In Spherical Coordinates, the Scale Factors are
,
,
, and
the separation functions are
,
,
, giving a Stäckel
Determinant of
. The Laplacian is
![\begin{displaymath}
\nabla^2 \equiv {1\over r^2} {\partial\over\partial r}\left(...
...artial\phi}\left({\sin\phi{\partial\over\partial\phi}}\right).
\end{displaymath}](h_1109.gif) |
(1) |
To solve the Helmholtz Differential Equation in Spherical
Coordinates, attempt Separation of Variables by writing
![\begin{displaymath}
F(r, \theta, \phi) = R(r)\Theta(\theta)\Phi(\phi).
\end{displaymath}](h_1110.gif) |
(2) |
Then the Helmholtz Differential Equation becomes
![\begin{displaymath}
{d^2R\over dr^2}\Phi\Theta + {2\over r}{dR\over dr}\Phi\Thet...
...hi }\Theta R + {1\over r^2}{d^2\Phi\over d\phi^2}\Theta R = 0.
\end{displaymath}](h_1111.gif) |
(3) |
Now divide by
,
![\begin{displaymath}
\left({{r^2\sin^2\phi\over R}{d^2R\over dr^2}+ {2r\sin^2\phi...
...phi} + {\sin^2\phi\over\Phi}{d^2\Phi\over d\phi^2}}\right)= 0.
\end{displaymath}](h_1116.gif) |
(5) |
The solution to the second part of (5) must be sinusoidal, so the differential equation is
![\begin{displaymath}
{d^2\Theta\over d\theta^2}{1\over\Theta}= -m^2,
\end{displaymath}](h_1117.gif) |
(6) |
which has solutions which may be defined either as a Complex function with
, ...,
![\begin{displaymath}
\Theta(\theta) = A_me^{im\theta},
\end{displaymath}](h_1120.gif) |
(7) |
or as a sum of Real sine and cosine functions with
, ...,
![\begin{displaymath}
\Theta(\theta) = S_m\sin(m\theta)+C_m\cos(m\theta).
\end{displaymath}](h_1121.gif) |
(8) |
Plugging (6) back into (7),
![\begin{displaymath}
{r^2\over R}{d^2R\over dr^2}+ {2r\over R}{dR\over dr} - {1\o...
...r d\phi }
+ {\sin^2\phi \over \Phi}{d^2\Phi\over d\phi^2}= 0.
\end{displaymath}](h_1122.gif) |
(9) |
The radial part must be equal to a constant
![\begin{displaymath}
{r^2\over R}{d^2R\over dr^2}+ {2r\over R}{dR\over dr}= l(l+1)
\end{displaymath}](h_1123.gif) |
(10) |
![\begin{displaymath}
r^2 {d^2R\over dr^2}+ 2r {dR\over dr}= l(l+1)R.
\end{displaymath}](h_1124.gif) |
(11) |
But this is the Euler Differential Equation, so we try a series solution of the form
![\begin{displaymath}
R = \sum_{n=0}^\infty a_nr^{n+c}.
\end{displaymath}](h_1125.gif) |
(12) |
Then
|
|
|
(13) |
|
|
|
(14) |
![\begin{displaymath}
\sum_{n=0}^\infty [(n+c)(n+c+1)-l(l+1)]a_nr^{n+c}= 0.
\end{displaymath}](h_1129.gif) |
(15) |
This must hold true for all Powers of
. For the
term (with
),
![\begin{displaymath}
c(c+1) = l(l+1),
\end{displaymath}](h_1131.gif) |
(16) |
which is true only if
and all other terms vanish. So
for
,
. Therefore, the
solution of the
component is given by
![\begin{displaymath}
R_l(r) = A_lr^l+B_lr^{-l-1}.
\end{displaymath}](h_1135.gif) |
(17) |
Plugging (17) back into (9),
![\begin{displaymath}
l(l+1)-{m^2\over\sin^2\phi}+ {\cos\phi\over\sin\phi}{1\over\Phi} {d\Phi\over d\phi}+ {1\over\Phi}{d^2\Phi\over d\phi^2}=0
\end{displaymath}](h_1136.gif) |
(18) |
![\begin{displaymath}
\Phi''+{\cos\phi\over\sin\phi}\Phi'+\left[{l(l+1)-{m^2\over\sin^2\phi}}\right]\Phi = 0,
\end{displaymath}](h_1137.gif) |
(19) |
which is the associated Legendre Differential Equation for
and
, ...,
. The general
Complex solution is therefore
|
|
|
(20) |
where
![\begin{displaymath}
Y_l^m(\theta, \phi) \equiv P_l^m(\cos \phi)e^{-im\theta}
\end{displaymath}](h_1142.gif) |
(21) |
are the (Complex) Spherical Harmonics. The general Real solution is
![\begin{displaymath}
\sum_{l=0}^\infty \sum_{m=0}^l (A_lr^l+B_lr^{-l-1})P_l^m(\cos \phi)[S_m\sin(m\theta)+C_m\cos(m\theta)].
\end{displaymath}](h_1143.gif) |
(22) |
Some of the normalization constants of
can be absorbed by
and
, so this equation may appear in the
form
|
|
|
(23) |
where
![\begin{displaymath}
Y_l^{m(o)}(\theta, \phi) \equiv P_l^m(\cos \theta)\sin(m\theta)
\end{displaymath}](h_1149.gif) |
(24) |
![\begin{displaymath}
Y_l^{m(e)}(\theta, \phi) \equiv P_l^m(\cos \theta)\cos(m\theta)
\end{displaymath}](h_1150.gif) |
(25) |
are the Even and Odd (real) Spherical Harmonics. If azimuthal symmetry is present, then
is constant and the solution of the
component is a Legendre Polynomial
. The
general solution is then
![\begin{displaymath}
F(r,\phi)=\sum_{l=0}^\infty (A_lr^l+B_lr^{-l-1})P_l(\cos \phi).
\end{displaymath}](h_1154.gif) |
(26) |
Actually, the equation is separable under the more general condition that
is of the form
![\begin{displaymath}
k^2(r,\theta,\phi)=f(r)+{g(\theta)\over r^2}+{h(\phi)\over r^2\sin\theta}+k'^2.
\end{displaymath}](h_1156.gif) |
(27) |
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 514 and 658, 1953.
© 1996-9 Eric W. Weisstein
1999-05-25