Questions about Rotating Dumbbell Homework?

[creepypasta13]

Homework Statement

I would like some help with the following problems

1. Consider in R^{j}={f : f = \Sigma^{l}_{m=-l} a^{m}f^{l}_{m}} the operator \stackrel{\rightarrow}{e}\bullet\stackrel{\rightarrow}{J}, where \stackrel{\rightarrow}{e} is a unit vector in 3-dimensional space.
(a) Calculate the probabilities for all eigenvalues of \stackrel{\rightarrow}{e}\bullet\stackrel{\rightarrow}{J} in the state W^{j} = Tr(\Lambda^{j})^{-1}\Lambda^{j}, where \Lambda^{j} is the projection operator onto Rj .
(b) Calculate the expectation value for the component J_{2} in the state W^{j} .

2. What spaces R^{l'}_{m'} are obtained when the operators (Q_{\stackrel{+}{-}})^{2} act on the space R^{l}_{m}?

3. Consider the rigidly rotating dumbbell molecule and let Q_{i}, J_{i}, i = 1, 2, 3 denote the position and angular momentum operators.
(a) Find a complete system of commuting observables.
(b) Explain the physical meaning of these observables and explain the meaning of their eigenvalues.
(c) Prove that the operators of your choice form a system of commuting observables.

The Attempt at a Solution

Number 1 is really confusing me since we need the probabilities for ALL eigenvalues of \stackrel{\rightarrow}{e}\bullet\stackrel{\rightarrow}{J}, and we don't know what 'j' is.
To find, say, the probabilities for the eigenvalues of J_{3}, is it just

\Sigma^{r}_{s=-r}\Sigma^{l}_{m=-l} |<a^{s}f^{r}_{s} | J_{3} | a^{m}f^{l}_{m}>| ^{2} = \Sigma^{l}_{m=-l}m^{2} ?

I am clueless as to how to solve #2

For #3, I found that because [J_{i}, Q_{j}] = i*h*\epsilon_{i,j,k}*Q_{k}, then they don't commute. Thus the CSCO is {Q_{I}, Q_{j}, Q_{k}}. Is this right?

 

[member 553083]

I don't understand how to explain the physical meaning of these observables and their eigenvalues. Any help would be greatly appreciated. Thanks!