Simple degenerate perturbation problem.

[Vkun]

Homework Statement

I have two particles with spin 1/2 and the following Hamiltonian:

H = J S1.S2 + S1.B

Now getting an exact solution is not a problem, however I need to use pertubation theory in the limit case when J is small.

Homework Equations

Just your standard perturbation theory.

The Attempt at a Solution

If the limit case was small B, then I can use standard perturbation theory for the 00 state, and degenerate theory for 10,11,1-1 states since

H = H0 + k H1 and in this case k=B and if H0 has degenerate eigenvalues I need to use the degenerate perturbation for the 3 states.

Now if J is small, how do I approach the problem? Do I need to check separate cases? Does J = k in this case?

 

[Jhero]

In the limit case when J is small, the perturbation theory approach will still be applicable. You will need to consider the perturbation Hamiltonian as H1 = J S1.S2 and the unperturbed Hamiltonian as H0 = S1.B. The parameter k in this case will be J, so you can follow the same steps as you did for the small B case. However, since S1.B does not have degenerate eigenvalues, you will not need to use degenerate perturbation theory. You can simply use first-order perturbation theory to calculate the energy shifts for the states. I hope this helps. Good luck with your calculations!