Differential force law and related integral theorems for a system ofNidentical interacting particles. I. General geometries

Abstract

Starting from the stationary Schrödinger equation for a system of identical interacting particles, the three-dimensional differential force law (DFL) is derived in terms of the kinetic energy density tensor with components tαβ(x), the particle density n(x), and the potential. The most general vector field h(x) is given such that integrating the scalar product of h with the DFL over an arbitrary volume Ω yields theorems involving in their volume integrals the tensor components only in the form t≡∑3α=1tαα (if at all) t being the positive definite density of kinetic energy. The procedure results in four integral theorems: (i) balance equation of forces, (ii) balance equation of torques, (iii) the generalized virial theorem, and (iv) a new exact theorem which can be regarded as vector theorem on the first moment of the kinetic energy density. The new theorem is shown to imply validity of the other three, and therefore is more comprehensive than they.