Quantum Mechanics in One Band of a Solid

In elementary quantum mechanics it is customary to describe the motion of a spinless particle by means of its coordinate r and momentum p operators. In such a description any physical operator is necessarily a function o f t and p, and in this sense r and p form a complete set of operators [1]. In solids one often works with a one-band approximation which considerably restricts the freedom of motion of the particle. One should expect that in describing physical phenomena in one band the set of r and p will turn out to be too broad and that a more restricted set of operators should suffice for this purpose. This was actually shown by Mclrvine and Overhauser [2] who derived the superlattice representation by localizing simultaneously the quasimomentum k and the Bravais lattice vector R for one energy band. In approaches to elementary dynamics of solids it is a common procedure to use one-band operators in the effective hamiltonian theories. This is shown in a comprehensive article by Blount [3] who has defined a mixed representation for dealing with band operators in the framework of Wigner's density matrix [4]. It is well known from both experiment and theory that an isolated band in a solid is a relatively stable entity under different perturbations and it seems that the physics of the problem takes place entirely in one particular band. It should be of interest to have a set of elementary operators complete with respect to one band in the same