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Journal of Physics A Mathematical and General

Wigner function for the radial equation of the Coulomb problem in Langer space

J Stanek, Jerzy Konarski

Published Dec 7, 2005• 1 citation
Publisher

Abstract

In this paper a theoretical study of the Coulomb problem in quantum phase-space is reported. The Langer coordinate transformation is used to map the Coulomb problem into a one-dimensional Morse oscillator. As a result the Wigner distribution functions for the Morse oscillator are obtained. The form of these functions is presented for a few principal quantum number n and the angular momentum quantum number l . The results obtained correspond to the solution of the Coulomb problem in the spherical coordinate system.

DOI
10.1088/0305-4470/39/1/015

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In this paper a theoretical study of the Coulomb problem in quantum phase-space is reported. The Langer coordinate transformation is used to map the Coulomb problem into a one-dimensional Morse oscillator. As a result the Wigner distribution functions for the Morse oscillator are obtained. The form of these functions is presented for a few principal quantum number n and the angular momentum quantum number l . The results obtained correspond to the solution of the Coulomb problem in the spherical coordinate system.
Abstract In this article, the rotating Kratzer oscillator in quantum phase space is studied. The Langer transformation is used to map the Kratzer oscillator with centrifugal term onto a one‐dimensional Morse oscillator. As a result, the Wigner distribution functions for the Morse oscillator are obtained. The quantum states of the system are visualized in the phase space for a few vibrational and rotational quantum numbers. The results obtained in the phase space correspond to those derived in the standard quantum theory. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010
In this work we present a theoretical study of the d-dimensional Coulomb problem in quantum phase space. A coordinate transformation in hyperspherical space is used that maps the d-dimensional Coulomb problem into the D-dimensional harmonic oscillator and the Wigner distribution function for the d-dimensional Coulomb problem is then obtained. This exactly soluble model can shed some light on finite-size features of Wigner’s distribution, which will be a vital experience for various dynamic problems.
The mapping of the Wigner distribution function (WDF) for a given bound state onto a semiclassical distribution function (SDF) satisfying the Liouville equation introduced previously by us is applied to the ground state of the Morse oscillator. The purpose of the present work is to obtain values of the potential parameters represented by the number of levels in the case of the Morse oscillator, for which the SDF becomes a faithful approximation of the corresponding WDF. We find that for a Morse oscillator with one level only, the agreement between the WDF and the mapped SDF is very poor but for a Morse oscillator of ten levels it becomes satisfactory. We also discuss the limit h --> 0 for fixed potential parameters.
The Morse oscillator, radial Coulomb and radial harmonic oscillator problems can be solved exactly using a variety of algebraic methods. These problems correspond to different realizations of the so(2,1) algebra and a comparison of the generators of the algebra may be used to identify mappings between each pair of systems. The resultant transition operators act as ladder, or energy changing, operators in the cases of the Coulomb and harmonic oscillator potentials, whereas they act as shift operators, acting at constant energy, in the case of the Morse potential. This is a consequence of the so(2,1) dynamical symmetry, whereby the Morse Hamiltonian is expressible solely in terms of the Casimir operator of the algebra. An alternative algebraic approach, the use of the method of supersymmetric quantum mechanics, or factorization, produces in each case a set of shift operators. Relations between the various ladder and shift operators may be identified by means of the appropriate mappings, and these results can be generalized so as to relate the one dimensional Morse oscillator to the radial Coulomb and radial harmonic oscillator potentials involving an arbitrary number of angular dimensions.
The bound-state solutions and the su(1,1) description of the d-dimensional radial harmonic oscillator, the Morse, and the D-dimensional radial Coulomb Schrödinger equations are reviewed in a unified way using the point canonical transformation method. It is established that the spectrum generating su(1,1) algebra for the first problem is converted into a potential algebra for the remaining two. This analysis is then extended to Schrödinger equations containing some position-dependent mass. The deformed su(1,1) construction recently achieved for a d-dimensional radial harmonic oscillator is easily extended to the Morse and Coulomb potentials. In the last two cases, the equivalence between the resulting deformed su(1,1) potential algebra approach and a previous deformed shape invariance one generalizes to a position-dependent mass background a well-known relationship in the context of constant mass.