The inhomogeneous Helmholtz Differential Equation is
![\begin{displaymath}
\nabla^2\psi({\bf r})+k^2\psi({\bf r})=\rho({\bf r}),
\end{displaymath}](g_2272.gif) |
(1) |
where the Helmholtz operator is defined as
. The Green's function is then defined by
![\begin{displaymath}
(\nabla^2+k^2)G({\bf r}_1,{\bf r}_2) = \delta^3({\bf r}_1-{\bf r}_2).
\end{displaymath}](g_2274.gif) |
(2) |
Define the basis functions
as the solutions to the homogeneous Helmholtz Differential Equation
![\begin{displaymath}
\nabla^2\phi_n({\bf r})+{k_n}^2\phi_n({\bf r})=0.
\end{displaymath}](g_2276.gif) |
(3) |
The Green's function can then be expanded in terms of the
s,
![\begin{displaymath}
G({\bf r}_1,{\bf r}_2)=\sum_{n=0}^\infty a_n({\bf r}_2)\phi_n({\bf r}_1),
\end{displaymath}](g_2277.gif) |
(4) |
and the Delta Function as
![\begin{displaymath}
\delta^3({\bf r}_1-{\bf r}_2)=\sum_{n=0}^\infty \phi_n({\bf r}_1)\phi_n({\bf r}_2).
\end{displaymath}](g_2270.gif) |
(5) |
Plugging (4) and (5) into (2) gives
![\begin{displaymath}
\nabla^2 \left[{\sum_{n=0}^\infty a_n({\bf r}_2)\phi_n({\bf ...
...bf r}_1)=\sum_{n=0}^\infty \phi_n({\bf r}_1)\phi_n({\bf r}_2).
\end{displaymath}](g_2278.gif) |
(6) |
Using (3) gives
![\begin{displaymath}
-\sum_{n=0}^\infty a_n({\bf r}_2){k_n}^2\phi_n({\bf r})+k^2\...
...\bf r}_1)=\sum_{n=0}^\infty \phi_n({\bf r}_1)\phi_n({\bf r}_2)
\end{displaymath}](g_2279.gif) |
(7) |
![\begin{displaymath}
\sum_{n=0}^\infty a_n({\bf r}_2)\phi_n({\bf r}_1)(k^2-{k_n}^2) = \sum_{n=0}^\infty \phi_n({\bf r}_1)\phi_n({\bf r}_2).
\end{displaymath}](g_2280.gif) |
(8) |
This equation must hold true for each
, so
![\begin{displaymath}
a_n({\bf r}_2)\phi_n({\bf r}_1)(k^2-{k_n}^2) = \phi_n({\bf r}_1)\phi_n({\bf r}_2)
\end{displaymath}](g_2281.gif) |
(9) |
![\begin{displaymath}
a_n({\bf r}_2) = {\phi_n({\bf r}_2)\over k^2-{k_n}^2},
\end{displaymath}](g_2282.gif) |
(10) |
and (4) can be written
![\begin{displaymath}
G({\bf r}_1,{\bf r}_2)=\sum_{n=0}^\infty {\phi_n({\bf r}_1)\phi_n({\bf r}_2)\over k^2-{k_n}^2}.
\end{displaymath}](g_2283.gif) |
(11) |
The general solution to (1) is therefore
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 529-530, 1985.
© 1996-9 Eric W. Weisstein
1999-05-25