Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Coherent-state analysis of the quantum bouncing ball

William Mather, Ronald F. Fox

Published Mar 23, 2006• 17 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.73.032109

Related papers

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

William Mather, Ronald F. Fox

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.

Physical Review A•Oct 5, 2006•0 citations

Generalized coherent states and quantum-classical correspondence

Ronald F. Fox, Mee Hyang Choi

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.

Physical Review A•Feb 14, 2000•31 citations

Madelung transformation of the quantum bouncer problem

Eyal Heifetz, I. Plochotnikov

The "quantum bouncer" problem of a quantum particle in a gravitational field, bouncing elastically from a horizontal surface, is mapped via the Madelung transformation into the dynamics of a 1D, compressible, inviscid fluid. Under this transformation the energy states of the quantum bouncer, given in terms of the Airy function, are hydrostatic equilibria states where gravity is balanced by the corresponding vertical pressure gradient force exerted by the fluid. The stability of the hydrostatic states, with respect to small perturbations, is expressed in terms of the positive definite pseudoenergy integral derived here. Pseudoenergy, which is a constant of motion of the linearized perturbation dynamics, is in this case the sum of the perturbation kinetic and available potential energies. The latter is found to be proportional to the square of the vertical derivative of the buoyancy perturbation rather than the square of the buoyancy itself, as is generally the case in classical fluids. For sinusoidal perturbations with vertical wavenumber k , the effective frequency of the buoyancy oscillation is found to be , which is twice the ratio between the gravitational force and the momentum of a free quantum particle.

Europhysics Letters (EPL)•Apr 30, 2020•7 citations

Quantum bouncer on a spring

John Lekner, Huyen Nguyen

The usual 'quantum bouncer' is a quantum particle bouncing elastically under the influence of gravity. Here we consider a particle bouncing off an impenetrable wall, and bound to it by the harmonic potential instead of by the gravitational potential mgx . An analytic solution is possible, easily accessible to classes in intermediate quantum mechanics. The solution is related to Schrödinger's oscillating wavepacket (the original coherent state ). Although periodic, it changes its form during the oscillation, in contrast to Schrödinger's packet. The uncertainties in the position (Δ x ) and the momentum (Δ p ) oscillate also, as does the product Δ x Δ p .

European Journal of Physics•Jul 13, 2009•7 citations

Caustics and Superenergy in the Quantum Bouncer

Marko. M. Ćosić, Andrew N. Jordan

We investigate the quantum interference and energetic phenomena associated with classical caustics in the quantum bouncing ball problem, we refer to as quantum caustics. By considering an initial Gaussian wavepacket, we show that caustics associated with the underlying classical trajectory families are exhibited. We connect the associated phase singularity chains in the vicinity of the caustic with the semiclassical Pearcey function built on the cusp catastrophe lines. We also quantify the amount of superenergy exhibited in these solutions - regions of space where the local energy exceeds the largest constituent energy eigenvalue. We give a complimentary description of the caustic and superenergy behavior using the Madelung/Bohm trajectories, which gives additional insight about the energy of the trajectories and how they traverse the phase singularity chains.

quant-pharXiv:2608.02427•Aug 4, 2026•0 citations

Generalized coherent states

Ronald F. Fox

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.

Physical Review A•May 1, 1999•25 citations
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.
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 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.
The "quantum bouncer" problem of a quantum particle in a gravitational field, bouncing elastically from a horizontal surface, is mapped via the Madelung transformation into the dynamics of a 1D, compressible, inviscid fluid. Under this transformation the energy states of the quantum bouncer, given in terms of the Airy function, are hydrostatic equilibria states where gravity is balanced by the corresponding vertical pressure gradient force exerted by the fluid. The stability of the hydrostatic states, with respect to small perturbations, is expressed in terms of the positive definite pseudoenergy integral derived here. Pseudoenergy, which is a constant of motion of the linearized perturbation dynamics, is in this case the sum of the perturbation kinetic and available potential energies. The latter is found to be proportional to the square of the vertical derivative of the buoyancy perturbation rather than the square of the buoyancy itself, as is generally the case in classical fluids. For sinusoidal perturbations with vertical wavenumber k , the effective frequency of the buoyancy oscillation is found to be , which is twice the ratio between the gravitational force and the momentum of a free quantum particle.
The usual 'quantum bouncer' is a quantum particle bouncing elastically under the influence of gravity. Here we consider a particle bouncing off an impenetrable wall, and bound to it by the harmonic potential instead of by the gravitational potential mgx . An analytic solution is possible, easily accessible to classes in intermediate quantum mechanics. The solution is related to Schrödinger's oscillating wavepacket (the original coherent state ). Although periodic, it changes its form during the oscillation, in contrast to Schrödinger's packet. The uncertainties in the position (Δ x ) and the momentum (Δ p ) oscillate also, as does the product Δ x Δ p .
We investigate the quantum interference and energetic phenomena associated with classical caustics in the quantum bouncing ball problem, we refer to as quantum caustics. By considering an initial Gaussian wavepacket, we show that caustics associated with the underlying classical trajectory families are exhibited. We connect the associated phase singularity chains in the vicinity of the caustic with the semiclassical Pearcey function built on the cusp catastrophe lines. We also quantify the amount of superenergy exhibited in these solutions - regions of space where the local energy exceeds the largest constituent energy eigenvalue. We give a complimentary description of the caustic and superenergy behavior using the Madelung/Bohm trajectories, which gives additional insight about the energy of the trajectories and how they traverse the phase singularity chains.
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.