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Physical Review A

Wigner phase-space distribution function for the hydrogen atom

S. Nouri

Published Mar 1, 1998• 20 citations
Publisher

Abstract

In this work we present a theoretical study of the hydrogen atom in quantum phase space. A coordinate transformation is used that maps the hydrogen atom into a four-dimensional harmonic oscillator and the Wigner distribution function for the hydrogen atom 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
10.1103/physreva.57.1526

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In this work we present a theoretical study of the hydrogen atom in quantum phase space. A coordinate transformation is used that maps the hydrogen atom into a four-dimensional harmonic oscillator and the Wigner distribution function for the hydrogen atom 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 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 hydrogen atom is a fundamental exactly soluble system for which the Wigner function, being a quantum analogue of the joint probability distribution of position and momentum, is unknown. In this paper, we present an effective method of calculating the Wigner function, for all bound states of the nonrelativistic hydrogen atom. The formal similarity between the eigenfunctions of the nonrelativistic hydrogen atom in the momentum representation and the Klein–Gordon propagator has allowed the calculation of the Wigner function for an arbitrary bound state of the hydrogen atom, using a simple atomic integral as a generator. These Wigner functions for some low-lying states are depicted and discussed.
A coordinate transformation which exhibits the rotational invariance of the hydrogen atom in four-dimensional Hilbert space is introduced. The coordinates are shown to be directly related to the spherical polar and parabolic coordinates in position space. With the use of the transformation, the Schr\"odinger equation for the hydrogen atom left-multiplied by 4r4r4r is transformed into one for a four-dimensional harmonic oscillator. Solutions are obtained and related to the hydrogenic wave functions. Group-theoretical implications of the transformation and its application to the hydrogen Stark problem are briefly discussed.