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Commutator problem (position, momentum)

  1. Oct 10, 2014 #1
    I'm having some difficulties with a certain commutator producing inconsistent results. Specifically I'm referring to

    [tex][p_x,x^3][/tex]

    Depending on how i expand this it seems i get different coefficients, i.e

    [tex][p_x,x^3]=[p_x,x]x^2+x^2[p_x,x]=-i\hbar x^2 -x^2i\hbar=-2i\hbar x^2[/tex]

    However

    [tex][p_x,x^3]=[p_x,x^2]x+x[p_x,x^2]=([p_x,x]x+x[p_x,x])x+x([p_x,x]x+x[p_x,x])=-i4\hbar x^2[/tex].

    Clearly I'm missing something here, but i can't quite figure out what.
     
  2. jcsd
  3. Oct 10, 2014 #2

    rubi

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    Science Advisor

    The rule is ##[X,AB] = [X,A]B + A[X,B]##. You seem to be using ##[X,AB] = [X,B]A + A[X,B]## instead.
     
  4. Oct 10, 2014 #3

    dextercioby

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    Homework Helper

    Put a wavefunction from the Schwartz space over the reals to the right of the commutator, use the position representation, don't use Leibniz, but only the definition and check which of the 2 results is right.
     
  5. Oct 10, 2014 #4
    Yes of course, it should be

    [tex][p_x,x^3]=[p_x,x]x^2+x[p_x,x^2]=-3i\hbar x^2.[/tex]

    Thank you!
     
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