- #1
jfy4
- 649
- 3
Hi,
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,
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,