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Bloch wave

From Academic Kids

A Bloch wave or Bloch state is the wavefunction of a particle (usually, an electron) placed in a periodic potential. It consists of the product of a plane wave and a periodic function unk(r) which has the same periodicity as the potential:

<math>\psi_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}.\mathbf{r}}u_{n\mathbf{k}}(\mathbf{r}).<math>

The plane wave wavevector k (multiplied by Planck's constant, this is the particle's crystal momentum) is unique only up to a reciprocal lattice vector, so one only needs to consider the wavevectors inside the Brillouin zone. For a given wavevector and potential, there are a number of solutions, indexed by n, to Schrodinger's equation for a Bloch electron. These solutions, called bands, are separated in energy by a finite spacing, called the band gap. The band structure is the collection of energy eigenstates within the first Brillouin zone. All the properties of electrons in a periodic potential can be calculated from this band structure, at least within the independent electron approximation.

It can be shown that the wavefunction of a particle in a periodic potential must have this form by proving that translation operators (by lattice vectors) commute with the Hamiltonian. This result is called Bloch's Theorem.

The concept of the Bloch state was developed by Felix Bloch in 1928.

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