(i) A simple model of a one-dimensional crystal consists of an infinite array of sites equally spaced with separation $a$. An electron occupies the $n$th site with a probability amplitude $c_{n}$. The time-dependent Schrödinger equation governing these amplitudes is

$$i \hbar \frac{d c_{n}}{d t}=E_{0} c_{n}-A\left(c_{n-1}+c_{n+1}\right)$$

where $E_{0}$ is the energy of an electron at an isolated site and the amplitude for transition between neighbouring sites is $A>0$. By examining a solution of the form

$$c_{n}=e^{i k a n-i E t / \hbar}$$

show that $E$, the energy of the electron in the crystal, lies in a band

$$E_{0}-2 A \leq E \leq E_{0}+2 A$$

Identify the Brillouin zone for this model and explain its significance.

(ii) In the above model the electron is now subject to an electric field $\mathcal{E}$ in the direction of increasing $n$. Given that the charge on the electron is $-e$ write down the norm of the time-dependent Schrödinger equation for the probability amplitudes. Show that it has a solution of the form

$$c_{n}=\exp \left\{-\frac{i}{\hbar} \int_{0}^{t} \epsilon\left(t^{\prime}\right) d t^{\prime}+i\left(k-\frac{e \mathcal{E} t}{\hbar}\right) n a\right\}$$

where

$$\epsilon(t)=E_{0}-2 A \cos \left(\left(k-\frac{e \mathcal{E} t}{\hbar}\right) a\right)$$

Explain briefly how to interpret this result and use it to show that the dynamical behaviour of an electron near the bottom of the energy band is the same as that for a free particle in the presence of an electric field with an effective mass $m^{*}=\hbar^{2} /\left(2 A a^{2}\right)$.