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arXiv:quant-ph/0212132 · Physical Review A

Hydrogen atom in phase space: The Kirkwood-Rihaczek representation

Ludmiła Praxmeyer, K. Wódkiewicz

Published May 30, 2003• 13 citations
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Abstract

We present a phase-space representation of the hydrogen atom using the Kirkwood-Rikaczek distribution function. This distribution allows us to obtain analytical results, which is unique because an exact analytical form of the Wigner functions corresponding to the atom states is not known. We show how the Kirkwood-Rihaczek distribution reflects the properties of the hydrogen atom wave functions in the position and momentum representations.

DOI
10.1103/physreva.67.054502

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We present a phase-space representation of the hydrogen atom using the Kirkwood-Rikaczek distribution function. This distribution allows us to obtain analytical results, which is unique because an exact analytical form of the Wigner functions corresponding to the atom states is not known. We show how the Kirkwood-Rihaczek distribution reflects the properties of the hydrogen atom wave functions in the position and momentum representations.
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.
The rigorous solutions of the stationary Schrodinger equation for hydrogen atom are solved with the wave-mechanics method within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The Fourier-like projection transformations of wave function from the phase space to position and momentum spaces are extended to three-dimensional systems. The eigenfunctions in general position and momentum spaces could be obtained through the transformations from eigenfunction in the phase space.
The wave functions in momentum space for discrete states of hydrogenlike atoms are discussed in terms of both the early contributions to the subject and their present usefulness in atomic physics. The derivation of some of their important properties is outlined, in particular the expectation values of various powers of momentum and kinetic energy, and the kinetic energy distribution. As an application, the cross section for ionization of hydrogen in the metastable 2S state by electron impact is calculated with the aid of the binary-encounter approximation, which requires the use of momentum-space wave functions.