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The Journal of Chemical Physics

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

Saul T. Epstein

Published Oct 15, 1975• 53 citations
Publisher

Abstract

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.
DOI
10.1063/1.431797

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Geometrical properties of space are described by relations between quantum observables, functions of generalized coordinate and momentum operators of a physical system. Space comprises the set of eigenvalues of the coordinate observables, which may be a discrete set of points instead of the continuum of classical geometry. Differentiation on coordinates and momenta is defined by commutation operations. Well-known formulas from classical geometry for gradient and divergence operators, connections, covariant differentiation, tangent vectors, geodesics, and the Riemann curvature tensor are derived as relations between quantum operators. The quantum formulation retains the symmetry between coordinate and momentum representations required by the principle of complementarity.
It is shown that a sufficient condition for an optimal energy variational wave function ${\ensuremath{\psi}}_{0}$ to satisfy the hypervirial relation $({\ensuremath{\psi}}_{0}, [H, W]{\ensuremath{\psi}}_{0})=0$ is for the trial function $\ensuremath{\psi}$ to admit variations of the form $\frac{\ensuremath{\partial}\ensuremath{\psi}}{\ensuremath{\partial}a}=(\frac{i}{\ensuremath{\hbar}})W\ensuremath{\psi}$. Here HHH is the Hamiltonian, WWW is a Hermitian operator, and aaa is a variational parameter. Explicit forms of such trial functions are exhibited for several W′sW'\mathrm{s}W′s. The case in which WWW generates a point transformation of the coordinates is discussed in detail. Conditions are given for the existence of simultaneous hypervirial theorems.
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.