Editing Bloch's theorem (section)

{{short description|Fundamental theorem in condensed matter physics}}
{{about|a theorem in quantum mechanics|the theorem used in complex analysis|Bloch's theorem (complex variables)}}
[[Image:BlochWave in Silicon.png|thumb|upright=1.2|[[Isosurface]] of the [[square modulus]] of a Bloch state in a silicon lattice]]
[[File:Bloch_function.svg|thumb|upright=1.7|Solid line: A schematic of the real part of a typical Bloch state in one dimension. The dotted line is from the factor {{math|''e''<sup>''i'''''k'''·'''r'''</sup>}}. The light circles represent atoms.]]

In [[condensed matter physics]], '''Bloch's theorem''' states that solutions to the [[Schrödinger equation#Time-independent equation|Schrödinger equation]] in a periodic potential can be expressed as [[plane wave]]s modulated by [[periodic function]]s. The theorem is named after the Swiss physicist [[Felix Bloch]], who discovered the theorem in 1929.<ref>Bloch, F. (1929). Über die quantenmechanik der elektronen in kristallgittern. Zeitschrift für physik, 52(7), 555-600.</ref> Mathematically, they are written<ref>{{cite book|last1= Kittel|author-link=Charles Kittel |title=[[Introduction to Solid State Physics]]|publisher=Wiley|location= New York|year=1996| first1=Charles|isbn= 0-471-14286-7}}</ref>
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|title='''Bloch function'''
|equation=<math>\psi(\mathbf{r}) = e^{i \mathbf{k}\cdot\mathbf{r}} u(\mathbf{r})</math>
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where <math>\mathbf{r}</math> is position, <math>\psi</math> is the [[wave function]], <math>u</math> is a [[periodic function]] with the same periodicity as the crystal, the [[wave vector]] <math>\mathbf{k}</math> is the [[crystal momentum|crystal momentum vector]], <math>e</math> is [[E (mathematical constant)|Euler's number]], and <math>i</math> is the [[imaginary unit]].

Functions of this form are known as '''Bloch functions''' or '''Bloch states''', and serve as a suitable [[Basis function|basis]] for the [[wave functions]] or [[quantum states|states]] of electrons in [[Crystal|crystalline solids]].

The description of electrons in terms of Bloch functions, termed '''Bloch electrons''' (or less often ''Bloch Waves''),  underlies the concept of [[electronic band structure]]s.

These eigenstates are written with subscripts as <math>\psi_{n\mathbf{k}}</math>, where <math>n</math> is a discrete index, called the [[energy band|band index]], which is present because there are many different wave functions with the same <math>\mathbf{k}</math> (each has a different periodic component <math>u</math>). Within a band (i.e., for fixed <math>n</math>), <math>\psi_{n\mathbf{k}}</math> varies continuously with <math>\mathbf{k}</math>, as does its energy. Also, <math>\psi_{n\mathbf{k}}</math> is unique only up to a constant [[reciprocal lattice]] vector <math>\mathbf{K}</math>, or, <math>\psi_{n\mathbf{k}}=\psi_{n(\mathbf{k+K})}</math>. Therefore, the wave vector <math>\mathbf{k}</math> can be restricted to the first [[Brillouin zone]] of the reciprocal lattice [[without loss of generality]].