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Physical Review A

Universal turning point behavior for Gaussian-Klauder states and an application for maximally eccentric Rydberg atoms

William Mather, Ronald F. Fox

Published Oct 5, 2006• 0 citations
Publisher

Abstract

Universal behavior of Gaussian-Klauder states emerges near soft classical turning points, as expressed through a complex-valued Airy transformation that approximates the wave function. Study of these classical turning points provides analytic evidence that Gaussian-Klauder states generally display recurrent localization for many classical orbital periods. Analytic position and momentum moments of the wave function are determined from this approximation, leading in part to connections with the traditionally chosen positional Gaussian wave functions as the limit of large energy uncertainty. Application of this procedure to hydrogenic states of maximal eccentricity leads to the classical limit of recurrent collisional bouncing in the Kepler problem, via the explicit construction of states that maintain phase space localization for many orbital periods.

DOI
10.1103/physreva.74.044101

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Universal behavior of Gaussian-Klauder states emerges near soft classical turning points, as expressed through a complex-valued Airy transformation that approximates the wave function. Study of these classical turning points provides analytic evidence that Gaussian-Klauder states generally display recurrent localization for many classical orbital periods. Analytic position and momentum moments of the wave function are determined from this approximation, leading in part to connections with the traditionally chosen positional Gaussian wave functions as the limit of large energy uncertainty. Application of this procedure to hydrogenic states of maximal eccentricity leads to the classical limit of recurrent collisional bouncing in the Kepler problem, via the explicit construction of states that maintain phase space localization for many orbital periods.
Gaussian-Klauder coherent states are applied to the bound ``quantum bouncer,'' a gravitating particle above an infinite potential boundary. These Gaussian-Klauder states, originally created for Rydberg atoms, provide an overcomplete set of wave functions that mimic classical trajectories for extended times through the utilization of energy localization. For the quantum bouncer, analytic methods are applied presently to compute first and second moments of position and momentum operators, and from these results, at least two scalings of Gaussian-Klauder parameters are highlighted, one of which tends to remains localized for markedly more bounces than comparable states that are Gaussian in position (by an order of magnitude in some cases). We close with a connection that compares Gaussian-Klauder states and positional Gaussian states directly for the quantum bouncer, relating the two through a known energy-position duality of Airy functions. Our results, taken together, ultimately reemphasize the primacy of energy localization as a key ingredient for long-lived classical correspondence in systems with smooth spectra.
Gaussian Klauder coherent states are constructed for the harmonic oscillator, the planar rotor, and the particle in a box. The standard harmonic oscillator coherent states are given by expansions in the eigenstates of the Hamiltonian in terms of a complex parameter \ensuremath{\alpha}. When the complex modulus of \ensuremath{\alpha} is large, these states are identical in behavior with a particular choice of Gaussian Klauder coherent state. When the angular momentum of a planar rotor is large compared with Planck's constant, the angle distribution associated with a Gaussian Klauder coherent state for this case remains sharply localized for many rotations. Similarly, for the particle in a box, it is possible to choose parameters in the Gaussian Klauder coherent state so that a localized particle bounces back and forth at constant velocity between the walls of the box for many periods without significant delocalization. Buried in this behavior is the Fourier series for a triangle wave. These examples show how Gaussian Klauder coherent states are of utility in understanding quantum-classical correspondence.
Generalized coherent states are constructed for the Coulomb problem. Following a construction procedure proposed by Klauder [J. Phys. A 29, L293 (1996)], Rydberg atom coherent states are defined and analyzed. The relationship between decorrelation in time and delocalization in space is elucidated. Keplerian orbits are discussed. The connection with sharp Gaussian wave packets used to explain pump-probe experiments is made. This is achieved by introducing genuine Gaussian Klauder coherent states that are overcomplete, and permit a resolution of the identity operator. They decorrelate comparatively slowly, and remain spatially localized for many Keplerian periods.
The quantum-mechanical motion of Gaussian wave packets on elliptical and hyperbolic orbits is calculated and shown in plots. The approximation scheme makes explicit use of energy and angular-momentum conservation thus guaranteeing controlled accuracy of the approximation over long-time intervals. This allows the study of the revivals, a phenomenon of particular interest in Rydberg atoms on elliptical orbits. A comparison with the behavior of a classical phase-space distribution of the same position and momentum widths is carried out.
Coherent states in the harmonic oscillator have long been thought of as a bridge from quantum mechanics to classical mechanics. Many of their important properties (most notably those involving time evolution) are only valid in the harmonic oscillator and are at best approximations elsewhere. This thesis is an investigation into several means of generalizing coherent states for other systems and examines the resultant states for some systems. One system of interest is the harmonic oscillator with centripetal barrier for which coherent states of several disparate definitions coincide. Also studied is this spherical rotator, a system which is particularly amenable to defining annihilation operator coherent states. A third system under investigation is the hydrogen atom. This system serves as an arena for the development of an extension to a generalization due to Klauder [J. Phys. A. 29(12):L293-L298. 1996]. Klauder's construction is only applicable to systems without energy degeneracies and must be extended for application where degeneracies are present. The author provides a means for this extension and applies the complete construction to the hydrogen atom problem. As a demonstration of how this construction may be adapted, the author constructs Rydberg wave packets which are initially localized and exhibit full and fractional revivals in the long time evolution.