A periodic potential is expressed as $V(\mathbf{x})=\sum_{\mathbf{g}} a_{\mathbf{g}} e^{i \mathbf{g} \cdot \mathbf{x}}$, where $\{\mathbf{g}\}$ are reciprocal lattice vectors and $a_{\mathbf{g}}{ }^{*}=a_{-\mathbf{g}}, a_{\mathbf{0}}=0$. In the nearly free electron model explain why it is appropriate, near the boundaries of energy bands, to consider a Bloch wave state

$$\left|\psi_{\mathbf{k}}\right\rangle=\sum_{r} \alpha_{r}\left|\mathbf{k}_{r}\right\rangle, \quad \mathbf{k}_{r}=\mathbf{k}+\mathbf{g}_{r},$$

where $|\mathbf{k}\rangle$ is a free electron state for wave vector $\mathbf{k},\left\langle\mathbf{k}^{\prime} \mid \mathbf{k}\right\rangle=\delta_{\mathbf{k}^{\prime} \mathbf{k}}$, and the sum is restricted to reciprocal lattice vectors $\mathbf{g}_{r}$ such that $\left|\mathbf{k}_{r}\right| \approx|\mathbf{k}|$. Obtain a determinantal formula for the possible energies $E(\mathbf{k})$ corresponding to Bloch wave states of this form.

[You may take $\mathbf{g}_{1}=\mathbf{0}$ and assume $e^{i \mathbf{b} \cdot \mathbf{x}}|\mathbf{k}\rangle=|\mathbf{k}+\mathbf{b}\rangle$ for any $\mathbf{b}$.]

Suppose the sum is restricted to just $\mathbf{k}$ and $\mathbf{k}+\mathbf{g}$. Show that there is a gap $2\left|a_{\mathbf{g}}\right|$ between energy bands. Setting $\mathbf{k}=-\frac{1}{2} \mathbf{g}+\mathbf{q}$, show that there are two Bloch wave states with energies near the boundaries of the energy bands

$$E_{\pm}(\mathbf{k}) \approx \frac{\hbar^{2}|\mathbf{g}|^{2}}{8 m} \pm\left|a_{\mathbf{g}}\right|+\frac{\hbar^{2}|\mathbf{q}|^{2}}{2 m} \pm \frac{\hbar^{4}}{8 m^{2}\left|a_{\mathbf{g}}\right|}(\mathbf{q} \cdot \mathbf{g})^{2}$$

What is meant by effective mass? Determine the value of the effective mass at the top and the bottom of the adjacent energy bands if $\mathbf{q}$ is parallel to $\mathrm{g}$.