Bloch Wave

A Bloch wave or Bloch state, named after Swiss physicist Felix Bloch, is the wavefunction of a particle (usually, an electron) placed in a periodic potential. Bloch's theorem states that the energy eigenfunction for such a system may be written as the product of a plane wave envelope function and a periodic function (periodic Bloch function) that has the same periodicity as the potential, giving:

The corresponding energy eigenvalues are ϵ_n(k) = ϵ_n(k + K), periodic with periodicity K of a reciprocal lattice vector. The energies associated with the index n vary continuously with wave vector k and form an energy band identified by band index n. The eigenvalues for given n are periodic in k; all distinct values of ϵ_n(k) occur for k-values within the first Brillouin zone of the reciprocal lattice.