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International Journal of Quantum Chemistry

The virial theorem with boundary conditions applications to the harmonic oscillator and to sine-shaped potentials

Edmond Weislinger, Gabriel Olivier

Published Jun 18, 2009• 5 citations
Publisher

Abstract

New objective formulations of the quantum virial theorem rise from a parallelism drawn with d'Alembert'sprinciple. In particular, boundary conditions correspond to constraints whose virial give a complementary term to the usual formulation of the quantum virial theorem. Two examples are examined. first the non-Coulombic and nonperiodical harmonic oscillator for which all the virials can be calculated and compared; second the periodical sine-shaped potential connected with Mathieu'sfunctions, for which only the complementary term due to the cyclic conditions is investigated, showing that its importance is varying from zero to twice the mean kinetic energy when the amplitude of the interaction is vanishing, in the limiting free-electron case. A remark is also made about normalization and the use of mean energy values over a part of space, when mathematical integration is stoped on a surface (partitioning of a molecule, for example) or on a cell in periodical solid-state problems.

DOI
10.1002/qua.560090852

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We investigate the particle and kinetic-energy densities for a system of NNN fermions confined in a potential $V(\bfr)$. In an earlier paper [J. Phys. A: Math. Gen. {\bf 36}, 1111 (2003)], some exact and asymptotic relations involving the particle density and the kinetic-energy density locally, i.e. at any given point $\bfr$, were derived for isotropic harmonic oscillators in arbitrary dimensions. In this paper we show that these {\it local virial theorems} (LVT) also hold exactly for linear potentials in arbitrary dimensions and for the one-dimensional box. We also investigate the validity of these LVTs when they are applied to arbitrary smooth potentials. We formulate generalized LVTs that are supported by a semiclassical theory which relates the density oscillations to the closed non-periodic orbits of the classical system. We test the validity of these generalized theorems numerically for various local potentials. Although they formally are only valid asymptotically for large particle numbers NNN, we show that they practically are surprisingly accurate also for moderate values of NNN.