(a) Consider the nearly free electron model in one dimension with mass $m$ and periodic potential $V(x)=\lambda U(x)$ with $0<\lambda \ll 1$ and

$$U(x)=\sum_{l=-\infty}^{\infty} U_{l} \exp \left(\frac{2 \pi i}{a} l x\right)$$

Ignoring degeneracies, the energy spectrum of Bloch states with wavenumber $k$ is

$$E(k)=E_{0}(k)+\lambda\langle k|U| k\rangle+\lambda^{2} \sum_{k^{\prime} \neq k} \frac{\left\langle k|U| k^{\prime}\right\rangle\left\langle k^{\prime}|U| k\right\rangle}{E_{0}(k)-E_{0}\left(k^{\prime}\right)}+\mathcal{O}\left(\lambda^{3}\right)$$

where $\{|k\rangle\}$ are normalized eigenstates of the free Hamiltonian with wavenumber $k$. What is $E_{0}$ in this formula?

If we impose periodic boundary conditions on the wavefunctions, $\psi(x)=\psi(x+L)$ with $L=N a$ and $N$ a positive integer, what are the allowed values of $k$ and $k^{\prime}$ ? Determine $\left\langle k|U| k^{\prime}\right\rangle$ for these allowed values.

(b) State when the above expression for $E(k)$ ceases to be a good approximation and explain why. Quoting any result you need from degenerate perturbation theory, calculate to $\mathcal{O}(\lambda)$ the location and width of the band gaps.

(c) Determine the allowed energy bands for each of the potentials

$$\begin{aligned} &\text { (i) } V(x)=2 \lambda \cos \left(\frac{2 \pi x}{a}\right) \text {, } \\ &\text { (ii) } V(x)=\lambda a \sum_{n=-\infty}^{\infty} \delta(x-n a) \text {. } \end{aligned}$$

(d) Briefly discuss a macroscopic physical consequence of the existence of energy bands.