(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A particle of massmis confined to move in one dimension. its wavefunction is periodic with period [tex]L\gg 1[/tex] - i.e. periodic boundary conditions are imposed.

a)Determine the eigenfunctions and eigenvalues of momentum. Normalise the eigenfunctions on the interval [tex][0,L)[/tex]

b)The wavefunction is the periodic function

[tex]\psi (x) = { 1-|x-mL}, |x-mL| < 1, m=0, \pm 1, \pm 2,...[/tex]

[tex]\psi (x) =0[/tex] otherwise

normalise [tex]\psi (x)[/tex] on the interval [tex][0,L)[/tex] and expand the normalised wavefunction in eigenfunctions of momentum.

2. Relevant equations

3. The attempt at a solution

I understand that periodic boundary conditions imply that [tex]\psi (0) = \psi (L)[/tex]

Now this is where I'm not sure if I've gone right:

I let [tex]\psi = e^{ikx}, [/tex] with [tex]k=\frac{2\pi n}{L}[/tex] and [tex]n=1,2,...[/tex]

Hence I quantised momentum in terms of n. My lecturer hinted that the solution would involve quantising momentum, since we are not considering the wavefunction over all of space, and by letting L go to infinity it would approach continuous values for p, my solution doesn't seem to do this though.

I can follow the maths through to get the eigenfunctions/values from here, I'm just not sure about my choice of [tex]\psi (x)[/tex]

In part b I'm not sure how to normalise the wavefunction, I understand it will involve splitting the integral up into sections with different boundaries, but I'm not sure how to tackle a function like this.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Momentum eigenfunctions with periodic boundary conditions

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**