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HNPS Advances in Nuclear Physics

The HVT Technique and the "Uncertainty" Relation for Central

M. E. Grypeos, C. G. Koutroulos, K. J. Oyewumi, Th. Petridou

Published Jul 27, 2021• 0 citations
Publisher

Abstract

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 $P_{nl}\equiv <r^2>_{nl}<p^2>_{nl}/\hbar^2$, 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.

DOI
10.12681/hnps.3351

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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.
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.
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.
Heretofore, the hypervirial theorems recently introduced by Hirschfelder have only been applied to bound-state systems. In this paper it is shown that, with certain modifications, these theorems can also be applied to free systems, and, in particular, to scattering problems. The new form of the general hypervirial theorem is derived, and the theory is illustrated with the problem of a particle scattered by a central field. The ordinary virial theorem is deduced, together with other results of physical interest. Both classical and quantum-mechanical formalisms are considered, and in some cases the semiclassical approximation links corresponding results.