Time Independent Perturbation Theory: Nondegenerate Case

So Jordan Glassman has decided to keep a blog to help him understand Jackson. I am hoping that if I do the same for quantum mechanics it will improve my performance in the class. I understand the concepts when we go over them during lecture, but then I don't do well on the exams. So here we go. I'm starting with section 5.1 in Sakurai.

The problem is thus: there are very few cases where the Schrodinger equation can be solved directly. Here we have a Hamiltionian H=H₀ + V, where H₀ is a Hamiltonian that can be solved analytically and V is a small perturbation. This is generally written as H=H₀ + λV, so that we can expand the energy eigenvalues and eigenstates in powers of λ.

I do not understand the next section at all. It is called "The Two-State Problem." It looks Sakurai has the perturbation as a mixing term. Then Sakurai uses the analogy of the spin-orientation problem, but I don't think I read that section, so that's not helpful. He then ends with a convergence condition for the perturbation.

I am too tired to continue right now, so hopefully I'll write about the rest of the section tomorrow. Also, I should probably do the problem set and read Baym, as well.

Now back to your regularly scheduled brooding.