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Commutator Problem

  1. Apr 14, 2015 #1
    1. The problem statement, all variables and given/known data

    Let the commutator [A,B] = cI, I the identity matrix and c some arbitrary constant.

    Show [A,Bn] = cnBn-1

    2. Relevant equations

    [A,B] = AB - BA

    3. The attempt at a solution

    So I have started off like this:

    [A,Bn] = ABn - BnA = cI

    I'm not sure where to go from here.
     
  2. jcsd
  3. Apr 14, 2015 #2

    HallsofIvy

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    I think you are misunderstanding the question. [A,B^n] = AB^n - B^nA = cI is NOT in general true. You seem to be thinking that "AB- BA= cI" is to be true for all A, B. It is not. In this problem AB- BA= cI is true for this specific A and B.

    You are told that AB- BA= cI. So [A, B^2]= AB^2- B^2A= AB^2- BAB+ BAB- B^2A= (AB- BA)B+ B(AB- BA)= cB+ Bc= 2cB. etc. Use proof by induction.
     
  4. Apr 14, 2015 #3

    Dick

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    For any operators you have the identity ##[D,EF]=[D,E]F+E[D,F]## that's handy and it's easy to prove. The case ##n=1## is obvious, so now try ##n=2##. Write ##[A,B^2]## as ##[A,BB]##. For the general case think about induction.
     
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