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HNPS Advances in Nuclear Physics

Approximate Determination of the ground-state Schroedinger Eigenfunction by Means of the Hypervirial Theorems Technique

M. E. Grypeos, Th. E. Liolios

Published Dec 5, 2019• 0 citations
Publisher

Abstract

The method of the Hypervirial Theorems is used to obtain analytically the ground-state· Schroedinger eigenfunction for a wide class of potentials in non-relativistic Quantum Mechanics.The whole scheme, despite its simplicity, yields in many cases a good approximation.
DOI
10.12681/hnps.2397

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Two applications of the quantum mechanical hypervirial theorem are discussed. This states that for any system having eigenstates A and B of Hamiltonian H, an operator W has the matrix element relation (A,(HW andminus; WH) B) = (EA andminus; EB) (A, W B) providing W is such that H is Hermitean between A and W B. The first application uses the operator W = eitx to derive a further relationship between certain matrix elements. For one dimensional systems in which the potential V(x) involves only positive powers of x, this relationship becomes a differential equation in f(t) = (A, eitx B) This differential equation has been solved analytically for the harmonic oscillator and in series for the quartic oscillator which were taken as example systems. The second application is the off-diagonal hypervirial method, which seeks to require a wave function A to satisfy a number of hypervirial relations. In particular the wave function A is forced to obey as many conditions as it has variable parameters. To accomplish this a computational algorithm was developed which allowed one of these parameters to be non-linear. The results for oscillator test systems are not encouraging. The systems tested were the quartic and sixth power oscillators with a trial basis of harmonic oscillator eigenfunctions. In general the results were not as good as those obtainable by the variational method.
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.
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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.