Lecture Summary 01/21/2007
Today Abers covered sections 7.1.1, 7.1.2 and 7.2.1 in his book. They are as follows.
Introduction to bound-state perturbation theoryGenerally, one uses bound-state perturbation theory to solve problems that don't have exact solutions, but are very close to problems that do have exact solutions. If you can represent something as a problem with an exact solution plus some small wiggle, you can use perturbation theory. For example, the infinite square well potential has an exact solution. The infinite square well with a lump at the bottom probably doesn't, but is probably amenable to perturbation theory.
The process goes like this :
- Write the Hamiltonian as the sum of the unperturbed Hamiltonian and some H', which is small (we hope).
- Note that the presence of the H' will change the energy eigenvalue, hopefully by some small amount. Call this delta.
- We want to solve for delta. It is usually not possible to do this directly. However, we can get a zeroth-order approximation of delta by taking the expectation value of the perturbed Hamiltonian H' with the unperturbed eigenstate. The error will be one order higher.
- To get the higher order terms, we need to expand the state vector (the state vector, not the unperturbed state vector) in powers of H' (or its matrix elements, if that makes any sense). Equation 7.8 in Abers does the job.
- Successive iterations of the process yields successfully higher order terms.
- Care must be given to the normalization of the perturbed eigenstates; different books (e.g. Griffiths) use slightly different conventions.
- There must be a one-to-one correspondence between eigenstates of the unperturbed Hamiltonian and the eigenstates of the full Hamiltonian.
- There is a second restriction, explained incoherently in the book and mysteriously in today's lecture, that has something to do with how one handles degeneracies.
Perturbation of the first excited state of HydrogenWe went through the derivation of the energy splitting due to the presence of a weak electric field (the Stark effect). There was much confusion resulting from the change in bases required to avoid dividing by zero when evaluating the formula for obtaining the first order terms. Abers noted that many textbooks treat "degenerate" perturbation theory separately, and that this was silly because all you had to do was switch into a basis that diagonalizes H', find the higher order terms, and then switch back into the original basis.
However, by the time we had returned from our basis-changing digression, it was rather unclear what had happened. Fortunately, it seemed to work, and we calculated something for the energy shift in terms of the energy eigenstates.
To solve for the shift directly, we integrated over all space in spherical coordinates. As usual, the spherical harmonics are separable, and the result for the perturbed radial wave function is -3*sqrt(3)a, where a is the Bohr radius. The energy shift is -3*e*E*a.