# Momentum eigenfunctions with periodic boundary conditions

1. Mar 13, 2008

### aabb009

1. The problem statement, all variables and given/known data
A particle of mass m is confined to move in one dimension. its wavefunction is periodic with period $$L\gg 1$$ - i.e. periodic boundary conditions are imposed.

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

b)The wavefunction is the periodic function
$$\psi (x) = { 1-|x-mL}, |x-mL| < 1, m=0, \pm 1, \pm 2,...$$
$$\psi (x) =0$$ otherwise
normalise $$\psi (x)$$ on the interval $$[0,L)$$ 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 $$\psi (0) = \psi (L)$$

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

I let $$\psi = e^{ikx},$$ with $$k=\frac{2\pi n}{L}$$ and $$n=1,2,...$$
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 $$\psi (x)$$

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.