Operator calculus in the electron theory of metals

Abstract

An operator calculus is developed applicable to problems in the electron theory of metals. It differs from the common operator calculus of the quantum theory in the fact that the wave function is defined in a finite space (the atomic polyhedron) bounded by a finite surface. This leads to the introduction of surface operators. The position operator x cannot be developed with respect to the proper functions of the Hamiltonian. Instead an operator ξ is introduced, which is essentially the Fourier development of x. Thus there are three fundamental types of operators: the differential operator p, the multiplication operator ξ and the surface operators. It is shown that with the help of these a consistent calculus can be developed.