Consider a one-dimensional crystal of lattice space $b$, with atoms having positions $x_{s}$ and momenta $p_{s}, s=0,1,2, \ldots, N-1$, such that the classical Hamiltonian is

$$H=\sum_{s=0}^{N-1}\left(\frac{p_{s}^{2}}{2 m}+\frac{1}{2} m \lambda^{2}\left(x_{s+1}-x_{s}-b\right)^{2}\right)$$

where we identify $x_{N}=x_{0}$. Show how this may be quantized to give the energy eigenstates consisting of a ground state $|0\rangle$ together with free phonons with energy $\hbar \omega\left(k_{r}\right)$ where $k_{r}=2 \pi r / N b$ for suitable integers $r$. Obtain the following expression for the quantum operator $x_{s}$

$$x_{s}=s b+\left(\frac{\hbar}{2 m N}\right)^{\frac{1}{2}} \sum_{r} \frac{1}{\sqrt{\omega\left(k_{r}\right)}}\left(a_{r} e^{i k_{r} s b}+a_{r}^{\dagger} e^{-i k_{r} s b}\right)$$

where $a_{r}, a_{r}^{\dagger}$ are annihilation and creation operators, respectively.

An interaction involves the matrix element

$$M=\sum_{s=0}^{N-1}\left\langle 0\left|e^{i q x_{s}}\right| 0\right\rangle .$$

Calculate this and show that $|M|^{2}$ has its largest value when $q=2 \pi n / b$ for integer $n$.

Disregard the case $\omega\left(k_{r}\right)=0$.

[You may use the relations

$$\sum_{s=0}^{N-1} e^{i k_{r} s b}= \begin{cases}N, & r=N b \\ 0 & \text { otherwise }\end{cases}$$

and $e^{A+B}=e^{A} e^{B} e^{-\frac{1}{2}[A, B]}$ if $[A, B]$ commutes with $A$ and with $\left.B .\right]$