Question from Haar's Problems in QM.

[MathematicalPhysicist]

Homework Statement

Determine the momentum proobability distribution function for particles in the n-th energy state in a potential well.

Homework Equations

a is the width of the well.
The answer is:for n odd:$$|a(p)|^2 = \frac{4an^2\pi}{\hbar}\frac{1}{(\frac{a^2p^2}{\hbar^2}-\pi^2n^2)^2} cos^2(\frac{pa}{2\hbar})$$
for n even:
:$$|a(p)|^2 = \frac{4an^2\pi}{\hbar}\frac{1}{(\frac{a^2p^2}{\hbar^2}-\pi^2n^2)^2} sin^2(\frac{pa}{2\hbar})$$

The Attempt at a Solution

What equations should I use here?

Sorry for my misunderstanding, it's more than a year from QM1, and I am bit rusty, thanks.

 

[gabbagabbahey]

You have a couple of options:

(1)Solve Schroedinger's equation in the momentum basis to find (a normalized) $\psi(p)$ (your problem statement seem to call this $a(p)$)...Since your potential is given in the position basis, you would need to first take its Fourier Transform to find its momentum basis representation.

(2)Solve Schroedinger's equation in the position basis to find (a normalized) $\psi(x)$ and then compute its Fourier transform to find $\psi(p)$

(3)If you have already solved Shroedinger's equation in either basis in your text/notes, just quote that solution and take the Fourier transform if necessary.

Once you've found $\psi(p)$, the momentum probability distribution is just its modulus squared.