Define the Floquet matrix for a particle moving in a periodic potential in one dimension and explain how it determines the allowed energy bands of the system.

A potential barrier in one dimension has the form

$$V(x)= \begin{cases}V_{0}(x), & |x|<a / 4 \\ 0, & |x|>a / 4\end{cases}$$

where $V_{0}(x)$ is a smooth, positive function of $x$. The reflection and transmission amplitudes for a particle of wavenumber $k>0$, incident from the left, are $r(k)$ and $t(k)$ respectively. For a particle of wavenumber $-k$, incident from the right, the corresponding amplitudes are $r^{\prime}(k)$ and $t^{\prime}(k)=t(k)$. In the following, for brevity, we will suppress the $k$-dependence of these quantities.

Consider the periodic potential $\tilde{V}$, defined by $\tilde{V}(x)=V(x)$ for $|x|<a / 2$ and by $\tilde{V}(x+a)=\widetilde{V}(x)$ elsewhere. Write down two linearly independent solutions of the corresponding Schrödinger equation in the region $-3 a / 4<x<-a / 4$. Using the scattering data given above, extend these solutions to the region $a / 4<x<3 a / 4$. Hence find the Floquet matrix of the system in terms of the amplitudes $r, r^{\prime}$ and $t$ defined above.

Show that the edges of the allowed energy bands for this potential lie at $E=\hbar^{2} k^{2} / 2 m$, where

$$k a=i \log \left(t \pm \sqrt{r r^{\prime}}\right)$$