Momentum eigenfunctions with periodic boundary conditions

[aabb009]

Homework Statement

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.

Homework Equations

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.

 

[Jhero]

Dear forum post author,

Thank you for your interesting question. I can provide some guidance on how to approach this problem.

Firstly, your choice of \psi(x) as e^{ikx} is a good starting point. This is a common choice for periodic boundary conditions as it satisfies the condition \psi(0) = \psi(L).

To quantize momentum, we can use the de Broglie relation p = \hbar k, where \hbar is the reduced Planck constant. This means that the allowed values for momentum are p = \frac{2\pi \hbar n}{L} with n=1,2,...

Now, for the normalization of the wavefunction in part b, we can use the fact that the integral of the squared modulus of the wavefunction over the entire space must be equal to 1. This means that we can split the integral into sections where the wavefunction is non-zero and then use the normalization condition to find the appropriate coefficients for each section.

For example, for the section |x-mL| < 1, we can write the normalization condition as:

1 = \int_{0}^{L} |\psi(x)|^2 dx = \int_{mL-1}^{mL+1} |\psi(x)|^2 dx

We can then use the given wavefunction \psi(x) to calculate the integral and solve for the coefficient for this section. We can repeat this process for the other sections where the wavefunction is non-zero.

Finally, to expand the normalized wavefunction in terms of eigenfunctions of momentum, we can use the fact that the eigenfunctions are orthogonal. This means that we can write the normalized wavefunction as a linear combination of the eigenfunctions with appropriate coefficients.

I hope this helps and provides some direction for your solution. Good luck with your calculations!