Why does Bloch function show that electrons in crystals do common motion?

Bloch function or Bloch (wave) function is the solution of Schrodinger equation of electrons moving in periodic potential field. Bloch function is an amplitude-modulated plane wave in the form ψ k (x) = uk(x) exp (ik x), where uk(x) has the periodicity of lattice.

The exponential part reflects the chemical movement of crystal electrons, and the lattice periodic function part reflects the movement of crystal electrons around the nucleus; The electrons it describes are so-called Bloch electrons (electrons in the lattice periodic potential field), which are electrons with chemical movement in the crystal, so the Bloch state of Bloch electrons is an extended state, which corresponds to the state of energy band electrons, that is, many quasi-continuous energy levels in the energy band.

Only the wave function of * * * chemical electrons in the crystal has the form of Bloch function, and the energy of the corresponding electrons presents as energy bands, not energy levels. Different from Bloch state, it is the so-called localized state of electrons confined near atoms, such as the bound state of electrons on impurities or defects; This local energy state is manifested as energy level binding energy level. The energy of these bound States has nothing to do with the energy of the extended state, so the energy level of bound States can be anywhere in the energy band, that is, it can be in both the energy band and the forbidden band. For example, donor impurity level, acceptor impurity level, recombination center level, trap center level, exciton level and so on are generally in the middle of the forbidden band.

In a certain complete crystal structure, Bloch wave vector is a conserved quantity (based on reciprocal lattice vector), that is, the group velocity of electron wave 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.