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Probability Density Integral

  1. Apr 25, 2012 #1

    I am trying to figure out the following integral. I have two normalized 1D harmonic osccilator wave functions [itex]\psi_{n}(x)[/itex] and [itex]\psi_{m}(x)[/itex] and I would like to integrate
    \int_{\text{all space}} |\psi_{n}(x)|^2 |\psi_{m}(x)|^2 dx
    for [itex]m\neq n [/itex]. I would also be interested in knowing for what conditions on [itex]m[/itex] and [itex]n[/itex] could this integral be approximated as
    \int_{\text{all space}} |\psi_{n}(x)|^2 |\psi_{m}(x)|^2 dx \approx \left( \int |\psi_{n}(x)|^2 dx \right) \left( \int |\psi_{m}(x)|^2 dx \right) =1
    I have tried integrating by parts and waded through a couple of identities but I haven't been able to make much progress. Any ideas would be appreciated.

  2. jcsd
  3. Apr 25, 2012 #2

    Jano L.

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    Gold Member

    I do not know how to calculate your integral, but the approximation you have indicated cannot work for any n,m because the right-hand side of the equality has different dimensions.
  4. Apr 25, 2012 #3


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    Science Advisor

    jfy4, I don't see any physical relevance of the integrals you are interested in
  5. May 1, 2012 #4
    Good point, thanks.
  6. May 1, 2012 #5
    now, neither do I...
  7. May 1, 2012 #6
    Last edited by a moderator: May 6, 2017
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