Issued Wed, Nov 20 Due Wed, Nov 27

Required reading for this week: Finish Ch. 10, start Ch. 11

1) Recall that Gas. Eq's 10-21 through 10-23 were derived assuming the special case V=V(r), i.e. a purely central potential. Drop this assumption, and prove that, in general, .

Also, show that in the special case V=V(r), this reduces to what Gas. had.

(N.B., this problem is the "Ehrenfest analogue" of the classical equation , i.e. torque gives the rate of change of angular momentum)

2a) Show that , and also that Find .

b) Using Gas. Eq. 10-62, expand to find approximate (polynomial) formulas valid for x<<1. (Expand to "next to leading order", i.e. giving the leading two powers of x in each case.) Verify that the j's are finite as x -> 0, but the n's are not.

Also, expand to "leading order", i.e. giving the leading power of x only, and verify that Gas 10-66 gives the correct answer.

c) We talked about the spherical harmonics in class already, and Gas. has them written down on pp. 196-7.

Reexpress the spherical harmonic with l=0 (i.e. ), all three spherical harmonics with l=1, (i.e. ), and also , purely in terms of x, y, and z rather than theta and phi. (See Fig. 10-1 on p. 174 for Gas' conventions on radial coordinates) (Gas. Eq'n 11-60 tells you his sign convention to get spherical harmonics with negative m's)

3a) Gas #10-6 (Ignore his question "what is it for ka large?" - I claim the answer doesn't really care whether ka is large, small, or whatever!)

3b) Determine (graphically) the allowed energies for a particle trapped in an infinite spherical well of radius a, when l=1.

Find a simple , analytic, approximate expression for if n>>1.

4) Gas #10-3

Extra Credit: Gas #10-7