An electron of mass $m$ moves in a $D$-dimensional periodic potential that satisfies the periodicity condition

$$V(\boldsymbol{r}+\boldsymbol{l})=V(\boldsymbol{r}) \quad \forall l \in \Lambda,$$

where $\Lambda$ is a D-dimensional Bravais lattice. State Bloch's theorem for the energy eigenfunctions of the electron.

For a one-dimensional potential $V(x)$ such that $V(x+a)=V(x)$, give a full account of how the "nearly free electron model" leads to a band structure for the energy levels.

Explain briefly the idea of a Fermi surface and its rôle in explaining the existence of conductors and insulators.