Free particle model in QM

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Zinc Potterman - 26 Sep 2005 10:35 GMT

In my text it states that a truncated TOP Hat function in one dimension

f(x)= exp(iKx) | x | < or = d

can be used as a simple model for a particle at time t=0 to explore the
subsequent time development.
After some exploration of this which I can follow it states

" the top hat function is not a momentum eigenfunction"

This rather pulled the carpet from under my understanding.

It seems that the point of the truncation was to make the starting
eigenfunction normalisable, but
in doing so it is no longer a momentum eigenfunction.

1. Does this mean that it is not a solution to Schroedinger's Time dependent
equation?
If it isn't then what is the point of exploring it in this context?

Regards
Zinc

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Chris Dams - 26 Sep 2005 19:04 GMT

Dear Zinc,

> In my text it states that a truncated TOP Hat function in one dimension
>
[quoted text clipped - 11 lines]
> eigenfunction normalisable, but
> in doing so it is no longer a momentum eigenfunction.

What is so mysterious about this? You start out with a momentum
eigenfunction and then you truncate it. Why would it remain a momentum
eigenfunction then? It just happens to be the case that momentum
eigenfunctions are not normalizable in the sense of
<psi_n|psi_m>=delta_n,m.

> 1. Does this mean that it is not a solution to Schroedinger's Time dependent
> equation?

For one thing, this wave function f does not contain a time variable t,
so unless H f=0, it is not a solution of the time dependent Schroedinger
equation.

> If it isn't then what is the point of exploring it in this context?

I suppose not. Perhaps the point is that this exploration was deemed
instructive and/or interesting by the author of your text.

Best wishes,
Chris

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carlip-nospam@physics.ucdavis.edu - 27 Sep 2005 00:45 GMT

> In my text it states that a truncated TOP Hat function in one dimension

> f(x)= exp(iKx) | x | < or = d

> can be used as a simple model for a particle at time t=0 to explore the
> subsequent time development.

Presumably it explains that this is an *initial value* for the
wave function -- that's what "at time t=0" means.

[...]

> 1. Does this mean that it is not a solution to Schroedinger's Time
> dependent equation?

Well, it doesn't have time in it, and it is described as the wave
function at t=0, so it's pretty unlikely that it's a solution of
a time-dependent equation.

You are asking the quantum mechanical analog of the following question
in classical mechanics: "My text describes a free particle as being at
position x=0 and having velocity v at time t=0. Is this a solution
of Newton's equations?" The answer, of course, is, "No, it's an initial
condition for Newton's equations. To get the time-dependent solution,
x=vt, you have to *solve* the equations."

Steve Carlip

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Natural Science Forum / Physics / Particle Physics / September 2005

Free particle model in QM