These eigenstates are written with subscripts as , where is a discrete index, called the band index, which is present because there are many different wave functions with the same (each has a different periodic component ). Within a band (i.e., for fixed ), varies continuously with , as does its energy. Also, , is unique only up to a constant reciprocal lattice vector , or, . Therefore, the wave vector can be restricted to the first Brillouin zone of the reciprocal lattice without loss of generality.
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 k1 (left) or k2 (right). The difference (k1−k2) 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
where u is periodic with the same periodicity as the crystal lattice. The actual quantum state of the electron is entirely determined by , not k or u directly. This is important because k and u are not unique. Specifically, if can be written as above using k, it can also be written using (k + K), where K is any 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 k with the property that no two of them are equivalent, yet every possible k is equivalent to one (and only one) vector in the first Brillouin zone. Therefore, if we restrict k to the first Brillouin zone, then every Bloch state has a unique 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.
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 vectorsa1, a2, a3. If the crystal is shifted by any of these three vectors, or a combination of them of the form
where ni 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 units of inverse length), with the property that ai · bi = 2π, but ai · bj = 0 when i ≠ j. (For the formula for bi, see reciprocal lattice vector.)
Let denote a translation operator that shifts every wave function by the amount n1a1 + n2a2 + n3a3 (as above, nj are integers). The following fact is helpful for the proof of Bloch's theorem:
Lemma: If a wave function is an eigenstate of all of the translation operators (simultaneously), then is a Bloch state.
Proof: Assume that we have a wave function which is an eigenstate of all the translation operators. As a special case of this,
for j = 1, 2, 3, where Cj are three numbers (the eigenvalues) which do not depend on r. It is helpful to write the numbers Cj in a different form, by choosing three numbers θ1, θ2, θ3 with e2πiθj = Cj:
Again, the θj are three numbers which do not depend on r. Define k = θ1b1 + θ2b2 + θ3b3, where bj are the reciprocal lattice vectors (see above). Finally, define
This proves that u has the periodicity of the lattice. Since , 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 denote a translation operator that shifts every wave function by the amount n1a1 + n2a2 + n3a3, where ni 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 (because they are eigenstates of the Hamiltonian), and they are also Bloch states (because they are eigenstates of the translation operators; see Lemma above).
The commutativity of the 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.
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 (i.e. the translation group here) 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
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 (as the invariants of the irreducible representations) can be treated as the fundamental building blocks instead of the irreducible representations themselves.
Velocity and effective mass of Bloch electrons
Given this is defined in a finite volume we expect an infinite family of eigenvalues, here
is a parameter of the Hamiltonian and therefore we arrive to a "continuous family" of eigenvalues dependent on the continuous parameter and therefore to the basic concept of an electronic band structure
^Alexander Mihailovich Lyapunov (1992). The General Problem of the Stability of Motion. London: Taylor and Francis. Translated by A. T. Fuller from Edouard Davaux's French translation (1907) of the original Russian dissertation (1892).
M.S.P. Eastham (1973). The Spectral Theory of Periodic Differential Equations. Texts in Mathematics. Edinburgh: Scottish Academic Press.
J. Gazalet; S. Dupont; J.C. Kastelik; Q. Rolland & B. Djafari-Rouhani (2013). "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". Wave Motion. 50 (3): 619–654. doi:10.1016/j.wavemoti.2012.12.010.