General method for solving SE

[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?

Thanks in advance.

 

[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.

 

[quZz]

Thank you!
This is indeed very helpful, I forgot about this method...

And the second question: how about numerical realization of it? Is it worth trying to solve integral equation numerically or there are better ways?

 

[TriTertButoxy]

If you have Matlab: put the whole system on a lattice, and use the finite-difference representation of the derivative. Then your radial equation will be a matrix in position space. Use the standard numerical routines to diagonalize to get the eigenvalues (energies) and eigenfunctions (wavefunctions). This method works well for stationary states, but quite poorly for scattering states.