Homework problems to work on ASAP: revised Sunday, more comments please

Many, if not all, of these problems are vague and not clearly articulated. I think they need clarification. Go to it. Clarifying comments and questions are needed, e.g., I think what Zack is asking for here is..., or I don't understand. Are we supposed to ...? That doesn't seem possible, doesn't make sense...

Also through discussion perhaps we can discuss HW priorities and relevance?

I am hoping for and expecting comments from each person actively involved in this class.

41. Suppose a hydrogen atom, for which the electron is initially in the ground state, is perturbed by an oscillating electric field polarized in the x direction, at a frequency which is resonant with the energy difference between the gs and 1st excited states. What are the probabilities, as a function of time, of transitions to any of the 1st-excited states (in the n,l.m basis, unless you prefer another one) and what are the branching ratios (relative probabilities)?
How would the branching ratios change for polarization along y or z?
How could you most effectively stimulate a transition to the 2,1,1 state?
(This is a question, i think, related to the electric field (of the "photon").

42. For variational problems involving two electrons, suppose you write the variational state as the product: Psi_a(r1) Psi_b(r2), where Psi_a and Psi_b are two states that differ in some way, and r1 and r2 are the vector positions of the two electrons, respectively. Discuss both the conceptual and practical significance of symmetrizing the wave function, which can be done via a 2x2 "determinant", i.e.,
Psi_sym = Psia*Psib+Psib*Psia .

You can explore this question with specific examples, e.g., wave-functions for H2 or He. Does symmetrization make a difference in calculating energy expectations values?
Please discuss here...

43. Regarding what we worked on in class today, perhaps you can think of the homework associated with that in terms of the question: Starting from the Schrodinger eqn, and the one-electron orbitals that follow as its solutions (for the -1/r potential): how can we account for and explain the patterns of bonding associated with carbon. Please explain with appropriate math and a few paragraphs of cogent discussion.

44. Consider equal strength delta functions symmetrically arranged around a hydrogen atom center (and of course equidistant from "r=0"). Determine their effect on the degeneracy and order of the 2nd excited states.

45. extra credit review problem: For the 2D harmonic oscillator, analyze the accuracy of the perturbation theory for either: 1) the 4 delta function perturbation from the midterm or 2) the problem with the perturbation V proportional xy problem that we did a long time ago. (hint: i only have some idea of how to actually do something for one of these choices. the other is sort of a decoy. Wouldn't it be funny if that one turned out to be more interesting, haha?