Bloch's Theorem

A Bloch wave (also called Bloch state or Bloch function or Bloch wave function), named after Swiss physicist Felix Bloch, is a type of wavefunction for a particle in a periodically-repeating environment, most commonly an electron in a crystal. A wavefunction ψ is a Bloch wave if it has the form:

where r is position, ψ is the Bloch wave, u is a periodic function with the same periodicity as the crystal, k is a vector of real numbers called the crystal wave vector, e is Euler's number, and i is the imaginary unit. In other words, if you multiply a plane wave by a periodic function, you get a Bloch wave.

Bloch waves are important because of Bloch's theorem, which states that the energy eigenstates for an electron in a crystal can be written as Bloch waves. (More precisely, it states that the electron wave functions in a crystal have a basis consisting entirely of Bloch wave energy eigenstates.) This fact underlies the concept of electronic band structures.

These Bloch wave energy eigenstates are written with subscripts as ψ_{n k}, where n is a discrete index, called the band index, which is present because there are many different Bloch waves with the same k (each has a different periodic component u). Within a band (i.e., for fixed n), ψ_{n k} varies continuously with k, as does its energy. Also, for any reciprocal lattice vector K, ψ_{n k} = ψ_n,(k+K). Therefore, all distinct Bloch waves occur for k-values within the first Brillouin zone of the reciprocal lattice.

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