Unit-cell scheme in solids

A unit-cell scheme for periodic systems is developed on the basis of the $\mathrm{kq}$ representation. In this scheme the wave function can be normalized so that it leads to a $k$-dependent probability density in a unit cell of the Bravais lattice. The quantum mechanics in the space of this unit cell is developed for periodic systems. An entirely elementary der vation is given of a variety of formulas in the dynamics of electrons in solids. The unit-cell scheme is also shown to be suitable for developing a perturbation theory in solids. The latter is, in particular, applied to the tight-binding band-structure calculation.