Hints for set 6.

1) To do unit analysis, first convince yourself how many powers of the small quantity

(in the first two questions, that would be "". What is it for part iii?) should appear in your answer. (I claim e.g. one power for part i, but not for part ii. Do you see why? ) Your final answer has to have units of energy, so figure out what combination of "natural" quantities (appearing in the unperturbed Hamiltonian) you could multiply times this power of to get an energy. That's it! You'll have to just think about the sign - sometimes you can't guess it, but in these two examples you really can. In part iii, be careful to remember what the UNITS of a delta function are! (Check how well the estimate does compared with the exact answer you got on HW 3, #6i)

2-i) We discussed in class (notes, P. 90) how to find the eigenvalues and vectors of block diagonal matrices. Just look at the little blocks! (The only annoying piece is the 3x3 block - just do it carefully, and don't forget to normalize your eigenvectors. I claim that the 3 eigenvalues should turn out to be )

2-i) The matrix is block diagonal, which should save a LOT of labor.

2-ii) Part 2-i was designed to make this part a lot easier!! If you have chosen your 9 basis vectors in some different order than I did, then your matrix won't look exactly like the one I gave you. Can you re-order your basis vectors to make your W matrix look just like mine? Here are some possibly helpful tips to connect with part i: First, you know that only brackets with EQUAL m values can be nonzero, so you should surely order your states such that equal m values sit right near to each other. Also, I defined

<3 1 +/-1 | z | 3 1+/- 1> = A, <3 0 0 | z | 3 1 0> = B, and <3 1 0 | z | 3 2 0 > = C.

3) The basic idea is the same as what Griffiths did for hydrogen (His trick of converting p^4 to (p^2)^2, and rewriting p^2 in terms of (total -potential energy) like he did in Equation 6.51, works in any problem)

Don't forget the virial theorem trick - Griffiths 4.192.

4) Note that e.g. is really a shorthand: the dot product L.S is a scalar (you can, and probably should, write it out in terms of LxSx+LySy+LzSz, but the second entry is a vector. So, really part a requires 3 answers, , , and . However, once you've gotten the first, you might just be able to see what the other two will be. (They are NOT zero, in part a!)