Perturbation Theory description of Positronium

[Magnesium]

Hi everyone,

I'm trying to work on Problem 5.32 from Sakurai's Modern Quantum Mechanics. In a nutshell, we need to use Perturbation Theory to find some of the energy levels of Positronium. Here's the full problem:

http://img714.imageshack.us/img714/6515/sakurai532.gif Problem 3.3 isn't much help. From what we have looked at so far, it looks like the first order correction is 0, so we need to go to second order. Needless to say, it gets very complicated real quick...especially when you throw in the Clebsh-Gordan Coefficients to transform from the $$S_{1}$$ and $$S_{2}$$ operators in $$H$$ to the $$S_{\mbox{total}}$$ you need for Perturbation Theory.

Any ideas on how to make it simpler, or a totally different way to do the problem?

Thanks,
Austin

 

[DrDu]

For two particles of spin 1/2 it is not difficult to write down the unperturbed spin singlet and triplet functions explicitly in terms of the eigenfunctions of S_z alpha and beta. I wouldn't bother with the Clebsch Gordan coefficients.