Editing Bloch's theorem (section)

== Applications and consequences ==

=== Applicability ===
The most common example of Bloch's theorem is describing electrons in a crystal, especially in characterizing the crystal's electronic properties, such as electronic band structure. However, a Bloch-wave description applies more generally to any wave-like phenomenon in a periodic medium. For example, a periodic [[dielectric]] structure in [[electromagnetism]] leads to [[photonic crystal]]s, and a periodic acoustic medium leads to [[phononic crystal]]s. It is generally treated in the various forms of the [[dynamical theory of diffraction]].

=== Wave vector ===
[[File:BlochWaves1D.svg|thumb|upright=1.75|A Bloch wave function (bottom) can be broken up into the product of a periodic function (top) and a plane-wave (center). The left side and right side represent the same Bloch state broken up in two different ways, involving the wave vector {{math|''k''<sub>1</sub>}} (left) or {{math|''k''<sub>2</sub>}} (right). The difference ({{math|''k''<sub>1</sub> − ''k''<sub>2</sub>}}) is a [[reciprocal lattice]] vector. In all plots, blue is real part and red is imaginary part.]]

Suppose an electron is in a Bloch state
<math display="block">\psi ( \mathbf{r} ) = e^{ i \mathbf{k} \cdot \mathbf{r} } u ( \mathbf{r} ) ,</math>
where {{math|''u''}} is periodic with the same periodicity as the crystal lattice. The actual quantum state of the electron is entirely determined by <math>\psi</math>, not {{math|'''k'''}} or {{math|''u''}} directly. This is important because {{math|'''k'''}} and {{math|''u''}} are ''not'' unique. Specifically, if <math>\psi</math> can be written as above using {{math|'''k'''}}, it can ''also'' be written using {{math|('''k''' + '''K''')}}, where {{math|'''K'''}} is any [[reciprocal lattice|reciprocal lattice vector]] (see figure at right). Therefore, wave vectors that differ by a reciprocal lattice vector are equivalent, in the sense that they characterize the same set of Bloch states.

The [[first Brillouin zone]] is a restricted set of values of {{math|'''k'''}} with the property that no two of them are equivalent, yet every possible {{math|'''k'''}} is equivalent to one (and only one) vector in the first Brillouin zone. Therefore, if we restrict {{math|'''k'''}} to the first Brillouin zone, then every Bloch state has a unique {{math|'''k'''}}. Therefore, the first Brillouin zone is often used to depict all of the Bloch states without redundancy, for example in a band structure, and it is used for the same reason in many calculations.

When {{math|'''k'''}} is multiplied by the [[reduced Planck constant]], it equals the electron's [[crystal momentum]]. Related to this, the [[group velocity]] of an electron can be calculated based on how the energy of a Bloch state varies with {{math|'''k'''}}; for more details see crystal momentum.

=== Detailed example ===
For a detailed example in which the consequences of Bloch's theorem are worked out in a specific situation, see the article [[Particle in a one-dimensional lattice (periodic potential)]].