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Homework Help: Expected value of x for quantum oscillator - integration help

  1. May 5, 2013 #1
    1. The problem statement, all variables and given/known data

    I have a wavefunction [itex]Cxe^{-ax^2}[/itex] and I have to find the expected value of x.

    2. Relevant equations

    [itex]∫_{-∞}^{∞} x^3 e^{-Ax^2} dx = 1/A^2 for A>0[/itex]

    3. The attempt at a solution

    I get an integral like this:

    [itex]<x>=|C|^2 ∫_{-∞}^{∞} x^3 e^{-Ax^2} dx[/itex]

    After trying integration by parts and failing (miserably), I took the coward's way out and went to an integration table. I found this:

    [itex]∫_{-∞}^{∞} x^3 e^{-Ax^2} dx = 1/A^2 for A>0[/itex]

    My question is this: my [itex]a=ωm/ \hbar[/itex] is always positive, right? I googled stuff on negative frequency, but I don't think that it applies in this situation (quantum oscillator).
    Last edited: May 5, 2013
  2. jcsd
  3. May 5, 2013 #2
    Actually, I think I got it using integration by parts, but I'd still like to know if my 'shortcut' is still valid.
  4. May 6, 2013 #3


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    Staff Emeritus
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    Gold Member

    If A<0, it's not that you need a different formula, it's that your integral doesn't converge at all.

    Although for this specific integral I think you either wrote it down wrong or did the problem wrong because you're integrating an odd function and should just get 0 regardless of what a is (again, as long as it's positive so the integral converges)
  5. May 6, 2013 #4
    I realized that now using Wolfram Alpha...that was my first thought, but then I thought that it would be too easy. I need to stop convincing myself that an easy solution is wrong solution...
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