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Commutation Relationships and Operator Functions

  1. Nov 3, 2012 #1
    There are 2 operators such that [A,B] = 0. Does [F(A),B]=0 ?

    Specifically, lets say we had the Hamiltonian of a 3-D oscillator H and L^2. We know that L^2 = Lx^2+Ly^2+Lz^2, and it is known that [H,Lz] = 0. Can we say that since H and Lz commute, H and Lz^2 also commute, by symmetry H and Lx^2,Ly^2 commute also and therefore H and L^2 commute?
  2. jcsd
  3. Nov 3, 2012 #2
    If the Taylor expansion of [itex]F(A)[/itex] converges, then you can essentially assume that it is a polynomial in [itex]A[/itex], so it will commute. Your argument about [itex]H[/itex] and [itex]L^2[/itex] sounds right.
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