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.

can be restricted to the first Brillouin zone of the reciprocal lattice without loss of generality.

However, a Bloch-wave description applies more generally to any wave-like phenomenon in a periodic medium.

It is generally treated in the various forms of the dynamical theory of diffraction.

The actual quantum state of the electron is entirely determined by

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.

When k 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 k; for more details see crystal momentum.

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 (periodic potential).

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.

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 ψ is an eigenstate of all of the translation operators (simultaneously), then ψ is a Bloch state.

Assume that we have a wave function ψ which is an eigenstate of all the translation operators.

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).

denote a translation operator that shifts every wave function by the amount n1a1 + n2a2 + n3a3, where ni are integers.

if we use the normalization condition over a single primitive cell of volume V

[6]: 365–367 [7] 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.

operators gives three commuting cyclic subgroups (given they can be generated by only one element) which are infinite, 1-dimensional and abelian.

This substituting in the time independent Schrödinger equation with a simple effective Hamiltonian

In this context, the wave vector serves as a quantum number for the translation operator.

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.

[9] If we apply the time-independent Schrödinger equation to the Bloch wave function we obtain

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

This equation is analogous to the de Broglie wave type of approximation[14]

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.

The same underlying mathematics, however, was also discovered independently several times: by George William Hill (1877),[16] Gaston Floquet (1883),[17] and Alexander Lyapunov (1892).

[18] As a result, a variety of nomenclatures are common: applied to ordinary differential equations, it is called Floquet theory (or occasionally the Lyapunov–Floquet theorem).

Mathematically Bloch's theorem is interpreted in terms of unitary characters of a lattice group, and is applied to spectral geometry.