How can you work out the average perturbation, from a normal hamiltonian, of all states that rely on the quantum numbers of s = __ and l = __, with the perturbation being proportional to the product of S.L, which if my memory serves is the J quantum number? i.e how can you work out all the average perturbation states of all the s=1/2 states? Thank you in advance, been puzzling over this for days. I've read that the S^2, L^2 and J^2 operators all commute with H, the normal hamiltonian.
H = p^2 /2m + V(r)
So if they commute then the eigenstates are zero, but how do you get past this?