The following are the two problems of your midterm. The idea is that everyone should work on their own, except that you can ask questions on this blog. Please feel free post comments and questions here.
1) Suppose that your unperturbed Hamiltonian is an isotropic 2D harmonic oscillator, and that your unperturbed eigenstates are chosen to be those made as products of 1D H.O. eigenstates.
Consider a perturbation potential which consists of 4 "repulsive" delta functions of equal strength located along the diagonals of an x-y coordinate system whose origin coincides with the center of symmetry for the original H.O.. Assume that they are all the same distance from the origin and that they are fairly far from it, e.g., 2a, 3a, 4a...something like that.
Calculate the effect this perturbation on the energies and eigenstates of the 2d HO, including the ground state, 1st excited states, and 2nd excited states.
Describe and discuss your results. What is the effect of this perturbation on the symmetry of the Hamiltonian and of the states? In presenting and discussing your results please engage your sense of physics aesthetics and emphasize aspects of the problem that seem most surprising, interesting or important. Please use a combination of text and figures (hand-written and hand-drawn is fine) to present your results in an insightful, clear and cogent manner.
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2) Again, suppose that your unperturbed Hamiltonian is an isotropic 2D harmonic oscillator, and that your unperturbed eigenstates are chosen to be those made as products of 1D H.O. eigenstates.
What is the effect on the energies and eigenstates of a perturbation k*Lz, where Lz is the z-component of the angular momentum and k is a small parameter.
Same comments as above regarding what states to consider and the nature of your discussion...
Regarding weighting of the problems, I am thinking that 1) is about 60% and that 2 is 40 % weighted, based on my feeling that 1) may be bit more challenging and interesting than 2) (though i could be wrong about that).
