Homework for Tuesday, Oct 21

I hope the GRE's went well for those who are taking them. I am not feeling so well today so any help here would be much appreciated. Does anyone have a good idea of what homework to work on for Tuesday? Thoughts and comments on that, or other stuff, are welcome here!

DegPert Probs17 and 19

Here are some solutions for problems 17 and 19 regarding the effect of perturbations on the 2-fold degenerate 1st-excited state manifolds for the 2D infinite square well and the 2D harmonic oscillator, respectively. These problems tend to start with:
1) establishing a basis of (2) degenerate eigenvectors which span the degeneracy "manifold",
which is followed by:
2) a calculation of the matrix of the perturbation in that basis,
3) determining the eigenvalues and eigenvectors of that matrix,
and
4a) using the eigenvalues to establish the perturbed energies.
Then the last part, which is very important, and conceptually and computationally difficult:
4b) using the eigen-vectors of the matrix to establish "new" spatial (x,y) eigenvectors.

Homework due Thursday, Oct 16

For Thursday Oct 16 (the day after tomorrow...) how about if you turn problem 18, 20 and the -eEx related problem. I'll just outline them briefly here, and if someone can describe them in more detail or from a different perspective, that would be most helpful and appreciated:

22. the -eEx problem: For this problem, since we have already gotten pretty far, due to the wonderful work of today's "presenters" and their friends, I would suggest focussing on the nature of the eigenvectors of the matrix:

0 0 1 1
0 0 0 0
1 0 0 0
1 0 0 0

and what they "tell us" in the context of this problem. Pictures would be important here. Would someone would post something about the basis for this matrix... and anything else relevant...

18. Design your own degenerate perturbation theory problem.

21. For a 2-D H.O., express the angular momentum in terms of raising and lowering operators. Get it as simple as you can. (Start from the definition of angular momentum.) What is the dimension of L in this case. Do the x and y raising and lowering operators commute?

23. (extra credit) Calculate the matrix of L in the basis of the lowest 3 or 6 states of the 2DHO.

WaveFunctions, 1D H.O. and hydrogen

Hmm. Not meaning to distract attention from the HW post preceding this, but here are some wave-functions for the 1D H.O. and H-atom. These are related to problems 6 and 8 (review).
I am not sure why, but I just LOVE the 1/sqrt(2) under the e{i phi}. How do you feel about it?

They may not be quite right, but let's try to establish and repair any errors through comments here. It is very helpful to have a "standard set" for these as part of our "common culture". If we all use the same, hopefully correct ones, in ongoing and upcoming problems, then, uh, well,... that is a good thing... right?