Why is my two-particle wavefunction not normalizing over time?

[htown1397]

Homework Statement

Hi, I've been working on this for a while but I just can't seem to figure this out. I have to solve a problem regarding a one-dimensional two-particle wavefunction psi(x1, x2, t) that is normalized at t=0, and the particles are not in spin. I have to show that the wavefunction remains normalized for all time. I would appreciate any help.

The Attempt at a Solution

I know that to normalize, Integral[|psi(x1, x2, t)|^2] =1. So, I have written out the wavefunction, how I think it is, for a two particle system:

(1/a) Sin[(n*Pi*x)/(2*a)]*Sin[(m*Pi*y)/(2*a)]*Exp[-I*w*t]

Then, I went ahead and found the complex conjugate and multiplied it by the original wavefunction. I tried to integrate it on mathematica from -Infinity to Infinity, but it said the integral does not converge.

 

[saman]

Hi,

I believe that you are using the wavefuction for a particle in a box of length 2a. Hence, you don't have to integrate over all space - only in the box (i.e. from -a to a).

Having a plane wave in unbound space is unphysical and not normalisable - since that would imply that the particle exists everywhere for all time.

 

[htown1397]

Right. I was working it out and that's the problem I ran into. So, if that's not how it's done, how would I go about solving this problem then?

 

[turin]

Actually, this problem is quite trivial if you know that wavefunction. You don't actually have to do the integral, because you can factorize e^iwt out.

 

[saman]

Right. I was working it out and that's the problem I ran into. So, if that's not how it's done, how would I go about solving this problem then?

Did you read my post? You have to integrate between "a" and "-a", and NOT from infinity to -infinity.