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Journal of Physics A Mathematical and General

A generalized quantum virial equation

Jan‐Åke Schweitz

Published Apr 1, 1977• 8 citations
Publisher

Abstract

A method, similar to the one previously used by the author in his derivation of a generalized classical virial theorem (see ibid., vol.10, no.4, p.507 (1977)), is used to derive a generalized quantum virial equation. This equation is applicable to any part of a larger system of particles. Furthermore, by the introduction of a flux density operator, it is possible to express the quantum surface flux virial in two alternative forms, as easily interpreted as the classical form. As an intermediate result, an equation of continuity for a general one-particle observable is obtained. In an appendix, an equation of motion of the reduced density matrix of the first order is derived.

DOI
10.1088/0305-4470/10/4/014

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A method, similar to the one previously used by the author in his derivation of a generalized classical virial theorem (see ibid., vol.10, no.4, p.507 (1977)), is used to derive a generalized quantum virial equation. This equation is applicable to any part of a larger system of particles. Furthermore, by the introduction of a flux density operator, it is possible to express the quantum surface flux virial in two alternative forms, as easily interpreted as the classical form. As an intermediate result, an equation of continuity for a general one-particle observable is obtained. In an appendix, an equation of motion of the reduced density matrix of the first order is derived.
An exact generalized master equation is derived for a large quantum-mechanical system in the form of a power series in the density. This derivation is a quantum-mechanical generalization of a previous work by the author for classical systems. The quantum equation can be viewed as a time-dependent analog of the virial expansion of the quantum-mechanical partition function---for both degenerate systems (Bose-Einstein or Fermi-Dirac statistics) and nondegenerate systems. The coefficients of the series, in the quantum master equation, are explicitly given in terms of operators (Green functions) which are determined by the dynamics of isolated groups of particles and are convergent functions of the interaction potential. Equations are obtained for the off-diagonal elements of the density matrix as well as for the diagonal elements. The equation for the diagonal elements is shown to reduce to a Markoffian master equation for asymptotically long times, and an explicit expression is obtained for the "scattering" operator of this asymptotic equation.
The hierarchy of the equations of motion for the reduced density matrices in quantum statistical mechanics is solved and the (cumulant) reduced density matrices at a time t are expressed in terms of those at an earlier time t0. Diagrams are introduced to express the results. With the aid of the technique of partial summations, the general term in the kinetic equation for the one-particle reduced density matrix or the generalized Boltzmann equation in quantum statistical mechanics is obtained. The equation is non-Markovian. A method of reducing the equation to Markovian is sketched.
A density-matrix formulation of the ab initio methods of nonrelativistic quantum mechanics is presented. The variational equations for the standard calculational techniques of molecular quantum mechanics are derived from a single expression for the total energy of an N-particle system. A generalization of the Hohenberg-Kohn theorem is proved, showing that the ground-state energy of a many-body Hamiltonian is a functional of the single-particle density matrix and hence of the first-order reduced-density function (the diagonal of which is the particle density of the standard theorem). The theorem is used to develop a new method to calculate the total energy and single-particle properties of the many-body problem.
Two distinct approaches have been followed to produce an approximate expression for the third virial coefficient of a quantum system in terms of the pertinent two-particle quantities (standard T matrix elements and bound state eigenvalues and eigenstates): One approach is based on the T approximation for the self-energy in the theory of temperature Green functions, and the other uses the diagrams which represent the various terms in the multiple-scattering expansion of the three-particle T operator. It is shown that the result given by the former method is reproduced exactly by the contribution of a small number of selected diagrams. The Green function method requires the evaluation of the Kadanoff and Baym generalized T matrix to one order in the activity higher than its familiar low-density limit. The common result can be written entirely in terms of the lowest order self-energy and should give a reasonable approximation to the third virial coefficient for systems with short-range forces.
Abstract The concepts of “open assembly” is defined, and the hypervirial theorem is derived in three steps: (i) An equation of motion of the reduced density matrix of the first order is derived. (ii) This equation is utilized to derive a general equation of continuity for single‐particle properties. (iii) The hypervirial equation for open assemblies is obtained as a long‐time average of the continuity equation. The hypervirial theorem is shown to be equivalent to a simple postulate: no physical property can keep on accumulating for ever in a limited volume. Finally, the hypervirial theorem is applied to the problems of force, torque, and virial balance in open dynamical assemblies, and the results are compared with the corresponding classical results.