A one-dimensional lattice has $N$ sites with lattice spacing $a$. In the tight-binding approximation, the Hamiltonian describing a single electron is given by

$$H=E_{0} \sum_{n}|n\rangle\langle n|-J \sum_{n}(|n\rangle\langle n+1|+| n+1\rangle\langle n|)$$

where $|n\rangle$ is the normalised state of the electron localised on the $n^{\text {th }}$lattice site. Using periodic boundary conditions $|N+1\rangle \equiv|1\rangle$, solve for the spectrum of this Hamiltonian to derive the dispersion relation

$$E(k)=E_{0}-2 J \cos (k a)$$

Define the Brillouin zone. Determine the number of states in the Brillouin zone.

Calculate the velocity $v$ and effective mass $m^{\star}$ of the particle. For which values of $k$ is the effective mass negative?

In the semi-classical approximation, derive an expression for the time-dependence of the position of the electron in a constant electric field.

Describe how the concepts of metals and insulators arise in the model above.