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- We remain with (en)
- All translations are unitary and abelian.
Translations can be written in terms of unit vectors
We can think of these as commuting operators
where
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 Schrödinger 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. (en)
- We evaluate the derivatives and
given they are the coefficients of the following expansion in where is considered small with respect to
Given are eigenvalues of
We can consider the following perturbation problem in q:
Perturbation theory of the second order states that
To compute to linear order in
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 to obtain
and we can reinsert the complete wave functions (en)
- The second order term
Again with
Eliminating and we have the theorem (en)
- Source:
We define the translation operator
with
We use the hypothesis 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 dividing 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
and where . Finally,
which is true for a Bloch wave i.e. for with (en)
- Source:
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 . If the crystal is shifted by any of these three vectors, or a combination of them of the form
where 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 , with the property that , but when .
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:
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 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 wave functions in this basis are energy eigenstates , and they are also Bloch states . (en)
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