Here is the HOMEWORK for this week

1) This problem has almost no work (calculations), but may require some thinking! In the end, you can just write down the answer with ease...

Gas Eqs 11-19 to11-21 gets you most of the way there, except for the "symmetry". (Once you know the energies, it is useful (necessary) to also know the corresponding wave functions, but that should be easy.

Draw yourself a classical little picture of the molecule. Pauli says your wave function must flip sign if you interchange the positions of your two fermions (preserve the sign if you interchange bosons). How do you interchange the positions of the particles in your picture, without using P12? (Is there a more physical way?) What does this do to YOUR wave function? Can you ensure that it gives the right sign behaviour?...

2a) 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?..)

3) This is a like Gas problem 11-4, only easier. The same tricks as you used for 2b 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...

4) Note the typo, the denominator in the second term is I2, not l2. This one can be a super quicky, and is really quite similar to #1.

5) If only we 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.... If you don't have that in front of you, Gas. Eqn 11-2 will help you reproduce what you need)