For a periodic potential $V(\mathbf{r})=V(\mathbf{r}+\ell)$, where $\ell$ is a lattice vector, show that we may write

$$V(\mathbf{r})=\sum_{\mathbf{g}} a_{\mathbf{g}} e^{i \mathbf{g} \cdot \mathbf{r}}, \quad a_{\mathbf{g}}^{*}=a_{-\mathbf{g}}$$

where the set of $g$ should be defined.

Show how to construct general wave functions satisfying $\psi(\mathbf{r}+\boldsymbol{\ell})=e^{i \mathbf{k} \cdot \boldsymbol{\ell}} \psi(\mathbf{r})$ in terms of free plane-wave wave-functions.

Show that the nearly free electron model gives an energy gap $2\left|a_{\mathbf{g}}\right|$ when $\mathbf{k}=\frac{1}{2} \mathbf{g}$.

Explain why, for a periodic potential, the allowed energies form bands $E_{n}(\mathbf{k})$ where $\mathbf{k}$ may be restricted to a single Brillouin zone. Show that $E_{n}(\mathbf{k})=E_{n}(\mathbf{k}+\mathbf{g})$ if $\mathbf{k}$ and $\mathbf{k}+\mathbf{g}$ belong to the Brillouin zone.

How are bands related to whether a material is a conductor or an insulator?