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##SO(3)## generators

  1. Jan 26, 2017 #1
    1. The problem statement, all variables and given/known data
    Using the commutation relations ##[x,p_x] = i\hbar## etc, together with the ##SO(3)## generators ##J_k (k=x,y,z)## in their operator form to calculate ##[J_x, \mathbf{r}]## and ##[J_x, \mathbf{p}]## where ##r = (x, y, z)## and ##p = (p_x, p_y, p_z)##.

    Then show that ##[J_k, r^2]=0##.
    2. Relevant equations


    3. The attempt at a solution
    I can't find the operator forms anywhere. I have looked on the internet and in textbooks, but nowhere does it specifically state that a particular form is the 'operator' form. Is it just these matrices:
    ##
    \left( \begin{array}{ccc}
    0& 0 & 0 \\
    0& 0 & 1 \\
    0& -1 & 0 \end{array} \right) ##
    ##
    \left( \begin{array}{ccc}
    0& 0 & -1 \\
    0& 0 & 0 \\
    1& & 0 \end{array} \right) ##
    ##
    \left( \begin{array}{ccc}
    0& 1 & 0 \\
    -1& 0 & 0 \\
    0& 0 & 0 \end{array} \right) ##
    Even if that's the case how could I show ##[J_k, r^2]=0##? ##J_k## could be any one of the three. Do I have to show it for all of them?
    Any help is much appreciated!
     
    Last edited: Jan 26, 2017
  2. jcsd
  3. Jan 27, 2017 #2

    strangerep

    User Avatar
    Science Advisor

    What textbook or lecture notes are you using? (I'd have thought the operator form of the ##J_k## would have been provided.)
    I'm guessing you're meant to use the angular momentum generators ##{\mathbf J} = {\mathbf r}\times{\mathbf p}##. In component form, this is ##J_i = \epsilon_{ijk} r_j p_k## (using implicit summation over repeated indices).
     
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