Differential Force Law Research Articles

Differential force law and related integral theorems for a system of N identical interacting particles. II. Spherical symmetry

The differential force law (DFL) and related integral theorems, derived in a previous paper for general geometries, are applied to spherical systems of identical interacting particles, e.g., electrons. From the special functional form of the first-order density matrix, induced solely by symmetry, the DFL occurs as a scalar, pointwise equation relating radial and rotational parts, trad and trot, of the kinetic energy density t to the force density f and to derivatives of the particle density n. Furthermore, an exact connection between the pressure p, trad, and n is established. Finally some theorems are derived which relate integrals, extending over an arbitrary concentric part of the system and involving t, p, and f, to values of p, n, and n′ at the surface of the sphere with radius R. One of these theorems is a generalized virial theorem tending to the usual well-known virial theorem in the limit R→∞.

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

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.

Coordinate invariance, the differential force law, and the divergence of the stress-energy tensor

Hermitian operators linear in momenta generate coordinate transformations. The associated hypervirial theorems are written in the form of moments of a differential force law, and a connection is made with the stress-energy tensor of the Schrödinger field in configuration space.