Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Questions about rigid rotating dumbbell molecule

  1. Sep 25, 2011 #1
    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.


    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?
     
    Last edited: Sep 25, 2011
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted