I'll start off by going back to the Two-State Problem. After attempting the homework problem on this, it makes a bit more sense, but not a lot. Basically Sakurai does perturbation using matrices. It still doesn't help me with the homework, though. Next Sakurai formally develops perturbation theory. He shows how to calculate the first order shift in the energy and the wavefunction. He then briefly does the higher orders. Next he renormalizes the wave-function, but I didn't understand this section at all. Basically, I don't understand Sakurai sometimes. This whole section is particularly dense. Then there are two examples.
I don't think I'll devote another post to the next section, which involves the degenerate case. Basically, you need to diagonalize the perturbation matrix. Only then will the degeneracy lift and you can solve the problem as in the non-degenerate case.
I think I will have to supplement with Townsend, discussion section, and banging through the homework as best I can. There is also Baym. Onward and upward.
Showing posts with label Sakurai. Show all posts
Showing posts with label Sakurai. Show all posts
Monday, January 21, 2008
Sunday, January 13, 2008
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.
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.
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