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!