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Commutator question.

  1. Nov 13, 2009 #1

    MathematicalPhysicist

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    1. The problem statement, all variables and given/known data
    Prove that if A is an operator which commutes with two components of the rotation generator operator, J, then it commute with its third component.


    2. Relevant equations
    [tex][A_{\alpha},J_{\beta}]=i \hbar \epsilon_{\alpha \beta \gamma} A_{\gamma}[/tex]
    (not sure about the sign of this commutator it might be minus.


    3. The attempt at a solution
    Ok, then I am given:
    [tex][A,J_{\alpha}]=[A,J_{\beta}]=0[/tex]
    thus also [tex][A^2,J^2]=[A^2_{\alpha}+A^2_{\beta}+A^2_{\gamma},J^2_{\alpha}+J^2_{\beta}+J^2_{\gamma}]=[A^2,J^2_{\gamma}]=0[/tex] I used the above relevant equations to get to the last equality, but here is where I am stuck, I also know that [tex][A^2,J_{\gamma}]=0[/tex], but I don't seem to get to the last punch line which is [tex][A,J_{\gamma}]=0[/tex].

    Any hints?
     
  2. jcsd
  3. Nov 13, 2009 #2

    George Jones

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    Did you really mean to write this?

    All directions are equivalennt, so, WLOG, you can pick J_1 and J_2 as the components that commute with A.
    Use [J_1 , J_2] = i \hbar J_3 in [A , J_3], and maybe use the Jacobi identity.
     
  4. Nov 13, 2009 #3

    MathematicalPhysicist

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    OK thanks, I glossed over Jacobi identity in Wiki, and it does the job.
     
  5. Nov 13, 2009 #4

    George Jones

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    A more transparent way might be to expand completely (i.e., get rid all commutators) [A, [J_1, J_2]], and use the fact that A commutes with J_1 and J_2 to move all the A's completely to right (or left) of all the terms.
     
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