Note: whenever I talk about a particle "with angular momentum L", remember L is a number, appearing in several places in the eigenvalue equation

(It's the first label of the state in Dirac notation, and also tells you what the eigenvalue is)

1) Remember that probability density is simply given by |Psi^2|.

2) See my notes, p. 28, for a useful formula for (L.S). In the third term, C(L.S)^2: if you have an eigenvector of some operator O, can you see that your state is automatically also an eigenvalue of O^2? (With what eigenvalue?)

(Also, be careful with the eigenvalue of the operator L^2. The answer is NOT exactly L^2, right?!)

Once you see how to get started on this problem, I believe you'll find the actual work pretty quick and simple.

3i) To add angular momentum of 3 particles, first add together a pair, and then for *each* possible outcome of that pair, see what happens if you add in the third. There is *no* need to use C-G tables, I'm not asking for probabilities, just asking what total angular momentum you can get. Again, once you see how to do this problem, you'll find it's very quick to actually work out the details.

3ii) The Pauli principle says that identical spin 1/2 particles (fermions) must be antisymmetric under interchange. This means if you have two indistinguishable protons, their combined wave function has to be the antisymmetric combination, it has to flip sign if you switch the labels of particles 1 and 2. (This is Griffith's Eq. 5.14)

4) This one is a harder problem, I think! You'll need C-G tables to answer part 1. Part ii you could have done on the first set - it's just the expansion theorem. Part iii involves a collapse of the wave function. For part iv, remember that if operators commute, measuring one won't affect the other, but if operators DON'T commute, measuring one will screw up the value of the other.

5) This is back to material from last week, just more review. All the questions fundamentally involve the expansion theorem, and collapsing of wave functions.

6) Both problems should be pretty straightforward applications of Griffiths' equation 6.9

6ii) See Griffiths Equation 4.192 for help with the matrix element of x^2.