# Probability Density Integral

1. Apr 25, 2012

### jfy4

Hi,

I am trying to figure out the following integral. I have two normalized 1D harmonic osccilator wave functions $\psi_{n}(x)$ and $\psi_{m}(x)$ and I would like to integrate
$$\int_{\text{all space}} |\psi_{n}(x)|^2 |\psi_{m}(x)|^2 dx$$
for $m\neq n$. I would also be interested in knowing for what conditions on $m$ and $n$ 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.

Thanks,

2. Apr 25, 2012

### Jano L.

jfy4,
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.

3. Apr 25, 2012

### tom.stoer

jfy4, I don't see any physical relevance of the integrals you are interested in

4. May 1, 2012

### jfy4

Good point, thanks.

5. May 1, 2012

### jfy4

now, neither do I...

6. May 1, 2012

### TomDance

Last edited by a moderator: May 6, 2017