Atoms of mass $m$ in an infinite one-dimensional periodic array, with interatomic spacing $a$, have perturbed positions $x_{n}=n a+y_{n}$, for integer $n$. The potential between neighbouring atoms is

$$\frac{1}{2} \lambda\left(x_{n+1}-x_{n}-a\right)^{2}$$

for positive constant $\lambda$. Write down the Lagrangian for the variables $y_{n}$. Find the frequency $\omega(k)$ of a normal mode of wavenumber $k$. Define the Brillouin zone and explain why $k$ may be restricted to lie within it.

Assume now that the array is periodically-identified, so that there are effectively only $N$ atoms in the array and the atomic displacements $y_{n}$ satisfy the periodic boundary conditions $y_{n+N}=y_{n}$. Determine the allowed values of $k$ within the Brillouin zone. Show, for allowed wavenumbers $k$ and $k^{\prime}$, that

$$\sum_{n=0}^{N-1} e^{i n\left(k-k^{\prime}\right) a}=N \delta_{k, k^{\prime}}$$

By writing $y_{n}$ as

$$y_{n}=\frac{1}{\sqrt{N}} \sum_{k} q_{k} e^{i n k a}$$

where the sum is over allowed values of $k$, find the Lagrangian for the variables $q_{k}$, and hence the Hamiltonian $H$ as a function of $q_{k}$ and the conjugate momenta $p_{k}$. Show that the Hamiltonian operator $\hat{H}$ of the quantum theory can be written in the form

$$\hat{H}=E_{0}+\sum_{k} \hbar \omega(k) a_{k}^{\dagger} a_{k}$$

where $E_{0}$ is a constant and $a_{k}, a_{k}^{\dagger}$ are harmonic oscillator annihilation and creation operators. What is the physical interpretation of $a_{k}$ and $a_{k}^{\dagger}$ ? How does this show that phonons have quantized energies?