Editing Bloch's theorem (section)

=== 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>