Consider a one-dimensional crystal lattice of lattice spacing $a$ with the $n$-th atom having position $r_{n}=n a+x_{n}$ and momentum $p_{n}$, for $n=0,1, \ldots, N-1$. The atoms interact with their nearest neighbours with a harmonic force and the classical Hamiltonian is

$$H=\sum_{n} \frac{p_{n}^{2}}{2 m}+\frac{1}{2} \lambda\left(x_{n}-x_{n-1}\right)^{2}$$

where we impose periodic boundary conditions: $x_{N}=x_{0}$. Show that the normal mode frequencies for the classical harmonic vibrations of the system are given by

$$\omega_{l}=2 \sqrt{\frac{\lambda}{m}}\left|\sin \left(\frac{k_{l} a}{2}\right)\right|,$$

where $k_{l}=2 \pi l / N a$, with $l$ integer and (for $N$ even, which you may assume) $-N / 2<l \leqslant$ $N / 2$. What is the velocity of sound in this crystal?

Show how the system may be quantized to give the quantum operator

$$x_{n}(t)=X_{0}(t)+\sum_{l \neq 0} \sqrt{\frac{\hbar}{2 N m \omega_{l}}}\left[a_{l} e^{-i\left(\omega_{l} t-k_{l} n a\right)}+a_{l}^{\dagger} e^{i\left(\omega_{l} t-k_{l} n a\right)}\right],$$

where $a_{l}^{\dagger}$ and $a_{l}$ are creation and annihilation operators, respectively, whose commutation relations should be stated. Briefly describe the spectrum of energy eigenstates for this system, stating the definition of the ground state $|0\rangle$ and giving the expression for the energy eigenvalue of any eigenstate.

The Debye-Waller factor $e^{-W(Q)}$ associated with Bragg scattering from this crystal is defined by the matrix element

$$e^{-W(Q)}=\left\langle 0\left|e^{i Q x_{0}(0)}\right| 0\right\rangle$$

In the case where $\left\langle 0\left|X_{0}\right| 0\right\rangle=0$, calculate $W(Q)$.