Here is the HOMEWORK for this week

1) Remember, to find the commutator of two operators, you let the commutator operate on some arbitrary f (and then pull out the f at the end, to leave a pure operator eq'n). You'll need to remember the "chain rule" involving the gradient acting on a product of scalar functions...

2a) When in doubt, you can always write out L_z explicitly in terms of x's, y's, d/dx's, and d/dys, and then, like in #1, let the commutators operate on some arbitrary function...

2b) This is a simple exercise in expanding sin's and cos's. The only trick is deciding how many terms you need in your starting expansions to get two left over at the end...

2c) Write (simple) formulas giving x, y, and z in terms of r, theta, phi. This should help! (You may also need the "double angle formulas" for cos(2x) and sin(2x), to do Y2,2(x,y,z))

3a) Since l=0, the radial equation (for u, not R) should be familiar (and easy! )What must u(r) be inside r<a, by inspection? What's the GENERAL solution outside r>a? There are many different ways to write this function, the generic form used by Gas in e.g. Eq. 10-89 should allow you to "read off" the phase shift without any real effort at all.

3b) Note that this is a bound state problem, whereas part a was a "continuum" problem! (The potentials are different, too!) I get if n is large. (Are the correct energies slightly less than this, or slightly larger? There's a definite, simple answer!)

4) Don't try to use Eq 10-85 directly!! Instead, do like problem 3a: since l=0, you can write down the radial eq'n (in terms of u), and then get the answers by inspection. (Remember, phase shift problems are always for E>0) The general form inside (r<a) is slightly different from the general form outside, because you know u(0)=0... Once again, if you cast the general solution for r>a in the form like Gas 10-89, you can read off the phase shift directly...

The case of a repulsive potential is mathematically similar, but NOT identical if E<V0, so you'll have to treat that one separately. Do your answers for the phase shift in the limiting cases Gas. suggests make any sense to you? They are all "physically reasonable" answers... (For the repulsive case when E<V0, some of the "limits" Gas asks for don't really make sense, if you think about it... E.g, if V0 is fixed, how can E -> infinity if E<V0?)