1 Answers
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. Mathematically, they are written
ψ = e i k ⋅ r u {\displaystyle \psi =e^{i\mathbf {k} \cdot \mathbf {r} }u}
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.
Functions of this form are known as Bloch functions or Bloch states, and serve as a suitable basis for the wave functions or states of electrons in crystalline solids.