Hints for set #7

1) This is straightforward algebra, I just want you to convince yourself there's no trickery going on! You should be able to do this even if we haven't quite gotten to it in lecture yet.

2) You'll want to look back at Griffiths Ch. 4 to remember some hydrogen atom stuff, like, e.g. what "J_max" means. Please explicitly check/verify that the answer you get does indeed agree with Griffiths Eq. 6.55

3) I noticed a lot of people were having troubles earlier this semester expanding an expression with a small number in it, so here's a doozy, for practice! This math technique is incredibly useful in a lot of applications. If you really don't care about it, there's no reason you couldn't use Mathematica (the "Series" command pretty much solves the problem for you! See my virtual office hours for another hint, involving "PowerExpand") If you want to do it yourself, here's the essential math:

In this problem, you need to decide (repeatedly) how many terms you *need* to expand out to, so that your final result is accurate at order alpha^4. (Not alpha^2, that's not good enough!) You have to work your way from "the inside out", expanding, truncating, and then doing it again... There's at least one place where you must expand an especially tricky quantity, which has generically the form . How do you do that, to order epsilon^2? Just use the formula above, again, but now let your "small parameter" be . (Notice that if you do that, the last term in the formula above will itself contain SOME higher order bits, which you can throw away, but keep anything of order or lower...

4) I hope this one is pretty straightforward Clebsch-Gordon-ology for you by now. Just thought you might like to see a concrete example once more.

5) Part i is a pure application of Griffiths Eq. 6.66 (or, my notes, P. 118) Again, you shouldn't have any trouble with this one even if we haven't fully covered the material in lecture. I just want you to get a feel for the numbers involved, and how people draw out the energy level diagrams. ( I suggest you keep about 3 significant figures in all your energy shifts, because you're going to be subracting them from each other in part ii)

Part ii (the Griffiths problem) is not especially *difficult*, but requires that you keep pretty organized. A lot of quantum books just draw the "energy level diagram" (like Fig 6.9), which is not what people really measure, and never really show you what the resulting spectrum looks like in the lab - that's what this problem does.