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Operator Theory Problem on Momentum Operator (QM)

  1. Jun 29, 2013 #1
    1. The problem statement, all variables and given/known data

    Given the operators [itex]\hat{x}=x\cdot[/itex] and [itex]\hat{p}=-i\hbar \frac{d}{dx}[/itex], prove that:

    [itex][\hat{x}, g(\hat{p})]=i\hbar \frac{dg}{d\hat p}[/itex]


    2. Relevant equations

    None.

    3. The attempt at a solution

    I have very little idea on how to begin this problem, but I don't want a solution, I simply want a hint in the right direction.

    Thanks, mates.
     
    Last edited: Jun 29, 2013
  2. jcsd
  3. Jun 29, 2013 #2

    Dick

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    Expand ##g(p)## in a power series. What's ##[x,p^n]##?
     
  4. Jun 29, 2013 #3
    Actually, I realized it was even easier than that, but thank you!

    Consider the case in momentum-space rather than position-space, and this reduces nicely using elementary operations.
     
  5. Jun 29, 2013 #4

    Dick

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    Sure, that works also. x is a differentiation operator in p space.
     
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