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Journal of Mathematical Physics

Determination of Wigner distribution function for the d-dimensional Coulomb problem

Saeid Nouri

Published Mar 1, 1999• 7 citations
Publisher

Abstract

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.

DOI
10.1063/1.532802

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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.
In this paper a set of generalized coherent states for the d-dimensional Coulomb problem in coordinate representation are constructed. A coordinate transformation in hyperspherical space is used that maps the d-dimensional Coulomb problem into the D-dimensional harmonic oscillator and the generalized coherent states for the d-dimensional Coulomb problem are then obtained. This exactly soluble model can provide an adequate means for a quantum coherency description of the Coulomb problem in arbitrary dimensions, specifically in the special case of the hydrogen atom, in many theoretical and applied related fields such as in coherent scattering.
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.
A set of generalized coherent states for the one-dimensional Coulomb problem in coordinate representation is constructed. At first, we obtain a mapping for proper transformation of the one-dimensional Coulomb problem into a nonrotating four-dimensional isotropic harmonic oscillator in the hyperspherical space, and the generalized coherent states for the one-dimensional Coulomb problem is then obtained in exact closed form. This exactly soluble model can provide an adequate means for a quantum coherency description of the Coulomb problem in one dimension, sample for coherent aspects of the exciton model in one-dimension example in high-temperature superconductivity, semiconductors, and polymers. Also, it can be useful for investigating the coherent scattering of the Coulomb particles in one dimension.
We define a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum. The distribution function is thus defined on discrete phase-space, i.e. on a finite discrete square grid. These discrete Wigner functions possess a number of properties similar to the Wigner function for a continuous quantum system such as the quantum harmonic oscillator. As an example, we consider the so-called oscillator model in dimension 2 j + 1, which is known to tend to the canonical oscillator when j tends to infinity. In particular, we compare plots of our discrete Wigner functions for the oscillator with the well known plots of Wigner functions for the canonical quantum oscillator. This comparison supports our approach to discrete Wigner functions.