# Integrating Laguerre Polynomials - Fine structure hydrogen

1. Jan 28, 2006

### sebhofer

Hi

I have the following problem:
To calculate the fine structure energy corrections for the hydrogen atom, one has to calculate the expectation value for (R,R/r^m), where R is the solution of the radial part of the schroedinger equation (i.e. essentially associated laguerre polynomial) and m=1,2,3.
$$\int_{0}^{\infty} {dx} {(\mathrm{L}_n^k)}^2e^{-x}x^{k+1-m}$$
Solving the integral for n=1 is easy because the laguerre polynomials are normalised that way.
For m=2,3 it is much harder. I have already spent a few hours trying to get a solution by using the generating function of the laguerre polynomials, but no luck yet. Can anybody give me a hint how to do it?

Thx
Sebastian

Edit:
Can anybody tell me how to post latex in this forum pls. Sorry for my stupidity, but I just don't get it.
Edit 2: Thx for the help on latex Statuts X. Added integral to the post.

Last edited: Jan 29, 2006
2. Jan 28, 2006