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In solid physics, Bloch wave is the wave function of particles (usually electrons) in periodic potential fields (such as crystals), also known as Bloch state.

Blochwave is named after its founder, Swiss-born American physicist felix bloch.

Bloch wave is obtained by multiplying plane wave by periodic function (Bloch wave packet). Which has that same periodicity as the potential field. The specific form of Bloch wave is:

Where k is the wave vector. The wave function expressed by the above formula is called Bloch function. When the potential field has lattice periodicity, the solution ψ of the wave equation satisfied by the particles in it exists:

This conclusion is called Bloch theorem, where is the lattice period vector. It can be seen that the wave function with the above properties can be written in the form of Bloch function.

Broadly speaking, Bloch wave can be used to describe any "wave-like phenomenon" in periodic media-such as electromagnetic phenomena in periodic dielectric (photonic crystal); Sound waves in periodic elastic media (phononic crystals), and so on.

Plane wave vector (also known as Bloch wave vector, whose product with reduced Planck constant is the crystal momentum of particles) represents the phase transition of electron wave function between different protocells, and its size only meets the one-to-one correspondence with wave function in a reciprocal lattice vector, so usually only the wave vector of the first Brillouin zone is considered. For a given wave vector and potential field distribution, the Schrodinger equation of electron motion has a series of solutions, called the energy band of electrons, which are usually distinguished by the subscript n of wave function. The energy of these energy bands has a finite gap on each single-valued boundary, which is called energy gap. The collection of all energy eigenstates in the first Brillouin zone constitutes the energy band structure of electrons. In the framework of single electron approximation, the macroscopic properties of electron motion in periodic potential field can be calculated according to energy band structure and corresponding wave function.

A corollary of the above results is that in a complete crystal structure, Bloch wave vector is a conserved quantity (based on reciprocal lattice vector), that is, the group velocity of electron waves is a conserved quantity. In other words, in a complete crystal, the movement of electrons can propagate without lattice scattering (so this model is also called near-free electron approximation), and the resistance of crystal conductor only comes from crystal defects that destroy the periodicity of potential field.

Starting from Schrodinger equation, it can be proved that the interaction order of Hamiltonian and translation operator satisfies the exchange law, so the intrinsic wave function of particles in periodic potential field can always be written in the form of Bloch function. More broadly, the symmetry relation of operator function satisfied by eigenfunction is a special case of representation theory in group theory.

The concept of Bloch wave was first put forward by felix bloch when he studied the conductivity of crystalline solids in 1928, but its mathematical basis was put forward by George William Hill (1877), gaston floquet (1883) and Alexander. Therefore, the concept of similarity has different names in various fields: in the theory of ordinary differential equations, it is called F Loki theory (some people call it "Lyapunov-F Loki theorem"); One-dimensional periodic wave equation is sometimes called Hill equation.