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MathematicalPhysicist
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Homework Statement
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.
Homework 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.
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?