Bloch's theorem

Fundamental theorem in condensed matter physics

In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential can be expressed as plane waves modulated by periodic functions. The theorem is named after the Swiss physicist Felix Bloch, who discovered the theorem in 1929. Mathematically, they are written where r {\displaystyle \mathbf {r} } is position, ψ {\displaystyle \psi } is the wave function, u {\displaystyle u} is a periodic function with the same periodicity as the crystal, the wave vector k {\displaystyle \mathbf {k} } is the crystal momentum vector, e {\displaystyle e} is Euler's number, and i {\displaystyle i} is the imaginary unit.

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