Derive the Bloch form of the wave function $\psi(x)$ of an electron moving in a onedimensional crystal lattice.

The potential in such an $N$-atom lattice is modelled by

$$V(x)=\sum_{n}\left(-\frac{\hbar^{2} U}{2 m} \delta(x-n L)\right)$$

Assuming that $\psi(x)$ is continuous across each lattice site, and applying periodic boundary conditions, derive an equation for the allowed electron energy levels. Show that for suitable values of $U L$ they have a band structure, and calculate the number of levels in each band when $U L>2$. Verify that when $U L \gg 1$ the levels are very close to those corresponding to a solitary atom.

Describe briefly how the band structure in a real 3-dimensional crystal differs from that of this simple model.