Normalization of a free particle quantum state

[JK423]

Can anyone explain to me why we use the periodic boundary condition
Ψ(x)=Ψ(x+L), in order to normalize the free particle's quantum state??

I've made 2 threads already on this some time ago, but haven't got an answer still..
I hope this time i`ll have because I am really curious about the physical significance of PBC!

John

 

[G01]

Hi John,

The issue with the free particle wavefunction, $e^{\pm ikx}$ and linear combinations of these functions, is that they are not square integrable, and thus not normalizable if you just try to integrate over all space.

There are several ways we can get around this:

1. Work with wavepackets. We can replace the free particle's momentum space wavefunction (a delta function), with some highly peaked, non delta distribution. This will result in a position space wavefunction that is a normalizable wave packet.

2. We talk only in terms of probability flux, and thus avoid the questions of normalization and the probability of finding the particle in any finite region.

3. We can think of free space as being periodic and impose periodic boundary conditions. Doing so, allows us to normalize the wavefunction. Since the periodicity of free space is arbitrary anyway (empty space is periodic with any periodicity), observable results for the free particle should not depend on it anyway.

 

[hokhani]

hi
thanks
but
i am not convinced about the free particle