# Quantum Mechanics. Normalised basis wave functions and Eigenvalues.

## Homework Statement

Consider a particle with periodic boundary conditions of length L. Write dwon the expression for the normalised basis wave functions and their eigenvalues. Find the eigen value of the momentum and the expectation value of the momentum with respect to one of the momentum with respect to one of these wave functions. Explain the result in terms of measurement outcomes of the momentum.

## The Attempt at a Solution

I haven't had any formal Linear algebre training so I always struggle with these questions.

I know that the wave function for periodic boundary conditions is;

$$\Psi$$ (x,t) = $$\frac{1}{\sqrt{L}}$$ eikx - iEt/h

Where h = h bar

and that the expetation value is;

<p> = $$\int$$ ($$\Psi$$* $$\hat{p}$$ $$\Psi$$ .dx

where p = -ih d/dx.

and that the measurement outcome is;

<$$\hat{p}$$> = lim n$$\rightarrow$$$$\infty$$ $$\frac{1}{N}$$ $$\sum$$ pi

What I don't understand is how to find the eigen values of momentum from these, and how to explain the result in terms of the measurement outcome.

Can anybody help me understand this?

Many Thanks