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Questions about rotating dumbbell

  1. Sep 26, 2011 #1
    1. The problem statement, all variables and given/known data

    I would like some help with the following problems

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

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

    3. Consider the rigidly rotating dumbbell molecule and let Q[itex]_{i}[/itex], J[itex]_{i}[/itex], 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.

    3. The attempt at a solution

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

    [itex]\Sigma^{r}_{s=-r}[/itex][itex]\Sigma^{l}_{m=-l}[/itex] |<a[itex]^{s}f^{r}_{s}[/itex] | J[itex]_{3}[/itex] | a[itex]^{m}f^{l}_{m}[/itex]>| [itex]^{2}[/itex] = [itex]\Sigma^{l}_{m=-l}[/itex]m[itex]^{2}[/itex] ?

    I am clueless as to how to solve #2

    For #3, I found that because [J[itex]_{i}[/itex], Q[itex]_{j}[/itex]] = i*h*[itex]\epsilon[/itex][itex]_{i,j,k}[/itex]*Q[itex]_{k}[/itex], then they don't commute. Thus the CSCO is {Q[itex]_{I}[/itex], Q[itex]_{j}[/itex], Q[itex]_{k}[/itex]}. Is this right?
  2. jcsd
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