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Commutation relations

  1. Oct 1, 2007 #1
    i need to find the commutation relation for:

    [tex] [x_i , p_i ^n p_j^m p_k^l] [/tex]

    I could apply a test function g(x,y,z) to this and get:

    [tex]=x_i p_i ^n p_j^m p_k^l g - p_i ^n p_j^m p_k^l x_i g [/tex]

    but from here I'm not sure where to go. any help would be appreciated.
     
  2. jcsd
  3. Oct 1, 2007 #2

    Gokul43201

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    You don't need a test function. All you need are the following:

    (i) [itex] [x_i,p_j] = i \hbar \delta_{i,j} [/itex]
    (ii) [itex] [AB,C]=A[B,C]+[A,C]B [/itex]
     
    Last edited: Oct 2, 2007
  4. Oct 2, 2007 #3
    should that be [itex] [x_i,p_j] = i \hbar \delta_{i,j} [/itex]?
     
  5. Oct 2, 2007 #4
    i guess a more reasonable question would i expand [itex][x_i,p_i^n][/itex]
     
  6. Oct 2, 2007 #5

    Gokul43201

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    If you use the second relationship in post #2 recursively, you will discover a general form for the commutator [itex][x_i,p_i^n] [/itex].
    Try p^2 and p^3 first - you'll see what I mean.

    PS: Yes, there was a "bad" minus sign which I've now fixed.
     
  7. Oct 2, 2007 #6
    how about: [itex][x_i,p_i^n]=ni \hbar p_i ^{n-1} [/itex]
     
  8. Oct 3, 2007 #7

    Gokul43201

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    Looks good. Now you're just a step or two away from the answer to the original question.
     
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