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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

[tex]

\int_{\text{all space}} |\psi_{n}(x)|^2 |\psi_{m}(x)|^2 dx

[/tex]

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

[tex]

\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

[/tex]

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.

Thanks,

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# Probability Density Integral

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