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Journal of Physics A Mathematical and General

The HVT technique and the treatment of two basic inequalities

M. E. Grypeos, C. G. Koutroulos, Th. Petridou

Published Feb 25, 2002• 2 citations
Publisher

Abstract

The quantum mechanical hypervirial theorems (HVT) technique is used in treating two basic inequalities relating the ground-state mean square radius of the orbit of a particle in a central potential and its kinetic energy, respectively, to the spacing of the two lowest energy levels Δ E . For quite a wide class of those potentials, the parameters of which also lead to a sufficiently small dimensionless quantity s , the difference between the two sides of each inequality is of order s 3 and higher (while Δ E is of order s and higher), and thus it is expected in general to be quite small.

DOI
10.1088/0305-4470/35/9/312

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The quantum mechanical hypervirial theorems (HVT) technique is used in treating two basic inequalities relating the ground-state mean square radius of the orbit of a particle in a central potential and its kinetic energy, respectively, to the spacing of the two lowest energy levels Δ E . For quite a wide class of those potentials, the parameters of which also lead to a sufficiently small dimensionless quantity s , the difference between the two sides of each inequality is of order s 3 and higher (while Δ E is of order s and higher), and thus it is expected in general to be quite small.
The quantum mechanical hypervirial theorems (HVT) technique is used to treat the so-called "uncertainty" relation for quite a wide class of central potential wells, including the (reduced) Poeschl-Teller and the Gaussian one.It is shown that this technique is quite suitable in deriving an approximate analytic expression in the form of a truncated power series expansion for the dimensionless product Pnl≡<r2>nl<p2>nl/ℏ2P_{nl}\equiv <r^2>_{nl}<p^2>_{nl}/\hbar^2Pnl​≡<r2>nl​<p2>nl​/ℏ2, for every (deeply) bound state of a particle moving non-relativistically in the well, provided that a (dimensionless) parameter s is sufficiently small. Numerical results are also given and discussed.
Sometimes it is useful and necessary to consider quantum-mechanical systems enclosed in "boxes" of different shapes with impenetrable or semipermeable walls, in contrast to the usual requirement that wave functions vanish at infinity. This kind of model has been widely applied to interpret, analyze, and discuss several physical phenomena. In this communication we describe and study the hypervirial relationships for finite systems in order to generalize some previous works related to the virial theorem. We apply our formal results to various simple models and discuss their potential applications to solid-state theory.