### 2.4.1.3 Band Structure Periodicity

For two wave vectors and the solutions of Schrödinger equation (2.68) are related to each other as . This leads to equal eigenvalues

 (2.70)

and equal wave functions

 (2.71)

It can be seen that each energy branch has the same period as the reciprocal lattice. As the functions are periodic, they have maxima and minima which determine the width of the bands.

It should be noted that the wave vector in (2.66) can always be chosen in a way to belong to the first Brillouin zone because any vector out of the first Brillouin zone can be represented as the sum , where is a vector of the reciprocal lattice. Using the equivalent form of Bloch's theorem:

 (2.72)

together with (2.71) and the equality one obtains (2.72) for vector . S. Smirnov: