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Behaviour of an Exponential Commutation

  1. Mar 31, 2013 #1
    1. The problem statement, all variables and given/known data
    Forgive the awkward title, it was hard to think how to describe the problem in such a short space. I'm following the proof of Proposition 1.2 of Nakahara's "Geometry, Topology and Physics", and the following commutation relation has been established:

    [itex]\partial_x^n e^{ikx} = e^{ikx}(ik + \partial_x )^n[/itex]

    The line immediately afterwards is:

    [itex]e^{-i\epsilon \{-\partial_x^2 / 2m + V(x)\}}e^{ikx} = e^{ikx}e^{-i\epsilon \{-(ik+\partial_x )^2 / 2m + V(x)\}}[/itex]

    I am not seeing how this follows from the previous relation, since the partial derivative is in the exponential so surely it doesn't act on [itex]e^{ikx}[/itex] in the same way? It seems dodgy, but I can only assume I'm missing some important result concerning exponentials of commutation relationships that hasn't otherwise been specified anywhere in the book as far as I can tell.

    2. Relevant equations
    (See above)

    3. The attempt at a solution
    (See above)
  2. jcsd
  3. Apr 1, 2013 #2


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    Homework Helper

    hint: try using the power series expansion of an exponential function.
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