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Commutation relation of operators involving momentum and position

  1. Sep 30, 2011 #1
    1. The problem statement, all variables and given/known data
    The problem is number 11, the problem statement would be in the first picture in the spoiler.
    Basically, I'm trying to find if two operators commute. They're not supposed to, since they involve momentum and position, but my work has been suggesting otherwise, so I'm doing something wrong.

    2. Relevant equations
    Are in problem 10 and written next to it. (2nd picture in spoiler.)
    x^ = x
    p^ = -iħ d/dx

    P^= p^/√(mωλ)
    Q^=x^ * (√(mω/ħ))


    3. The attempt at a solution
    Also in the picture.
    I think I'm messing up where the operators operate on each other and new terms are created, and I'm not sure where or how to fix it.

    2011-09-30161342.jpg
    2011-09-30161246.jpg
     
  2. jcsd
  3. Sep 30, 2011 #2
    In (10) you have already proved that [P,Q] = -i[itex]\sqrt{hbar/\lambda}[/itex]
    Now re (11), do not substitute for the operators P and Q but leave them as P and Q. Hence the resulting expression will be in terms of P and Q. Then use [P,Q] = -i[itex]\sqrt{hbar/\lambda}[/itex]
     
  4. Sep 30, 2011 #3
    But be very careful since P and Q do not commute!
     
  5. Oct 1, 2011 #4
    Thank you grzz, that worked out much easier than what I was trying.
    :smile:
     
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