Bloch's theorem

In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential can be expressed as plane waves modulated by periodic functions. The theorem is named after the Swiss physicist Felix Bloch, who discovered the theorem in 1929. Mathematically, they are written
where is position, is the wave function, is a periodic function with the same periodicity as the crystal, the wave vector is the crystal momentum vector, is Euler's number, and is the imaginary unit.
Functions of this form are known as Bloch functions or Bloch states, and serve as a suitable basis for the wave functions or states of electrons in crystalline solids.
The description of electrons in terms of Bloch functions, termed Bloch electrons, underlies the concept of electronic band structures.
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 . Within a band, 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.

Applications and consequences

Applicability

The most common example of Bloch's theorem is describing electrons in a crystal, especially in characterizing the crystal's electronic properties, such as electronic band structure. However, a Bloch-wave description applies more generally to any wave-like phenomenon in a periodic medium. For example, a periodic dielectric structure in electromagnetism leads to photonic crystals, and a periodic acoustic medium leads to phononic crystals. It is generally treated in the various forms of the dynamical theory of diffraction.

Wave vector

Suppose an electron is in a Bloch state
where is periodic with the same periodicity as the crystal lattice. The actual quantum state of the electron is entirely determined by, not or directly. This is important because and are not unique. Specifically, if can be written as above using, it can also be written using, where is any reciprocal lattice vector. 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 with the property that no two of them are equivalent, yet every possible is equivalent to one vector in the first Brillouin zone. Therefore, if we restrict to the first Brillouin zone, then every Bloch state has a unique. 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.
When is multiplied by the reduced Planck constant, it equals the electron's crystal momentum. Related to this, the group velocity of an electron can be calculated based on how the energy of a Bloch state varies with ; for more details see crystal momentum.

Detailed example

For a detailed example in which the consequences of Bloch's theorem are worked out in a specific situation, see the article Particle in a one-dimensional lattice.

Statement

A second and equivalent way to state the theorem is the following

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.

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.

Using operators

In this proof all the symmetries are encoded as commutation properties of the translation operators

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 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 it 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.

Velocity and effective mass

If we apply the time-independent Schrödinger equation to the Bloch wave function we obtain
with boundary conditions
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 at a "continuous family" of eigenvalues dependent on the continuous parameter and thus at the basic concept of an electronic band structure.
This shows how the effective momentum can be seen as composed of two parts,
a standard momentum and a crystal momentum. More precisely the crystal momentum is not a momentum but it stands for the momentum in the same way as the electromagnetic momentum in the minimal coupling, and as part of a canonical transformation of the momentum.
For the effective velocity we can derive
For the effective mass
The quantity on the right multiplied by a factor is called effective mass tensor and we can use it to write a semi-classical equation for a charge carrier in a band
where is an acceleration. This equation is analogous to the de Broglie wave type of approximation
As an intuitive interpretation, both of the previous two equations resemble formally and are in a semi-classical analogy with Newton's second law for an electron in an external Lorentz force.

Mathematical caveat

Mathematically, a rigorous theorem such as Bloch's theorem cannot exist in Quantum Mechanics: The spectral values of a band structure in a solid crystal or lattice system belong to the continuous spectrum, for which no finite norm eigenstates in the Hilbert space exist, i.e, no eigenstates with finite energy or finite probability can exist – cf. decomposition of spectrum –, because eigenvalues belong to the point spectrum by definition.
Therefore, all physicists' calculations in Bloch's theorem with eigenstate decompositions in a Hilbert space are in some sense purely formal: The decomposition series do not converge in Hilbert space, and no proper spatially periodic function can be a finite norm state in the full Hilbert space.
Decompositions of periodic continuous functions – similarly to Bloch – can possibly be performed in spaces of bounded or bounded continuous functions, but not in spaces of functions square integrable over full x-space, which would be the required Hilbert space setting for Quantum Mechanics.
In Mathematical Physics, as a substitute, different rigorous decompositions can be obtained which also provide the band structure, by exploiting lattice symmetry based on a Hilbert space direct integral decomposition. By that method, the Hamiltonian operator is decomposed into a parameter dependent family of so-called reduced Hamiltonian operators on a corresponding family of Hilbert spaces and with corresponding domains of definitions. Each of these Hamiltonians has a discrete point spectrum with finite eigenstates of finite multiplicity, corresponding to the physicist's eigenvalue computations.
Superposing these states with the direct integral would throw the states out of the original Hilbert space, but the spectra of these Hamiltonians combine into the continuous band spectrum of the original Hamiltonian.

History and related equations

The concept of the Bloch state was developed by Felix Bloch in 1928 to describe the conduction of electrons in crystalline solids. The same underlying mathematics, however, was also discovered independently several times: by George William Hill, Gaston Floquet, and Alexander Lyapunov. As a result, a variety of nomenclatures are common: applied to ordinary differential equations, it is called Floquet theory. The general form of a one-dimensional periodic potential equation is Hill's equation:
where is a periodic potential. Specific periodic one-dimensional equations include the Kronig–Penney model and Mathieu's equation.
Mathematically, various theorems similar to Bloch's theorem are for instance interpreted in terms of unitary characters of a lattice group, and applied to spectral geometry.
Floquet theory is usually not done in a Hilbert space of functions square integrable with respect to the periodic independent variable, but in Banach spaces of continuous or differentiable functions, or in Frechet or nuclear spaces. So the methods used there do not directly apply to the Hilbert space setting required in Quantum Mechanics and require proper adaptation, such as using a Hilbert space direct integral.