Bloch wave

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 three-dimensional crystal has three primitive lattice vectors a₁, a₂, a₃. If the crystal is shifted by any of these three vectors, or a combination of them of the form
where n_i 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 vectors. These are three vectors b₁, b₂, b₃, with the property that a_i · b_i = 2π, but a_i · b_j = 0 when i ≠ j.

Lemma about translation operators

Let denote a translation operator that shifts every wave function by the amount . The following fact is helpful for the proof of Bloch's theorem:
Proof: Assume that we have a wavefunction which is an eigenstate of all the translation operators. As a special case of this,
for j = 1, 2, 3, where C_j are three numbers which do not depend on r. It is helpful to write the numbers C_j in a different form, by choosing three numbers θ₁, θ₂, θ₃ with :
Again, the θ_j are three numbers which do not depend on r. Define, where b_j are the reciprocal lattice vectors. Finally, define
Then
This proves that u has the periodicity of the lattice. Since, that proves that the state is a Bloch wave.

Proof

Finally, we are ready for the main proof of Bloch's theorem which is as follows.
As above, let denote a translation operator that shifts every wave function by the amount, where n_i 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 simultaneous eigenbasis of the Hamiltonian operator and every possible operator. This basis is what we are looking for. The wavefunctions in this basis are energy eigenstates, and they are also Bloch waves.

We define the translation operator
We use the hyphotesis of a mean periodic potential
and the independent electron approximation with an hamiltonian
Given the Hamiltonian is invariant for translations it shall commute with the translation operator
and the two operators shall have a common set of eigenfunctions.
Therefore we start to look at the eigen-functions of the translation operator:
Given is an additive operator
If we substitute here the eigenvalue equation and diving both sides for we have
This is true for
where
if we use the normalization condition over a single primitive cell of volume V
and therefore
Finally
Which is true for a block wave i.e for
with

All Translations are unitary and Abelian.
Translations can be written in terms of unit vectors
We can think of these as commuting operators
The commutativity of the operators gives
three commuting cyclic subgroups which are infinite, 1-dimensional and abelian. All irreducible representations of Abelian groups are one dimensional
Given they are one dimensional the matrix representation and the character are the same. The character is the representation over the complex numbers of the group or also the trace of the 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 which shall obey to, and therefore the character. Note that this is straightforward in the finite cyclic group case but in the countable infinite case of the infinite cyclic group there is a limit for where the character remains finite.
Given the character is a root of unity, for each subgroup the character can be then written as
If we introduce the Born–von Karman boundary condition on the potential:
Where L is a macroscopic periodicity in the direction
that can also be seen as a multiple of where
This substituting in the time independent schroedinger equation with a simple effective Hamiltonian
induces a periodicity with the wave function:
And for each dimension a translation operator with a period L
From here we can see that also the character shall be invariant by a translation of :
and from the last equation we get for each dimension a periodic condition:
where is an integer and
The wave vector identify the irreducible representation in the same manner as,
and is a macroscopic periodic length of the crystal in direction. In this context, the wave vector serves as a quantum number for the translation operator.
We can generalize this for 3 dimensions
and the generic formula for the wave function becomes:
i.e. specializing it for a translation
and we have proven Bloch’s theorem.
A part 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 groups 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 basis.
In this proof it is also possible to notice how is key that the extra point group is driven by a symmetry in the effective potential but it shall commute with the Hamiltonian.
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 character expansion of the wave function where the characters are given from the specific finite point group.
Also here is possible to see how the characters can be treated as the fundamental building blocks instead of the irreducible representations themselves.

We remain with

We evaluate the derivatives and
given they are the coefficients of the following expansion in q where q is considered small with respect to k
Given are eigenvalues of
We can consider the following perturbation problem in q:
Perturbation theory of the second order tells that:
To compute to linear order in q
Where the integrations are over a primitive cell or the entire crystal, given if the integral:
is normalized across the cell or the crystal.
We can simplify over q and remain with
And we can reinsert the complete wave functions

The second order term
Again with
And getting rid of and we have the theorem