile{ylm1.eps}}
\end{figure}

All of the $l=0$, $l=1$, and $l=2$ spherical harmonics are listed below:

$\displaystyle Y_{0,0}$ $\textstyle =$ $\displaystyle \frac{1}{\sqrt{4\pi}},$ (595)
$\displaystyle Y_{1,0}$ $\textstyle =$ $\displaystyle \sqrt{\frac{3}{4\pi}}\,\cos\theta,$ (596)
$\displaystyle Y_{1,\pm1}$ $\textstyle =$ $\displaystyle \mp \sqrt{\frac{3}{8\pi}}\,\sin\theta\,{\rm e}^{\pm{\rm i}\,\phi},$ (597)
$\displaystyle Y_{2,0}$ $\textstyle =$ $\displaystyle \sqrt{\frac{5}{16\pi}}\,(3\,\cos^2\theta - 1),$ (598)
$\displaystyle Y_{2,\pm 1}$ $\textstyle =$ $\displaystyle \mp\sqrt{\frac{15}{8\pi}}\,\sin\theta\,\cos\theta\,{\rm e}^{\pm{\rm i}\,\phi},$ (599)
$\displaystyle Y_{2,\pm 2}$ $\textstyle =$ $\displaystyle \sqrt{\frac{15}{32\pi}}\,\sin^2\theta\,{\rm e}^{\pm 2\,{\rm i}\,\phi}.$ (600)

The $\theta $ variation of these functions is illustrated in Figs. 15 and 16.

Figure 16: The $\vert Y_{l,m}(\theta ,\phi )\vert^{\,2}$ plotted as a functions of $\theta $. The solid, short-dashed, and long-dashed curves correspond to $l,m=2,0$, and $2,\pm 1$, and $2,\pm 2$, respectively.
\begin{figure}
\epsfysize =4in
\centerline{\epsffile{ylm2.eps}}
\end{figure}



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next up previous contents
Next: Problems Up: Orbital angular momentum Previous: Eigenvalues of   Contents
Richard Fitzpatrick 2006-12-12