1) At the bottom of my notes on p. 116 is a formula for the total energy shift due to fine structure. I explicitly showed (on p. 117 of my notes) that when j=l+1/2, this simplifies to the boxed equation at the top of my notes p. 118. I argued (p. 117) that the exact same simplification is obtained when j=l-1/2. Verify this claim.

2) Read Griffiths 6.27 part a (you do not have to prove it), and then use this result to do problem 6.28b, only.

3) Griffiths 6.17.

4) An electron in hydrogen has spin 1/2, and orbital angular momentum l.

As usual, total angular momentum is given by J=L+S. Also as usual, our notation is

i) Given some definite l, what possible values can j have?

ii) Suppose I tell you l=2, j=3/2, and, let's say, m_j = +1/2.

a) What possible values can m_l have, with what probabilities?

b) What possible values can m_s have, with what probabilities?

c) Does this state have a definite value of L.S? If so, what is it? If not, what are the possibilities with what probabilities?

iii) Suppose instead I tell you l=2, m_l =+1, and m_s = -1/2.

a) What possible values can j have, with what probabilities?

b) What possible values can m_j have, with what probabilities?

c) Does this state have a definite value of L.S? If so, what is it? If not, what are the possibilities with what probabilities?

5i) Neatly sketch an energy level diagram for hydrogen's n=2 and n=3 levels, including the fine structure corrections. Make it nice and big, fill up a page! Label the energy shifts (calculate them out in eV, give simple formulas too) and clearly label all states. (What quantum numbers do you need to give?) Also, indicate the degeneracy of all your levels. Organize your diagram however you like - see Griffiths Fig. 6.9 for a fairly neat way. (Do you understand that figure better now?)

5ii) Griffiths 6.16. (Having done i) above should make this go more easily!)