info prev up next book cdrom email home

Green's Function--Helmholtz Differential Equation

The inhomogeneous Helmholtz Differential Equation is

\nabla^2\psi({\bf r})+k^2\psi({\bf r})=\rho({\bf r}),
\end{displaymath} (1)

where the Helmholtz operator is defined as $\tilde L\equiv \nabla^2+k^2$. The Green's function is then defined by
(\nabla^2+k^2)G({\bf r}_1,{\bf r}_2) = \delta^3({\bf r}_1-{\bf r}_2).
\end{displaymath} (2)

Define the basis functions $\phi_n$ as the solutions to the homogeneous Helmholtz Differential Equation
\nabla^2\phi_n({\bf r})+{k_n}^2\phi_n({\bf r})=0.
\end{displaymath} (3)

The Green's function can then be expanded in terms of the $\phi_n$s,
G({\bf r}_1,{\bf r}_2)=\sum_{n=0}^\infty a_n({\bf r}_2)\phi_n({\bf r}_1),
\end{displaymath} (4)

and the Delta Function as
\delta^3({\bf r}_1-{\bf r}_2)=\sum_{n=0}^\infty \phi_n({\bf r}_1)\phi_n({\bf r}_2).
\end{displaymath} (5)

Plugging (4) and (5) into (2) gives

\nabla^2 \left[{\sum_{n=0}^\infty a_n({\bf r}_2)\phi_n({\bf ... r}_1)=\sum_{n=0}^\infty \phi_n({\bf r}_1)\phi_n({\bf r}_2).
\end{displaymath} (6)

Using (3) gives

-\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} (7)

\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} (8)

This equation must hold true for each $n$, so
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} (9)

a_n({\bf r}_2) = {\phi_n({\bf r}_2)\over k^2-{k_n}^2},
\end{displaymath} (10)

and (4) can be written
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} (11)

The general solution to (1) is therefore
$\displaystyle \psi({\bf r}_1)$ $\textstyle =$ $\displaystyle \int G({\bf r}_1,{\bf r}_2)\rho({\bf r}_2)\,d^3{\bf r}_2$  
  $\textstyle =$ $\displaystyle \sum_{n=0}^\infty \int {\phi_n({\bf r}_1)\phi_n({\bf r}_2)\rho({\bf r}_2)\over k^2-{k_n}^2} \, d^3{\bf r}_2.$ (12)


Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 529-530, 1985.

info prev up next book cdrom email home

© 1996-9 Eric W. Weisstein