1a) This should be straightforward partial derivitive practice...

b) There's a trick here, analogous to the trick we used in Gas #7-3! It should be relatively easy to figure out matrix elements of L+ or L- between the given states (using Eqn 11-48, and orthonormality of the Ylm's) So, if you know the matrix elements of L+ and L- easily, can't you get the matrix elements of Lx and Ly from those?

(Here's a question for you to ponder - your answer for Ly will have an "i" explicitly floating around in it. But Ly is Hermitian, so how come this matrix element is allowed to be imaginary?..)

2) This is a lot like the old hw problem Gas #7-8 that you did. The same tricks as you used for 1b help here, only now you must also think about combinations of L+ and L- that end up yielding Lx^2, and Ly^2. There is a fairly hard way of doing this one, and a pretty quick and easy way too!...

b) Make use of commutation relations Gas has already proven or claimed...

3) Note the minor typo, the denominator in the second term is I2, not l2. This one can be a super quicky, and is really quite similar to the example problem Gas did on p. 191.

4) If only he had given you the wave function as an expansion in Ylm's, this would be so easy! (Last week, you wrote your first few Ylm's in terms of x,y, and z. Hmmm.... Gas. Eqn 11-2 will help you reproduce what you need)

Back to this week's homework.