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American Journal of Physics

A Momentum Representation Treatment of the Hydrogen Atom Problem

Eugene V. Ivash

Published Aug 1, 1972• 13 citations
Publisher

Abstract

Energy levels and wave functions are obtained for the quantum mechanical, nonrelativistic, hydrogen atom problem for the l = 0 case, using the momentum representation. The result for the energy levels is extended to the general case of arbitraryl, using the factorization method.
DOI
10.1119/1.1986767

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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.
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The radial form of Hylleraas’ equation for the hydrogen atom, Λl‖El〉=4ℏ4a−2‖El〉 (a=Bohr radius), is considered and it is shown that the operator Λl can be factorized. Hence ladder operators P±l are derived that are linear in the position operator r and are nonlinear functions of the momentum operator p. It is proven that P±l‖El〉 =2ℏ2a−1[1+(l+ (1)/(2) ± (1)/(2) )2 ×(2Mℏ−2a2E)]1/2‖E,l±1〉. In the momentum representation of wave mechanics the solutions to these equations are the radial momentum-space wavefunctions for the hydrogen atom. Thus a simple method of calculating these wavefunctions is obtained. The results complement the familiar operator solution for the hydrogen atom that is based on factorization of the radial Hamiltonian and yields operators that are linear in p and are nonlinear functions of r.
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.