Editing Bloch's theorem (section)

== 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>