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World Scientific series in 20th century physics

APPROXIMATE SOLUTIONS OF THE DENSITY MATRIX EQUATION FOR A LOCAL AVERAGE FIELD

N. H. March, W. H. Young

Published Sep 1, 2009• 0 citations
Publisher

Abstract

Abstract It is shown that a condition derived by Naqvi from the density matrix equation, and used by him to generalize the statistical theory, can be regarded as constituting a differential form of the virial theorem. This suggests that Naqvi's condition is satisfied by the customary treatments, and this is demonstrated directly.
DOI
10.1142/9789814271783_0069

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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.
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The virial theorem is derived in quantum statistical mechanics for a time-dependent Hamiltonian that describes a charged particle in a time-dependent electromagnetic field. The time derivative of the expectation value of the Clausius operator satisfies an equation which is the same form as the classical equation for the Clausius function (r⋅mv). By taking a time average, the virial theorem of quantum mechanics for time-dependent states is obtained. By taking a statistical average and a time average, the virial theorem of quantum statistical mechanics is obtained.