2.4.1 Electron in a Periodic Potential

In an ideal crystal the ions occupy positions which form a regular periodic structure. The potential $U(\vec {r})$ is thus a periodic function with the period equal to the period of the corresponding Bravais lattice:

$$U(\vec{r}+\vec{R})=U(\vec{r}),$$ (2.64)

where $\vec{R}$ are the vectors which belong to the Bravais lattice. The period of the potential is of the same order as the de Broglie wave length which requires quantum mechanical consideration of the problem. As the total Hamiltonian for solids contains electron-electron interaction terms, the problem represents the many-body system. Within the theory of independent electrons an effective single-electron potential $U(\vec {r})$ is introduced. In the case of the ideal periodic crystal this potential must satisfy property (2.64). The main purpose is to analyze the periodicity^2.20 induced properties of the single-electron Schrödinger equation:

$$\biggl(-\frac{\hbar^{2}}{2m_{0}}\nabla^{2}+U(\vec{r})\biggr)\psi(\vec{r})=\epsilon\psi(\vec{r}).$$ (2.65)

Due to the potential periodicity the solution of this equation has several remarkable properties shortly given below.
S. Smirnov: