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## Homework Statement

A free particle (de Broglie wave) may be represented by the wave-function

[itex]\psi[/itex](x)=Ae

^{ikx}Show that this is an eigenstate of the momentum operator [itex]\hat{p}[/itex]=-[itex]\hbar[/itex][itex]\frac{\delta}{\deltax}[/itex]

## Homework Equations

[itex]\hat{p}[/itex]u

_{n}(x)=a

_{n}u

_{n}(x)

a

_{n}is the eigenvalue

u

_{n}(x) is the corresponding eigenfunction

## The Attempt at a Solution

Ok so first I have

[itex]\hat{p}[/itex]=-[itex]\hbar[/itex][itex]\frac{\delta}{\deltax}[/itex]

which has the boundary condition that u_{n}is periodic in the range L. Then the eigenvalue equation is

-i[itex]\hbar\frac{\delta}{\deltax}[/itex][itex]\psi[/itex](x)=a

where [itex]\psi[/itex](x) is the eigenfunction u_{n}[itex]\psi[/itex](x)_{n}(x)

The solution of the equation is then

[itex]\psi[/itex](x)=[itex]\hbar[/itex]e

=[itex]\psi[/itex](x)=Ae

where k is the eigenvalue a^{ianx}=[itex]\psi[/itex](x)=Ae

^{ikx}_{n}and A is [itex]\hbar[/itex] (Plancks Constant)

Ok so what I would like to know is:

Is the above correct?

If so why is it correct?

If its wrong why is it wrong?

What exactly is an eigenfunction and eigenvalue?

What is this range L?

As you can probably guess from the above questions I know absolutely nothing on this topic. I have a sort of understanding on what an eigenfunction and eigenvalue are but after going through revision notes on this stuff numerous times I still can't get my head round the mathematics of it, how everything changes to the way it does. The only reason why I have the above working out is because the revision notes on the topic is very similar to the question but is quite vague on how one thing transitions to the next.

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