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Spherical Harmonics
The simultaneous eigenstates,
, of
and
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}](img1436.png) |
(591) |
since there is no state for which
has a larger value than
.
Writing
![\begin{displaymath}
Y_{l,l}(\theta,\phi) = \Theta_{l,l}(\theta) {\rm e}^{ {\rm i} l \phi}
\end{displaymath}](img1438.png) |
(592) |
[see Eqs. (570) and (574)], and making use of
Eq. (555), 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}](img1439.png) |
(593) |
This equation yields
![\begin{displaymath}
\frac{d\Theta_{l,l}}{d\theta} - l \cot\theta \Theta_{l,l} = 0.
\end{displaymath}](img1440.png) |
(594) |
which can easily be solved to give
![\begin{displaymath}
\Theta_{l,l}\sim (\sin\theta)^l.
\end{displaymath}](img1441.png) |
(595) |
Hence, we conclude that
![\begin{displaymath}
Y_{l,l}(\theta,\phi) \sim (\sin\theta)^l {\rm e}^{ {\rm i} l \phi}.
\end{displaymath}](img1442.png) |
(596) |
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}](img1443.png) |
(597) |
Once we know
, we can obtain
by operating
on
with the lowering operator
. 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}](img1446.png) |
(598) |
where use has been made of Eq. (555).
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}](img1447.png) |
(599) |
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}](img1448.png) |
(600) |
where
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}](img1450.png) |
(601) |
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}](img1451.png) |
(602) |
We can now obtain
by operating on
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}](img1453.png) |
(603) |
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}](img1454.png) |
(604) |
Finally, making use of Eq. (600), 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}](img1455.png) |
(605) |
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}](img1456.png) |
(606) |
A comparison of Eqs. (596), (601), and (605)
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}](img1457.png) |
(607) |
Here,
is assumed to be non-negative. Making the substitution
, 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}](img1459.png) |
(608) |
Finally, it is clear from Eqs. (597), (602), and (606)
that
![\begin{displaymath}
Y_{l,-m} \sim Y^{ \ast}_{l,m}.
\end{displaymath}](img1460.png) |
(609) |
Figure 18:
The
plotted as a functions of
. The solid, short-dashed, and long-dashed curves correspond to
, and
, and
, respectively.
![\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter08/fig01.eps}}
\end{figure}](img1461.png) |
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}](img1462.png) |
(610) |
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} \f...
...right]^{1/2} P_{l,m}(\cos\theta) {\rm e}^{ {\rm i} m \phi}
\end{displaymath}](img1463.png) |
(611) |
for
, where the
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}](img1466.png) |
(612) |
for
. 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}](img1467.png) |
(613) |
for
.
The spherical harmonics characterized by
can be calculated from those characterized by
via the identity
![\begin{displaymath}
Y_{l,-m} = (-1)^m Y^{ \ast}_{l,m}.
\end{displaymath}](img1470.png) |
(614) |
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}](img1471.png) |
(615) |
and also form a complete set. In other words,
any function of
and
can be represented as
a superposition of spherical harmonics. Finally, and most importantly,
the spherical harmonics are the simultaneous eigenstates of
and
corresponding to the eigenvalues
and
,
respectively.
Figure 19:
The
plotted as a functions of
. The solid, short-dashed, and long-dashed curves correspond to
, and
, and
, respectively.
![\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter08/fig02.eps}}
\end{figure}](img1472.png) |
All of the
,
, and
spherical harmonics are listed below:
![$\displaystyle Y_{0,0}$](img1473.png) |
![$\textstyle =$](img122.png) |
![$\displaystyle \frac{1}{\sqrt{4\pi}},$](img1474.png) |
(616) |
![$\displaystyle Y_{1,0}$](img1475.png) |
![$\textstyle =$](img122.png) |
![$\displaystyle \sqrt{\frac{3}{4\pi}} \cos\theta,$](img1476.png) |
(617) |
![$\displaystyle Y_{1,\pm1}$](img1477.png) |
![$\textstyle =$](img122.png) |
![$\displaystyle \mp \sqrt{\frac{3}{8\pi}} \sin\theta {\rm e}^{\pm{\rm i} \phi},$](img1478.png) |
(618) |
![$\displaystyle Y_{2,0}$](img1479.png) |
![$\textstyle =$](img122.png) |
![$\displaystyle \sqrt{\frac{5}{16\pi}} (3 \cos^2\theta - 1),$](img1480.png) |
(619) |
![$\displaystyle Y_{2,\pm 1}$](img1481.png) |
![$\textstyle =$](img122.png) |
![$\displaystyle \mp\sqrt{\frac{15}{8\pi}} \sin\theta \cos\theta {\rm e}^{\pm{\rm i} \phi},$](img1482.png) |
(620) |
![$\displaystyle Y_{2,\pm 2}$](img1483.png) |
![$\textstyle =$](img122.png) |
![$\displaystyle \sqrt{\frac{15}{32\pi}} \sin^2\theta {\rm e}^{\pm 2 {\rm i} \phi}.$](img1484.png) |
(621) |
The
variation of these functions is illustrated in Figs. 18 and
19.
Subsections
Next: Exercises
Up: Orbital Angular Momentum
Previous: Eigenvalues of
Richard Fitzpatrick
2010-07-20