View source for Bloch's theorem

{{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>
{{Equation box 1
|indent=:
|title='''Bloch function'''
|equation=<math>\psi(\mathbf{r}) = e^{i \mathbf{k}\cdot\mathbf{r}} u(\mathbf{r})</math>
|cellpadding
|border
|border colour = rgb(80,200,120)
|background colour = rgb(80,200,120,10%)}}
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]].

== 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)]].

== Statement ==
{{math theorem | name = Bloch's theorem | math_statement = 
For electrons in a perfect crystal, there is a [[basis (linear algebra)|basis]] of wave functions with the following two properties:
* each of these wave functions is an energy eigenstate,
* each of these wave functions is a Bloch state, meaning that this wave function <math>\psi</math> can be written in the form <math>\;\psi(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} u(\mathbf{r}),</math> where <math>u(\mathbf{r})</math> has the same periodicity as the atomic structure of the crystal, such that <math display="block">u_{\mathbf{k}}(\mathbf{x}) = u_{\mathbf{k}}(\mathbf{x} + \mathbf{n} \cdot \mathbf{a}).</math>
}}

A second and equivalent way to state the theorem is the following<ref name="ziman:1">{{cite book |last=Ziman  |first=J. M. |date=1972 |edition=2nd |title=Principles of the theory of solids |publisher=Cambridge University Press |isbn=0521297338 |pages=17–20}}</ref>

{{math theorem | name = Bloch's theorem | math_statement = 
For any wave function that satisfies the Schrödinger equation and for a translation of a lattice vector <math>\mathbf{a}</math>, there exists at least one vector <math>\mathbf{k}</math> such that: 
<math display="block">\psi_{\mathbf{k}}(\mathbf{x}+\mathbf{a}) = e^{i\mathbf{k}\cdot\mathbf{a}}\psi_{\mathbf{k}}(\mathbf{x}).</math>
}}

== Proof ==

=== Using lattice periodicity ===
Bloch's theorem, being a statement about lattice periodicity, all the symmetries in this proof are encoded as translation symmetries of the wave function itself.
{{math proof | title = Proof Using lattice periodicity | proof = 
Source:<ref name=":3">{{Harvnb|Ashcroft|Mermin|1976|p=134}}</ref>

==== Preliminaries: Crystal symmetries, lattice, and reciprocal lattice ====
The defining property of a crystal is translational symmetry, which means that if the crystal is shifted an appropriate amount, it winds up with all its atoms in the same places. (A finite-size crystal cannot have perfect translational symmetry, but it is a useful approximation.)

A three-dimensional crystal has three ''primitive lattice vectors'' {{math|'''a'''<sub>1</sub>, '''a'''<sub>2</sub>, '''a'''<sub>3</sub>}}. If the crystal is shifted by any of these three vectors, or a combination of them of the form
<math display="block">n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3,</math>
where {{mvar|n<sub>i</sub>}} are three integers, then the atoms end up in the same set of locations as they started.

Another helpful ingredient in the proof is the ''[[reciprocal lattice vector]]s''. These are three vectors {{math|'''b'''<sub>1</sub>, '''b'''<sub>2</sub>, '''b'''<sub>3</sub>}} (with units of inverse length), with the property that {{math|1='''a'''<sub>''i''</sub> · '''b'''<sub>''i''</sub> = 2''π''}}, but {{math|1='''a'''<sub>''i''</sub> · '''b'''<sub>''j''</sub> = 0}} when {{math|''i'' ≠ ''j''}}. (For the formula for {{math|'''b'''<sub>''i''</sub>}}, see [[reciprocal lattice vector]].)

==== Lemma about translation operators ====
Let <math> \hat{T}_{n_1,n_2,n_3} </math> denote a [[Translation operator (quantum mechanics)|translation operator]] that shifts every wave function by the amount {{math|''n''<sub>1</sub>'''a'''<sub>1</sub> + ''n''<sub>2</sub>'''a'''<sub>2</sub> + ''n''<sub>3</sub>'''a'''<sub>3</sub>}} (as above, {{mvar|n<sub>j</sub>}} are integers). The following fact is helpful for the proof of Bloch's theorem:
{{math theorem | name = Lemma | math_statement = If a wave function {{mvar|&psi;}} is an [[eigenfunction|eigenstate]] of all of the translation operators (simultaneously), then {{mvar|&psi;}} is a Bloch state.}}
{{math proof | title = Proof of Lemma | proof = Assume that we have a wave function {{mvar|&psi;}} which is an eigenstate of all the translation operators. As a special case of this, 
<math display="block">\psi(\mathbf{r}+\mathbf{a}_j) = C_j \psi(\mathbf{r})</math> 
for {{math|1=''j'' = 1, 2, 3}}, where {{mvar|C<sub>j</sub>}} are three numbers (the [[eigenvalue]]s) which do not depend on {{math|'''r'''}}. It is helpful to write the numbers {{mvar|C<sub>j</sub>}} in a different form, by choosing three numbers {{math|''θ''<sub>1</sub>, ''θ''<sub>2</sub>, ''θ''<sub>3</sub>}} with {{math|1=''e''<sup>2''πiθ''<sub>''j''</sub></sup> = ''C''<sub>''j''</sub>}}:
<math display="block">\psi(\mathbf{r}+\mathbf{a}_j) = e^{2 \pi i \theta_j} \psi(\mathbf{r})</math> 
Again, the {{mvar|θ<sub>j</sub>}} are three numbers which do not depend on {{math|'''r'''}}. Define {{math|1='''k''' = ''θ''<sub>1</sub>'''b'''<sub>1</sub> + ''θ''<sub>2</sub>'''b'''<sub>2</sub> + ''θ''<sub>3</sub>'''b'''<sub>3</sub>}}, where {{math|'''b'''<sub>''j''</sub>}} are the reciprocal lattice vectors (see above). Finally, define 
<math display="block">u(\mathbf{r}) = e^{-i \mathbf{k}\cdot\mathbf{r}} \psi(\mathbf{r})\,.</math>
Then
<math display="block">\begin{align}
u(\mathbf{r} + \mathbf{a}_j) &= e^{-i\mathbf{k} \cdot (\mathbf{r} + \mathbf{a}_j)} \psi(\mathbf{r}+\mathbf{a}_j) \\
&= \big( e^{-i\mathbf{k} \cdot \mathbf{r}} e^{-i\mathbf{k}\cdot \mathbf{a}_j} \big) \big( e^{2\pi i \theta_j} \psi(\mathbf{r}) \big) \\
&= e^{-i\mathbf{k} \cdot \mathbf{r}} e^{-2\pi i \theta_j} e^{2\pi i \theta_j} \psi(\mathbf{r}) \\
&= u(\mathbf{r}).
\end{align}</math>
This proves that {{mvar|u}} has the periodicity of the lattice. Since <math>\psi(\mathbf{r}) = e^{i \mathbf{k}\cdot\mathbf{r}} u(\mathbf{r}),</math> that proves that the state is a Bloch state.}}
Finally, we are ready for the main proof of Bloch's theorem which is as follows.

As above, let <math> \hat{T}_{n_1,n_2,n_3} </math> denote a ''translation operator'' that shifts every wave function by the amount {{math|''n''<sub>1</sub>'''a'''<sub>1</sub> + ''n''<sub>2</sub>'''a'''<sub>2</sub> + ''n''<sub>3</sub>'''a'''<sub>3</sub>}}, where {{mvar|n<sub>i</sub>}} are integers. Because the crystal has translational symmetry, this operator commutes with the [[Hamiltonian operator]]. Moreover, every such translation operator commutes with every other. Therefore, there is a [[Commuting matrices|simultaneous eigenbasis]] of the Hamiltonian operator and every possible <math> \hat{T}_{n_1,n_2,n_3} \!</math> operator. This basis is what we are looking for. The wave functions in this basis are energy eigenstates (because they are eigenstates of the Hamiltonian), and they are also Bloch states (because they are eigenstates of the translation operators; see Lemma above).
}}

=== Using operators ===
In this proof all the symmetries are encoded as commutation properties of the translation operators
{{math proof | title = Proof using operators | proof =
Source:<ref name=":4">{{Harvnb|Ashcroft|Mermin|1976|p=137}}</ref>

We define the translation operator
<math display="block">\begin{align}
\hat{\mathbf{T}}_{\mathbf{n}}\psi(\mathbf{r})&= \psi(\mathbf{r}+\mathbf{T}_{\mathbf{n}}) \\
&= \psi(\mathbf{r}+n_1\mathbf{a}_1+n_2\mathbf{a}_2+n_3\mathbf{a}_3) \\
&= \psi(\mathbf{r}+\mathbf{A}\mathbf{n})
\end{align}</math>
with
<math display="block">
\mathbf{A} = \begin{bmatrix} \mathbf{a}_1 & \mathbf{a}_2 & \mathbf{a}_3 \end{bmatrix}, \quad \mathbf{n} = \begin{pmatrix} n_1 \\ n_2 \\ n_3 \end{pmatrix}
</math>
We use the hypothesis of a mean periodic potential
<math display="block">U(\mathbf{x}+\mathbf{T}_{\mathbf{n}})= U(\mathbf{x})</math>
and the [[independent electron approximation]] with an Hamiltonian
<math display="block">\hat{H}=\frac{\hat{\mathbf{p}}^2}{2m}+U(\mathbf{x})</math>
Given the Hamiltonian is invariant for translations it shall commute with the translation operator
<math display="block">[\hat{H},\hat{\mathbf{T}}_{\mathbf{n}}] = 0</math>
and the two operators shall have a common set of eigenfunctions.
Therefore, we start to look at the eigen-functions of the translation operator:
<math display="block">\hat{\mathbf{T}}_{\mathbf{n}}\psi(\mathbf{x})=\lambda_{\mathbf{n}}\psi(\mathbf{x})</math>
Given <math>\hat{\mathbf{T}}_{\mathbf{n}}</math> is an additive operator 
<math display="block">
\hat{\mathbf{T}}_{\mathbf{n}_1} \hat{\mathbf{T}}_{\mathbf{n}_2}\psi(\mathbf{x}) =
\psi(\mathbf{x} + \mathbf{A} \mathbf{n}_1 + \mathbf{A} \mathbf{n}_2) = \hat{\mathbf{T}}_{\mathbf{n}_1 + \mathbf{n}_2} \psi(\mathbf{x})
</math>
If we substitute here the eigenvalue equation and dividing both sides for <math>\psi(\mathbf{x})</math> we have
<math display="block">
\lambda_{\mathbf{n}_1} \lambda_{\mathbf{n}_2} =
\lambda_{\mathbf{n}_1 + \mathbf{n}_2}
</math>

This is true for 
<math display="block">\lambda_{\mathbf{n}} = e^{s \mathbf{n} \cdot \mathbf{a} } </math>
where <math>s \in \Complex </math> if we use the normalization condition over a single primitive cell of volume V
<math display="block">
1 = \int_V |\psi(\mathbf{x})|^2 d \mathbf{x} = 
\int_V \left|\hat\mathbf{T}_\mathbf{n} \psi(\mathbf{x})\right|^2 d \mathbf{x} = 
|\lambda_{\mathbf{n}}|^2 \int_V |\psi(\mathbf{x})|^2 d \mathbf{x}
</math>
and therefore
<math display="block">1 = |\lambda_{\mathbf{n}}|^2</math> and <math display="block">s = i k </math> where <math>k \in \mathbb{R}</math>. Finally,
<math display="block">
\mathbf{\hat{T}_n}\psi(\mathbf{x})=
\psi(\mathbf{x} + \mathbf{n} \cdot \mathbf{a} ) = 
e^{i k \mathbf{n} \cdot \mathbf{a} }\psi(\mathbf{x})
,</math>
which is true for a Bloch wave i.e. for <math>\psi_{\mathbf{k}}(\mathbf{x}) = e^{i \mathbf{k} \cdot \mathbf{x} } u_{\mathbf{k}}(\mathbf{x})</math> with <math>u_{\mathbf{k}}(\mathbf{x}) = u_{\mathbf{k}}(\mathbf{x} + \mathbf{A}\mathbf{n})</math>
}}

=== Using group theory ===
Apart from the group theory technicalities this proof is interesting because it becomes clear how to generalize the Bloch theorem for groups that are not only translations.
This is typically done for [[space group]]s which are a combination of a [[translation]] and a [[point group]] and it is used for computing the band structure, spectrum and specific heats of crystals given a specific crystal group symmetry like FCC or BCC and eventually an extra [[Bravais lattice|basis]].<ref name="Dresselhaus2002"/>{{rp|pp=365–367}}<ref>The vibrational spectrum and specific heat of a face centered cubic crystal, Robert B. Leighton [https://authors.library.caltech.edu/47755/1/LEIrmp48.pdf]</ref>
In this proof it is also possible to notice how it is key that the extra point group is driven by a symmetry in the effective potential but it shall commute with the Hamiltonian.
{{math proof
| title = Proof with character theory<ref name="Dresselhaus2002">{{Cite web|last=Dresselhaus|first=M. S. | author-link=Mildred Dresselhaus |date=2002|title=Applications of Group Theory to the Physics of Solids|url=http://web.mit.edu/course/6/6.734j/www/group-full02.pdf | url-status=live | archive-url=https://web.archive.org/web/20191101074639/http://web.mit.edu/course/6/6.734j/www/group-full02.pdf | archive-date=1 November 2019|access-date=12 September 2020 | website=MIT}}</ref>{{rp|pp=345–348}}
| proof =
All [[Translation operator (quantum mechanics)|translations]] are [[Unitary operator|unitary]] and [[Abelian group|abelian]].
Translations can be written in terms of unit vectors
<math display="block">\boldsymbol{\tau} = \sum_{i=1}^3 n_i \mathbf{a}_i</math>
We can think of these as commuting operators
<math display="block">
\hat{\boldsymbol{\tau}} =
\hat{\boldsymbol{\tau}}_1 \hat{\boldsymbol{\tau}}_2 \hat{\boldsymbol{\tau}}_3
</math> where <math display="block">\hat{\boldsymbol{\tau}}_i = n_i \hat{\mathbf{a}}_i</math>

The commutativity of the <math>\hat{\boldsymbol{\tau}}_i</math> operators gives three commuting cyclic subgroups (given they can be generated by only one element) which are infinite, 1-dimensional and abelian. All irreducible representations of abelian groups are one dimensional.<ref>{{cite web |last=Roy |first=Ricky |title=Representation Theory |date=May 2, 2010 |url=http://buzzard.pugetsound.edu/courses/2010spring/projects/roy-representation-theory-ups-434-2010.pdf |publisher=University of Puget Sound}}</ref>

Given they are one dimensional the matrix representation and the [[Character (mathematics)#Character of a representation|character]] are the same. The character is the representation over the complex numbers of the group or also the [[Trace (matrix)|trace]] of the [[Group representation|representation]] which in this case is a one dimensional matrix. 
All these subgroups, given they are cyclic, they have characters which are appropriate [[roots of unity]]. In fact they have one generator <math>\gamma</math> which shall obey to <math>\gamma^n = 1</math>, and therefore the character <math>\chi(\gamma)^n = 1</math>. Note that this is straightforward in the finite cyclic group case but in the countable infinite case of the infinite [[cyclic group]] (i.e. the translation group here) there is a limit for <math>n \to \infty</math> where the character remains finite.

Given the character is a root of unity, for each subgroup the character can be then written as
<math display="block">\chi_{k_1}(\hat{\boldsymbol{\tau}}_1 (n_1,a_1)) = e^{i k_1 n_1 a_1}</math>

If we introduce the [[periodic boundary condition|Born–von Karman boundary condition]] on the potential:
<math display="block">V \left(\mathbf {r} +\sum_i N_{i} \mathbf {a}_{i}\right) = V (\mathbf {r} +\mathbf{L}) = V (\mathbf {r} )</math>
where ''L'' is a macroscopic periodicity in the direction <math>\mathbf{a}</math> that can also be seen as a multiple of <math>a_i</math> where <math display="inline">\mathbf{L} = \sum_i N_{i}\mathbf {a}_{i}</math>

This substituting in the time independent [[Schrödinger equation]] with a simple effective Hamiltonian
<math display="block">\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})</math>
induces a periodicity with the wave function:
<math display="block">\psi \left(\mathbf {r} + \sum_i N_{i}\mathbf {a}_{i}\right) = \psi (\mathbf {r} )</math>

And for each dimension a translation operator with a period ''L''
<math display="block">\hat{P}_{\varepsilon|\tau_i + L_i} = \hat{P}_{\varepsilon|\tau_i}</math>

From here we can see that also the character shall be invariant by a translation of <math>L_i</math>:
<math display="block">e^{i k_1 n_1 a_1} = e^{i k_1 ( n_1 a_1 + L_1)}</math>
and from the last equation we get for each dimension a periodic condition:
<math display="block"> k_1 n_1 a_1 = k_1 ( n_1 a_1 + L_1) - 2 \pi m_1</math>
where <math>m_1 \in \mathbb{Z}</math> is an integer and <math>k_1=\frac {2 \pi m_1}{L_1}</math>

The wave vector <math>k_1</math> identify the irreducible representation in the same manner as <math>m_1</math>, and <math>L_1</math> is a macroscopic periodic length of the crystal in direction <math>a_1</math>. In this context, the wave vector serves as a quantum number for the translation operator.

We can generalize this for 3 dimensions
<math>\chi_{k_1}(n_1,a_1)\chi_{k_2}(n_2,a_2)\chi_{k_3}(n_3,a_3) = e^{i\mathbf{k} \cdot \boldsymbol{\tau}}</math>
and the generic formula for the wave function becomes:
<math display="block">\hat{P}_R\psi_j = \sum_{\alpha} \psi_{\alpha} \chi_{\alpha j}(R)</math>
i.e. specializing it for a translation
<math display="block">\hat{P}_{\varepsilon|\boldsymbol{\tau}} \psi(\mathbf{r}) =\psi(\mathbf{r}) e^{i \mathbf{k} \cdot \boldsymbol{\tau}} = \psi(\mathbf{r} + \boldsymbol{\tau})</math>
and we have proven Bloch’s theorem.
}}

In the generalized version of the Bloch theorem, the Fourier transform, i.e. the wave function expansion, gets generalized from a [[discrete Fourier transform]] which is applicable only for cyclic groups, and therefore translations, into a [[Fourier transform on finite groups|character expansion]] of the wave function where the [[Character theory|characters]] are given from the specific finite [[point group]].

Also here is possible to see how the [[Character theory|characters]] (as the invariants of the irreducible representations) can be treated as the fundamental building blocks instead of the irreducible representations themselves.<ref>Group Representations
and Harmonic Analysis from Euler to Langlands, Part II [https://web.archive.org/web/20190305032503/http://pdfs.semanticscholar.org/ce73/4a226c19a412148dadbc2094fb75a7a609a4.pdf]</ref>

== Velocity and effective mass ==
If we apply the time-independent [[Schrödinger equation]] to the Bloch wave function we obtain
<math display="block">\hat{H}_\mathbf{k} u_\mathbf{k}(\mathbf{r}) = 
\left[ \frac{\hbar^2}{2m} \left( -i \nabla + \mathbf{k} \right)^2 + U(\mathbf{r}) \right] u_\mathbf{k}(\mathbf{r}) =
\varepsilon_\mathbf{k} u_\mathbf{k}(\mathbf{r})
</math>
with boundary conditions
<math display="block">u_\mathbf{k}(\mathbf{r}) = u_\mathbf{k}(\mathbf{r} + \mathbf{R})</math>
Given this is defined in a finite volume we expect an infinite family of eigenvalues; here <math>{\mathbf{k}}</math> is a parameter of the Hamiltonian and therefore we arrive at a "continuous family" of eigenvalues <math>\varepsilon_n(\mathbf{k})</math> dependent on the continuous parameter <math>{\mathbf{k}}</math> and thus at the basic concept of an electronic band structure.

{{math proof
| title = Proof<ref name=":1">{{Harvnb|Ashcroft|Mermin|1976|p=140}}</ref>
| proof =
<math display="block">
E_\mathbf{k} \left(e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x})\right) 
= 
\left[\frac{- \hbar^2}{2m} \nabla^2 + U(\mathbf{x} ) \right]
\left(e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x})\right)
</math>

We remain with
<math display="block">\begin{align}
E_\mathbf{k} e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x})
&=
\frac{- \hbar^2}{2m} \nabla \cdot 
\left( i \mathbf{k} e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x}) 
+ e^{i \mathbf{k} \cdot \mathbf{x} } \nabla u_\mathbf{k}(\mathbf{x}) \right) 
+ U(\mathbf{x}) e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x}) \\[1.2ex]

E_\mathbf{k} e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x})
&=
\frac{- \hbar^2}{2m} 
\left( i \mathbf{k} \cdot 
\left( i \mathbf{k} e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x}) 
+ e^{i \mathbf{k} \cdot \mathbf{x} } \nabla u_\mathbf{k}(\mathbf{x}) \right)
+ i \mathbf{k} \cdot e^{i \mathbf{k} \cdot \mathbf{x} } \nabla u_\mathbf{k}(\mathbf{x}) 
+ e^{i \mathbf{k} \cdot \mathbf{x} } \nabla^2 u_\mathbf{k}(\mathbf{x}) \right) 
+ U(\mathbf{x}) e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x}) \\[1.2ex]

E_\mathbf{k} e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x})
&=
\frac{ \hbar^2}{2m} 
\left(\mathbf{k}^2 e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x}) 
- 2i \mathbf{k} \cdot e^{i \mathbf{k} \cdot \mathbf{x} } \nabla u_\mathbf{k}(\mathbf{x}) 
- e^{i \mathbf{k} \cdot \mathbf{x} } \nabla^2 u_\mathbf{k}(\mathbf{x}) \right) 
+ U(\mathbf{x}) e^{i \mathbf{k} \cdot \mathbf{x} } u_\mathbf{k}(\mathbf{x}) \\[1.2ex]

E_\mathbf{k} u_\mathbf{k}(\mathbf{x})
&= 
\frac{ \hbar^2}{2m} 
\left(-i \nabla + \mathbf{k}\right)^2 u_\mathbf{k}(\mathbf{x}) 
+ U(\mathbf{x}) u_\mathbf{k}(\mathbf{x})
\end{align}</math>
}}

This shows how the effective momentum can be seen as composed of two parts,
<math display="block">\hat{\mathbf{p}}_\text{eff} =  -i \hbar \nabla + \hbar \mathbf{k} ,</math>
a standard momentum <math>-i \hbar \nabla</math> and a crystal momentum <math>\hbar \mathbf{k}</math>. More precisely the crystal momentum is not a momentum but it stands for the momentum in the same way as the electromagnetic momentum in the [[minimal coupling]], and as part of a [[canonical transformation]] of the momentum.

For the effective velocity we can derive
{{Equation box 1
|indent=:
|title='''mean velocity of a Bloch electron'''
|equation=<math>\frac{\partial \varepsilon_n}{\partial \mathbf{k}} = \frac {\hbar^2}{m} \int 
d\mathbf{r}\, \psi^{*}_{n\mathbf{k}} (-i \nabla)\psi_{n\mathbf{k}} = \frac {\hbar}{m}\langle\hat{\mathbf{p}}\rangle = \hbar \langle\hat{\mathbf{v}}\rangle</math>
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|background colour = rgb(80,200,120,10%)}}
{{math proof
| title = Proof<ref name=":2">{{Harvnb|Ashcroft|Mermin|1976|p=765 Appendix E}}</ref>
| proof =
We evaluate the derivatives <math>\frac{\partial \varepsilon_n}{\partial \mathbf{k}}</math> and <math>\frac{\partial^2 \varepsilon_n(\mathbf{k})}{\partial k_i \partial k_j}</math>
given they are the coefficients of the following expansion in {{math|'''q'''}} where {{math|'''q'''}} is considered small with respect to {{math|'''k'''}}
<math display="block">
\varepsilon_n(\mathbf{k} + \mathbf{q}) = \varepsilon_n(\mathbf{k}) + 
\sum_i \frac{\partial \varepsilon_n}{\partial k_i} q_i + 
\frac{1}{2} \sum_{ij} \frac{\partial^2 \varepsilon_n}{\partial k_i \partial k_j} q_i q_j +
O(q^3)
</math>
Given <math>\varepsilon_n(\mathbf{k}+\mathbf{q})</math> are eigenvalues of <math>\hat{H}_{\mathbf{k}+\mathbf{q}}</math>
We can consider the following perturbation problem in q:
<math display="block">
\hat{H}_{\mathbf{k}+\mathbf{q}} = 
\hat{H}_\mathbf{k} + 
\frac{\hbar^2}{m} \mathbf{q} \cdot ( -i\nabla + \mathbf{k} ) +
\frac{\hbar^2}{2m} q^2
</math>
Perturbation theory of the second order states that
<math display="block">
E_n =E^0_n + \int d\mathbf{r}\, \psi^{*}_n \hat{V} \psi_n + 
\sum_{n' \neq n} \frac{|\int d\mathbf{r} \,\psi^{*}_n \hat{V} \psi_n|^2}{E^0_n - E^0_{n'}} + ...
</math>
To compute to linear order in {{math|'''q'''}}
<math display="block">
\sum_i \frac{\partial \varepsilon_n}{\partial k_i} q_i = 
\sum_i \int d\mathbf{r}\, u_{n\mathbf{k}}^{*} \frac{\hbar^2}{m} ( -i\nabla + \mathbf{k} )_i q_i u_{n\mathbf{k}}
</math>
where the integrations are over a primitive cell or the entire crystal, given if the integral 
<math display="block">\int d\mathbf{r}\, u_{n\mathbf{k}}^{*} u_{n\mathbf{k}}</math>
is normalized across the cell or the crystal.

We can simplify over {{math|'''q'''}} to obtain
<math display="block">
\frac{\partial \varepsilon_n}{\partial \mathbf{k}} = 
\frac{\hbar^2}{m} \int d\mathbf{r} \, u_{n\mathbf{k}}^{*}( -i\nabla + \mathbf{k} ) u_{n\mathbf{k}}
</math>
and we can reinsert the complete wave functions
<math display="block">
\frac{\partial \varepsilon_n}{\partial \mathbf{k}} = 
\frac{\hbar^2}{m} \int d\mathbf{r} \, \psi_{n\mathbf{k}}^{*}( -i\nabla) \psi_{n\mathbf{k}}
</math>
}}

For the [[Effective mass (solid-state physics)|effective mass]]
{{Equation box 1
|indent=:
|title='''effective mass theorem'''
|equation=<math>
\frac{\partial^2 \varepsilon_n(\mathbf{k})}{\partial k_i \partial k_j} = 
\frac {\hbar^2}{m} \delta_{ij} + 
\left(
\frac {\hbar^2}{m}
\right)^2
\sum_{n' \neq n}
\frac{
\langle n\mathbf{k} | -i \nabla_i | n'\mathbf{k} \rangle 
\langle n'\mathbf{k} | -i \nabla_j | n\mathbf{k} \rangle 
+
\langle n\mathbf{k} | -i \nabla_j | n'\mathbf{k} \rangle 
\langle n'\mathbf{k} | -i \nabla_i | n\mathbf{k} \rangle
}{
\varepsilon_n(\mathbf{k}) - \varepsilon_{n'}(\mathbf{k})
}
</math>
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{{math proof
| title = Proof<ref name=":2"/>
| proof = 
The second order term
<math display="block">
\frac{1}{2} \sum_{ij} \frac{\partial^2 \varepsilon_n}{\partial k_i \partial k_j} q_i q_j =
\frac {\hbar^2}{2m} q^2 + 
\sum_{n' \neq n} 
\frac{|
\int d\mathbf{r} \,
u_{n\mathbf{k}}^{*} 
\frac{\hbar^2}{m} \mathbf{q} \cdot (-i\nabla + \mathbf{k}) 
u_{n'\mathbf{k}}
|^2} 
{\varepsilon_{n\mathbf{k}} - \varepsilon_{n'\mathbf{k}}}
</math>
Again with <math> \psi_{n\mathbf{k}} =| n\mathbf{k}\rangle = e^{i\mathbf{k}\mathbf{x}} u_{n\mathbf{k}}</math> 
<math display="block">
\frac{1}{2} \sum_{ij} \frac{\partial^2 \varepsilon_n}{\partial k_i \partial k_j} q_i q_j =
\frac {\hbar^2}{2m} q^2 + 
\sum_{n' \neq n} 
\frac{|
\langle n\mathbf{k} | 
\frac{\hbar^2}{m} \mathbf{q} \cdot (-i\nabla) 
| n'\mathbf{k}\rangle
|^2} 
{\varepsilon_{n\mathbf{k}} - \varepsilon_{n'\mathbf{k}}}
</math>
Eliminating <math>q_i</math> and <math>q_j</math> we have the theorem
<math display="block">
\frac{\partial^2 \varepsilon_n(\mathbf{k})}{\partial k_i \partial k_j} = 
\frac {\hbar^2}{m} \delta_{ij} + 
\left(
\frac {\hbar^2}{m}
\right)^2
\sum_{n' \neq n}
\frac{
\langle n\mathbf{k} | -i \nabla_i | n'\mathbf{k} \rangle 
\langle n'\mathbf{k} | -i \nabla_j | n\mathbf{k} \rangle 
+
\langle n\mathbf{k} | -i \nabla_j | n'\mathbf{k} \rangle 
\langle n'\mathbf{k} | -i \nabla_i | n\mathbf{k} \rangle
}{
\varepsilon_n(\mathbf{k}) - \varepsilon_{n'}(\mathbf{k})
}
</math>
}}
The quantity on the right multiplied by a factor<math>\frac{1}{\hbar^2}</math> is called effective mass tensor <math>\mathbf{M}(\mathbf{k})</math><ref name=":5">{{Harvnb|Ashcroft|Mermin|1976|p=228}}</ref> and we can use it to write a semi-classical equation for a [[charge carrier]] in a band<ref name=":6">{{Harvnb|Ashcroft|Mermin|1976|p=229}}</ref>
{{Equation box 1
|indent=:
|title='''Second order semi-classical equation of motion for a [[charge carrier]] in a band'''
|equation=<math>
\mathbf{M}(\mathbf{k}) \mathbf{a} = \mp e \left(\mathbf {E} + \mathbf{v}(\mathbf{k}) \times \mathbf{B}\right)
</math>
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where <math>\mathbf{a}</math> is an [[acceleration]]. This equation is analogous to the [[Matter wave|de Broglie wave]] type of approximation<ref name=":7">{{Harvnb|Ashcroft|Mermin|1976|p=227}}</ref>
{{Equation box 1
|indent=:
|title='''First order semi-classical equation of motion for electron in a band'''
|equation=<math>
\hbar \dot{k} = - e \left(\mathbf {E} +  \mathbf{v} \times \mathbf{B}\right)
</math>
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|background colour = rgb(80,200,120,10%)}}
As an intuitive interpretation, both of the previous two equations resemble formally and are in a semi-classical analogy with [[Newton's laws of motion#Newton's second law|Newton's second law]] for an electron in an external [[Lorentz force]].

== History and related equations ==

The concept of the Bloch state was developed by Felix Bloch in 1928<ref>{{cite journal|author=Felix Bloch|author-link=Felix Bloch|title=Über die Quantenmechanik der Elektronen in Kristallgittern|journal=Zeitschrift für Physik| volume=52 | issue=7–8| pages=555–600 |year=1928|doi=10.1007/BF01339455|bibcode = 1929ZPhy...52..555B |s2cid=120668259|language=de}}</ref> to describe the conduction of electrons in crystalline solids. The same underlying mathematics, however, was also discovered independently several times: by [[George William Hill]] (1877),<ref>{{cite journal|doi=10.1007/BF02417081| author=George William Hill|author-link=George William Hill|title=On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon|journal=Acta Math.|volume=8|pages=1–36 |year=1886|url=https://zenodo.org/record/1691491|doi-access=free}} This work was initially published and distributed privately in 1877.</ref> [[Gaston Floquet]] (1883),<ref>{{cite journal|author=Gaston Floquet|author-link=Gaston Floquet | title=Sur les équations différentielles linéaires à coefficients périodiques|journal= Annales Scientifiques de l'École Normale Supérieure|volume=12|pages=47–88 |year=1883|doi=10.24033/asens.220|doi-access=free}}</ref> and [[Alexander Lyapunov]] (1892).<ref>{{cite book|author=Alexander Mihailovich Lyapunov|author-link=Aleksandr Lyapunov|title=The General Problem of the Stability of Motion|location=London|publisher= Taylor and Francis|year= 1992}} Translated by A. T. Fuller from Edouard Davaux's French translation (1907) of the original Russian dissertation (1892).</ref> As a result, a variety of nomenclatures are common: applied to [[ordinary differential equations]], it is called [[Floquet theory]] (or occasionally the ''Lyapunov–Floquet theorem''). The general form of a one-dimensional periodic potential equation is [[Hill differential equation|Hill's equation]]:<ref name=Magnus_Winkler>
{{cite book 
|first1=W|last1= Magnus |author-link=Wilhelm Magnus|first2=S|last2= Winkler
|title=Hill's Equation
|year= 2004 
|page=11 
|publisher=Courier Dover
|isbn=0-486-49565-5
|url=https://books.google.com/books?id=ML5wm-T4RVQC&q=%22hill's+equation%22}}
</ref>
<math display="block">\frac {d^2y}{dt^2}+f(t) y=0, </math>
where {{math|''f''(''t'')}} is a periodic potential. Specific periodic one-dimensional equations include the [[Kronig–Penney model]] and [[Mathieu function|Mathieu's equation]].

Mathematically Bloch's theorem is interpreted in terms of unitary characters of a lattice group, and is applied to [[spectral geometry]].<ref>Kuchment, P.(1982), ''Floquet theory for partial differential equations'', RUSS MATH SURV., 37, 1–60</ref><ref>{{cite journal |author=Katsuda, A. |author2=Sunada, T |author2-link=Toshikazu Sunada |year=1987 |title=Homology and closed geodesics in a compact Riemann surface |journal=Amer. J. Math. |volume=110 |issue=1 |pages=145–156 |doi=10.2307/2374542| jstor=2374542 }}</ref><ref>{{cite journal |author=Kotani M |author2=Sunada T. |year=2000 |title=Albanese maps and an off diagonal long time asymptotic for the heat kernel |journal=Comm. Math. Phys. |volume=209 |issue=3 |pages=633–670 |doi=10.1007/s002200050033 | bibcode = 2000CMaPh.209..633K |s2cid=121065949 }}</ref>

== See also ==
{{colbegin}}
* [[Bloch oscillations]]
* [[Bloch wave – MoM method]]
* [[Electronic band structure]]
* [[Nearly free electron model]]
* [[Periodic boundary conditions]]
* [[Symmetries in quantum mechanics]]
* [[Tight-binding model]]
* [[Wannier function]]
{{colend}}

== References ==
{{reflist}}

== Further reading ==
{{Refbegin}}
* {{cite book|last1=Ashcroft|first1=Neil|author-link=Neil Ashcroft|last2=Mermin|first2=N. David|author-link2=N. David Mermin|title=Solid State Physics|publisher=Holt, Rinehart and Winston|location=New York|year=1976|isbn=978-0-03-083993-1|url=https://archive.org/details/solidstatephysic00ashc}}
* {{Cite book|last=Dresselhaus, M. S.|url=http://worldcat.org/oclc/692760083|title=Group theory: application to the physics of condensed matter|date=2010|publisher=Springer-Verlag|isbn=978-3-642-06945-1|oclc=692760083}}
* {{cite web|url=http://www.tf.uni-kiel.de/matwis/amat/semi_en/kap_2/backbone/r2_1_4.html |title=Periodic Potentials and Bloch's Theorem – lectures in "Semiconductors I" |publisher=The University of Kiel|author=H. Föll}}
* {{cite book|author=M.S.P. Eastham|title=The Spectral Theory of Periodic Differential Equations|location=Edinburgh|publisher= Scottish Academic Press|series= Texts in Mathematics|year= 1973}}
* {{cite journal|doi=10.1016/j.wavemoti.2012.12.010|author=J. Gazalet|author2=S. Dupont|author3=J.C. Kastelik|author4=Q. Rolland|author5=B. Djafari-Rouhani|name-list-style=amp|title=A tutorial survey on waves propagating in periodic media: Electronic, photonic and phononic crystals. Perception of the Bloch theorem in both real and Fourier domains|journal=Wave Motion|volume=50|issue=3|pages=619–654 |year=2013|bibcode=2013WaMot..50..619G |url=https://hal.archives-ouvertes.fr/hal-00808153 }}
{{Refend}}

{{Condensed matter physics topics}}
{{Authority control}}

[[Category:Eponymous theorems of physics]]
[[Category:Theorems in quantum mechanics]]
[[Category:Condensed matter physics]]