What does not normalizable ential

Re: What does "not normalizable" ential

When we talk about the "free particle wave function" we usually mean this (in the case of one-dimensional motion):

[tex]\Psi(x,t) = A e^{i(k x - \omega t)}[/tex]

By "not normalizable" we mean that there is no value of A that makes the following integral true:

[tex]\int^{+\infty}_{-\infty}{\Psi^* \Psi dx} = 1[/tex]

which makes the total probability of finding the particle somewhere equal to 1.

It simply means that the wave function given above is not actually a valid wave function for a free particle, strictly speaking. Physically, it means that it is not possible for a particle to have a completely definite, exact value of momentum [itex]p = \hbar k[/itex].

To get a valid wave function for a free particle, that is localized in space and is normalizable, you have to superpose wave functions that span a continuous range of momentum values:

[tex]\Psi(x,t) = \int^{+\infty}_{-\infty} {A(k) e^{i(k x - \omega t) } dk}[/tex]

where a function A(k) gives the amplitude of the wave that has momentum [itex]p = \hbar k[/itex]. This is called a wave packet. One common example (because it's relatively easy to analyze) is the Gaussian wave packet:

http://musr.physics.ubc.ca/~jess/hr/skept/GWP/

 

Re: What does "not normalizable" ential

Oops! I fixed it. Thanks!