How do you apply perturbation theory to a magnetic field Hamiltonian?

[Danielk010]

TL;DR

Given,
Take with . Determine the energy eigenvalues exactly and
compare with the results of perturbation theory through second order in .

I understand that the first order results would be $\langle \phi_n^0 | -\mu * B | \phi_n^0 \rangle$ = $\langle \phi_n^0 | -\frac{gq}{2mc}\hat{S} * B | \phi_n^0 \rangle$, the second order results would be $\sum_{k \ne n} \frac{|\langle \phi_n^0 | -\mu * B | \phi_n^0 \rangle|^2} {E_n^{(0)} - E_k^{(0)}}$, and the unperturbed hamiltonian would $E_n^{(0)} = (n + \frac{1}{2})\hbar\omega$.

I am confused on how to evaluate braket of phi as the problem is asking for the exact energy eigenvalues. I tried plugging in $B = \frac{-Zev \times r}{cr^3}$, but then I would have the Z, v and r term, which would not give me an exact solution.


Am I on the right track? Is there an equation I am missing? Thank you for any help on this problem. Also do any of you all know how to add equations to the TL;DR?

 

[renormalize]

TL;DR: Given,
Take with . Determine the energy eigenvalues exactly and
compare with the results of perturbation theory through second order in .

Can you edit your post to very clearly state the problem you are trying to solve?