How Many Eigenstates of J and Jz Exist for Particles in p-state and d-state?

[padark]

Homework Statement

Given a system of 2 particles, one in p-state and the other in d-state,

(a) how many eigenstates of J and $$J{z}$$ are there?

(b) what are the possible values of J and $$J{z}$$?
(J = $$l{1}$$ + $$l{2}$$)

(c) Find the eigenstates for the subspace with maximum value of j
(give their expansion in terms of |$$l{1}$$$$m{1}$$>|$$l{2}$$$$m{2}$$> representation)

(d) Assume that we measure $$J{z}$$ = 0. If we now measure $$m{1}$$, what is the possibility that it will be 1?

(e) If we now measure $$J{z}$$ again, will we always measure 0 again?


I seriously have no idea how to approach these questions... :( someone help me...
(Sorry for the lousy latex failures with subscripts...)

 

[nickjer]

You will need to look up addition of angular momentum to even begin this.

 

[Mike Pemulis]

This problem is about addition of angular momentum and Clebsch-Gordon coefficients. Can I ask what book you're using? It should have a section on how to add angular momentum.

I'll say a couple of things to get you started:

(1) In adding angular momenta, we are just adding two vectors: J = L₁ + L₂, and J = |J|. What are the magnitudes of L₁ and L₂? What does that tell you about the range of J? (What if L₁ and L₂ are in the same direction? In opposite directions?) Once you know the range of J, ordinary angular momentum quantization rules tell you range of J_z and the possible values of J and J_z. All this gives you the answers to (a) and (b).

(2) Part (c) is more difficult, and involves Clebsch-Gordon coefficients (whereas (a) and (b) do not). You need to answer those first two first.

(3) Once you answer (c), (d) and (e) are pretty easy.

(4) Question -- does it say you can use a Clebsch-Gordon coefficient table? Or do you have to derive them from scratch?