State and prove Bloch's theorem for the electron wave functions for a periodic potential $V(\mathbf{r})=V(\mathbf{r}+\mathbf{l})$ where $\mathbf{l}=\sum_{i} n_{i} \mathbf{a}_{i}$ is a lattice vector.

What is the reciprocal lattice? Explain why the Bloch wave-vector $\mathbf{k}$ is arbitrary up to $\mathbf{k} \rightarrow \mathbf{k}+\mathbf{g}$, where $\mathbf{g}$ is a reciprocal lattice vector.

Describe in outline why one can expect energy bands $E_{n}(\mathbf{k})=E_{n}(\mathbf{k}+\mathbf{g})$. Explain how $\mathbf{k}$ may be restricted to a Brillouin zone $B$ and show that the number of states in volume $d^{3} k$ is

$$\frac{2}{(2 \pi)^{3}} \mathrm{~d}^{3} k$$

Assuming that the velocity of an electron in the energy band with Bloch wave-vector $\mathbf{k}$ is

$$\mathbf{v}(\mathbf{k})=\frac{1}{\hbar} \frac{\partial}{\partial \mathbf{k}} E_{n}(\mathbf{k})$$

show that the contribution to the electric current from a full energy band is zero. Given that $n(\mathbf{k})=1$ for each occupied energy level, show that the contribution to the current density is then

$$\mathbf{j}=-e \frac{2}{(2 \pi)^{3}} \int_{B} \mathrm{~d}^{3} k n(\mathbf{k}) \mathbf{v}(\mathbf{k})$$

where $-e$ is the electron charge.