# General method for solving SE

1. May 29, 2010

### quZz

Hi!

What is the general method for solving Schroedinger equation
$$\nabla^2 \psi(\textbf{r}) + (p^2 - 2mU(\textbf{r}))\psi(\textbf{r}) = 0,$$
with arbitrary potential $U(\textbf{r})$ that is not singular and decreases rapidly at infinity.
I'm interested in scattering problem, so
$$\psi(\textbf{r}) \sim e^{i\textbf{p}\textbf{r}} + f(p,\textbf{r}/r)\frac{e^{ipr}}{r}$$
as $r\to\infty$.

What are the corresponding boundary/initial conditions?

2. May 29, 2010

### Haelfix

Its been awhile since i've done this or worked this out, but i'd venture to guess the general solution strategy would best be served by going down the Green's function path. So rewrite schrodingers equation as an integral equation with a Greens function (you probably know the drill).

The potential needs to satisfy rather complicated boundary conditions (say something like e^ikr + solutions to the homogenous part of the SE) eg outgoing spherical waves.

Then you guess that the homogenous solutions are seperable alla spherical harmonics (so something like a function of the radial part * spherical harmonic part) subject to the singularity conditions.

Anyway, you have to play with it a bit. Good luck.

3. May 29, 2010

Thank you!