a) Consider a particle moving in one dimension subject to a periodic potential, $V(x)=V(x+a)$. Define the Brillouin zone. State and prove Bloch's theorem.

b) Consider now the following periodic potential

$$V=V_{0}(\cos (x)-\cos (2 x))$$

with positive constant $V_{0}$.

i) For very small $V_{0}$, use the nearly-free electron model to compute explicitly the lowest-energy band gap to leading order in degenerate perturbation theory.

ii) For very large $V_{0}$, the electron is localised very close to a minimum of the potential. Estimate the two lowest energies for such localised eigenstates and use the tight-binding model to estimate the lowest-energy band gap.