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:
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.