Questions from Ph235-08 Students
Q1. Friday's lecture cleared up how to construct the operators for the two electron system, and I understood how to find the energy eigenvalues for the hyperfine interaction, but I'm having trouble with the remaining problems. I think I understand derivation of the Clebsch-Gordan Coefficients for two electrons, but I don't see how to generalize to more complicated spins.
Answer: See “Notes on Lect 2” and “Notes on Lect 3” on the web page. Different conventions are used for the notation, but the key idea is that the CG coefficient is equal to the projection of a basis vector in one eigenbasis onto an eigenvector in another. Using these, the eigenvectors in either basis can be expanded in terms of the eigenvectors in the other – and this, of course, allows us to transform from one basis to the other.
Also see “Example 1” and “Example 2” in Week 1, which go through a couple of specific cases.
Q2. Can you elaborate on how to express a direct product in terms of kets? For example, how does one express 1/2 circle-cross 1 = 1/2 circle-plus 3/2 in terms of kets?
Answer: Yes. The direct product space is derived solely from the single-particle eigenstates of the two particles, and its definition and derivation have nothing to do with total J. Take the full set of "1/2" states (there are 2 of them), and take the full set of "1" states (there are 3 of them). Take all possible combinations of one ket from the first set times one ket from the second (there are 6 possible combinations). This set of 6 direct-product states are the basis vectors of the product basis for the 2-particle system.
As we have seen, we can also write a basis in the |J,M> representation. The possible values of J are J=3/2 (there are 4 such states) and J=1/2 (there are 2 such states). So either way we get 6 basis vectors. We write that as in Shankar’s notation, or as in HEP notation.
Q3. I and several of others are having problems with homework #4 in set 1. So far our understanding of a projection operator is that it looks like |e><e|, where e are basis elements -- but the book answer looks nothing like a sum of basis elements. Unfortunately, we didn't have discussion on Wednesday so I hope that you are able to provide some guidance.
Answer
Q4. I was wondering if by any chance you can offer non-.doc versions of what you make available on the your web page. I run linux and donot have Microsoft Word so when I open any lecture note all the equations become jumbled and illegible.
Answer: Sorry about that. I’m working on it, but it may take some time. For now, I’ve added pdf links to the most important material.
Q5. I have a question about 5.12b. Is it possible to have J=3 for the C atom? That would require S=1 and L=2. S=1 implies that the spin wavefunction is symmetric. L=2 implies that l_1 and l_2 are either both +1 or both -1. But that means that the full wavefunction is symmetric. Would you let me know where I went wrong?
Answer: You didn't go wrong, you're absolutely correct. I listed J=3 as a possibility for Carbon in the notes for Lecture 6, in the last section -- but note the "would appear" in that sentence! Nevertheless, I guess I should change that. Thanks for pointing it out.
Q6. Thanks for the response. I'm afraid I still don't quite understand the whole picture. Griffiths lists 3P_0 as the angular momentum for C. That means that the spin wavefunction is symmetric (because S=1) and the spatial wavefunction is also symmetric (because L=1). Or does L=1 not imply that the spatial wavefunction is symmetric?
More generally, how do I tell whether a state with a given L is symmetric or not (aside from states where l_1=l_2)?
Answer
Q7. Why must Symmetric Spatial WF correspond to Antisymmetric Spin WF, and Antisymmetric Spatial WF correspond to Symmetric Spin WF?
Answer: Since the electron is a fermion, the whole wavefunction has to be antisymmetric under particle exchange; i.e., it must change sign under exchange. But the total wavefunction is just the product of the 2-particle spin wf and the 2-particle spatial wf, which are also eigenfunctions of the exchange operator.
The only way for the total wf to have negative exchange symmetry is for the space and spin functions to have opposite exchange symmetry, one positive and one negative.
If the electron were a boson, then the space and spin wf of a 2-particle system would have to have the same exchange symmetry, either both positive or both negative.
Q8. Several people asked about this in class, but I don't think there was time to answer it. How do you decide if a given orbital angular momentum (S, P, D...) is consistent with the overall symmetrization condition? Is there a fast algorithmic way to do it without looking at Clebsch-Gordan coefficients?
Answer: (here)
Q9: (asked in class) When we are trying to find the possible states of a multi-electron atom, you require that the valence electrons be in states that are totally antisymmetric under particle exchange, but you ignore the non-valence electrons in closed shells. Don’t these have to be antisymmetric as well? How can we just ignore them?
Answer: (here)
Q10: In deriving Hunds Rule 1, you showed that maximizing S gives a total spin of when the shell is half full (so a p-shell has ). When we add another spin, it has to have spin down, but why does this decrease S? Why can’t 4 spin ½ particles give S=2? For that matter, why can’t the total L of a filled p-shell be 6, since we have 6 L=1 particles?
Answer: (here)