2.4.1.2 Energy Bands

In (2.66) $n$ denotes the band index as there are several independent states for a given vector $\vec{k}$. This can be seen by substituting equation (2.66) into the Schrödinger equation (2.65) which gives:

$$\biggl\{\frac{\hbar^{2}}{2m_{0}}(-i\nabla+\vec{k})^{2}+U(\vec{r})\biggr\}u_{\vec{k}}(\vec{r})=\epsilon_{\vec{k}}u_{\vec{k}}(\vec{r}),$$ (2.68)

with the periodic boundary condition:

$$u_{\vec{k}}(\vec{r})=u_{\vec{k}}(\vec{r}+\vec{R}).$$ (2.69)

This periodic boundary condition is very important as it allows to consider equation (2.68) as an eigenvalue problem for a finite volume, which leads to a discrete set of eigenvalues. The wave vector is only a parameter in this problem. Therefore, there are several branches^2.21 of the electron energy $\epsilon_{n}(\vec{k})$. S. Smirnov: