2.4.1.1 Bloch's Theorem

Bloch's theorem states that the solution of equation (2.65) has the form of a plane wave multiplied by a function with the period of the Bravais lattice:

$$\psi_{n\vec{k}}(\vec{r})=\exp(i\vec{k}\cdot\vec{r})u_{n\vec{k}}(\vec{r}),$$ (2.66)

where the function $u_{n\vec{k}}(\vec{r})$ satisfies the following condition:

$$u_{n\vec{k}}(\vec{r}+\vec{R})=u_{n\vec{k}}(\vec{r}),$$ (2.67)

for all vectors lattice $\vec{R}$. Note that Bloch's theorem uses a vector $\vec{k}$. In the periodic potential this vector plays the role analogous to that of the wave vector in the theory of free electrons. S. Smirnov: