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Physical Review A

Radial coherent states for the Coulomb problem

Christopher C. Gerry, Jerry Kiefer

Published Feb 1, 1988• 43 citations
Publisher

Abstract

Using the SO(2,1) radial (or spectrum-generating) subgroup of the SO(4,2) dynamical group of the Coulomb problem, we construct the associated coherent states that follow the classical radial motion. These wave packets evolve in a fictitious time variable proportional to the eccentric anomaly. The packets consist of a superposition of Coulomb Sturmian functions and do not disperse as they evolve although the width of the packets changes periodically.

DOI
10.1103/physreva.37.665

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Using the SO(2,1) radial (or spectrum-generating) subgroup of the SO(4,2) dynamical group of the Coulomb problem, we construct the associated coherent states that follow the classical radial motion. These wave packets evolve in a fictitious time variable proportional to the eccentric anomaly. The packets consist of a superposition of Coulomb Sturmian functions and do not disperse as they evolve although the width of the packets changes periodically.
We formulate radial coherent states for the isotropic harmonic oscillator from the unitary irreducible representations associated with the radial spectrum-generating group Sp(2) which is locally isomorphic to SU(1,1) and SO(2,1). We further consider the time evolution of a coherent-state wave packet and show that, in contrast to the ordinary oscillator coherent states, this wave packet does change its shape (periodically) as it evolves. This change of shape may be interpreted in terms of a generalized squeezing.
We study the radial part of the Dunkl–Coulomb problem in two dimensions and show that this problem possesses the SU(1,[Formula: see text]1) symmetry. We introduce two different realizations of the su(1,[Formula: see text]1) Lie algebra and use the theory of irreducible representations to obtain the energy spectrum and eigenfunctions. For the first algebra realization, we apply the Schrödinger factorization to the radial part of the Dunkl–Coulomb problem to construct the algebra generators. In the second realization, we introduce three operators, one of them proportional to the Hamiltonian of the radial Schrödinger equation. Finally, we use the SU(1,[Formula: see text]1) Sturmian basis to construct the radial coherent states in a closed form.
The Kustaanheino-Stiefel transformation is used to transform the Coulomb problem to a four-dimensional isotropic harmonic oscillator with a constraint. Ordinary coherent states are introduced over this oscillator and are shown to evolve in a fictitious time. When projected back into the physical space, the states follow the classical Kepler orbits in physical time. The resulting quasiclassical picture is obtained without the Bohr correspondence principle of taking only contributions from high principal quantum numbers.
The minimum-uncertainty coherent-states formalism is extended to higher-dimensional systems. Specifically, for spherically symmetric three-dimensional potentials the formalism looks for coherent states which are products of an angular wave function times a radial wave function. After reviewing the many studies on angular coherent states, I concentrate on the physically distinguishing radial coherent states. The radial formalism is explained in detail and contrasted with the effective one-dimensional formalism. The natural classical variables in the radial formalism are those which vary sinusoidally as $g(E, L)\ensuremath{\theta}(t)$, where $\ensuremath{\theta}(t)$ is the real azimuthal angular variable and g(E,L)g(E, L)g(E,L) is the number of oscillations between apsidal distances per classical orbit. When changed to natural quantum operators, these operators can be given as the Hermitian sums and differences of the "lll" raising and lowering operators. The formalism is applied to the three-dimensional harmonic-oscillator and Coulomb problems.