# Russell's Blog

## Lecture Summary 01/21/2007

Posted by Russell on January 22, 2007 at 3:42 p.m.
To make sure I pay attention in class, I've decided to write summaries of each lecture. This lead immediately to the oddly difficult question of where to put them. Should I write them in my notebook? Write them on my computer at school? Scrawl them on the wall of the men's room? Since I have this blog thing that will otherwise go mostly unused this quarter, I decided to write them here. That way, maybe people will notice if I slack off, and admonish me. So, lectures are on Monday and Wednesday. If you see that a summary is missing, you can help me keep the slack away by sending me a finger-wagging email.

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 theory

Generally, 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.
The perturbation method has two restrictions:
1. There must be a one-to-one correspondence between eigenstates of the unperturbed Hamiltonian and the eigenstates of the full Hamiltonian.
2. 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 Hydrogen

We 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.

anthon on July 11, 2013 at 3:11 a.m.

All this summary made that certain organization how this will work for. - YOR Health

sdsafsafsaf on December 06, 2013 at 11:07 a.m.

Excellent Blog! I would like to thank for the efforts you have made in writing this post. I am hoping the same best work from you in the future as well. I wanted to thank you for this websites! Thanks for sharing. Great websites.

may tap co bung ab rocket twister
may tap co bung black power
may tap co bung ad rocket
may tap bung fitness
may tap co bung six pack care
may tap co bung
may tap co bung
may tap co bung
may tap co bung
may tap co bung
may tap co bung
may tap co bung

rivertrees residences on February 01, 2014 at 6:34 a.m.

this is one of the best articles on this website. definitely excellent information and hope to read more. awesome. this is the information i have been searching. Ecopolitan Ecopolitan Executive Condominium Tembusu Tembusu Condo Tembusu Kovan Vue 8 Sea Horizon Sea Horizon Executive Condominium Rivertrees Rivertrees Residences Rivertrees Condo Rivertrees Condominium Rivertrees Residences Fernvale Rivertrees Fernvale Ladies Bags Ladies Bag Ladies Handbags Designer Bags Handbags Shoulder Bag Designer Handbags Thanks for sharing information that is actually helpful

gazete on March 03, 2014 at 3:06 p.m.

but is probably amenable to perturbation theory. lpg

pollen and bleu on March 06, 2014 at 12:07 p.m.

this is one of the best articles on this website. definitely excellent information and hope to read more. awesome. this is the information i have been searching. Bags | Bag | handbags | Ladies bags | Shoulder bags | Pollen & Bleu | Pollen and Bleu | farrer condominium Thanks for sharing information that is actually helpful

 Name: Email: optional; will not be displayed URL: optional Comment: URLs auto-link and some tags are allowed:

.