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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Exact solution of the wave dynamics of a particle bouncing chaotically on a periodically oscillating wall

Jorge F. Willemsen

Published Oct 1, 1994• 10 citations
Publisher

Abstract

An exact solution to the Schr\"odinger equation corresponding to the problem of a particle falling under gravitational acceleration and bouncing on a harmonically oscillating surface is derived. Stationary phase approximations to the transition amplitudes from an initial state to eigenfunctions of the problem with a fixed surface are worked out and illustrated.

DOI
10.1103/physreve.50.3116

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Reply to ‘‘Comment on ‘Exact solution of the wave dynamics of a particle bouncing chaotically on a periodically oscillating-wall’’’

Jorge F. Willemsen

A reexamination of the results of my earlier paper [Phys. Rev. E 50, 3116 (1994)] is supplied using entirely straightforward methods, i.e., without appealing to operator formalism. This reexamination confirms the consistency of the work in that paper by extracting a term proportional to time that Dembrinski et al. [Phys. Rev. E 53, 4243 (1996)] claim is not present. The arguments of Dembinski et al. are not refuted, but it is pointed out that they are based upon methods within which one readily encounters paradoxes such [p${\mathrm{^}}^{2}$,p^]\ensuremath{\ne}0 and [x^,p^]\ensuremath{\ne}i^. The present exposition avoids such methods. Unfortunately I am not prepared to address their numerical results. This will have to await further investigation. However, an example of numerical results of my own is presented in order to illustrate the richness of the dynamics of the model. \textcopyright{} 1996 The American Physical Society.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jun 1, 1996•0 citations

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A trajectory of a harmonic oscillator obeying the Schrodinger equation is exactly derived and illustrated. The trajectory resembles well the classical orbit between the turning points, and also runs through the tunneling region. The dynamics of the 'particle' motion and the wave function associated with the motion are proposed. The period of a round trip on the trajectory is exactly equal to that obtained in classical mechanics.

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Bouncing in an oscillatory gravitational field

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Abstract We apply the time-dependent supersymmetry methods of Bagrov, Samsonov and Shekoyan to the Schrödinger equation for a quantum bouncer. A new family of potentials, that correspond to the linear gravitational potential with a small oscillatory component superimposed upon it, is produced. Both the frequency and amplitude of the oscillatory part can be controlled and the corresponding eigenfunctions are found. These solutions are explored and basic observables are calculated. In particular we examine how motion in such a potential can be distinguished from motion in a linear gravitational field. We also point out some pedagogical aspects of this project.

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It is shown here that the non-stationary Schrodinger equation can be solved exactly for two quantum models subject to Dirichlet boundary conditions. One of them is a modified problem of a quantum bouncer, i.e. the problem of a particle falling down in the gravitational field on a moving (oscillating) platform such as a loudspeaker. The second model is a 'cuff-off oscillator' with a moving infinite potential wall and a time-dependent frequency. In both cases exact solutions are given in closed forms, easy to use. Their possible applications are also indicated. In each of the models extra coordinate- and time-dependent phase factors are generated by moving boundaries in the former case giving rise to a non-local effect in quantum mechanics.

Journal of Physics A Mathematical and General•Jun 7, 1992•46 citations
An exact solution to the Schr"odinger equation corresponding to the problem of a particle falling under gravitational acceleration and bouncing on a harmonically oscillating surface is derived. Stationary phase approximations to the transition amplitudes from an initial state to eigenfunctions of the problem with a fixed surface are worked out and illustrated.
We consider time-dependent Gaussian wave packet solutions of the Schrödinger equation, with arbitrary initial central position, x 0 , and momentum, p 0 , for an otherwise free particle, but with an infinite wall at x = 0, so-called bouncing wave packets . We show how difference or mirror solutions of the form ψ( x , t ) − ψ(− x , t ) can, in this case, be normalized exactly, allowing for the evaluation of a number of time-dependent expectation values and other quantities in closed form. For example, we calculate ⟨ p 2 ⟩ t explicitly which illustrates how the free-particle kinetic (and hence total energy) is affected by the presence of the distant boundary. We also discuss the time dependence of the expectation values of position, ⟨ x ⟩ t , and momentum, ⟨ p ⟩ t , and their relation to the impulsive force during the `collision' with the wall. Finally, the x 0 , p 0 → 0 limit is shown to reduce a special case of a non-standard free-particle Gaussian solution. The addition of this example to the literature then expands of the relatively small number of Gaussian solutions to quantum mechanical problems with familiar classical analogs (free particle, uniform acceleration, harmonic oscillator, unstable oscillator, and uniform magnetic field) available in closed form.
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 .
A reexamination of the results of my earlier paper [Phys. Rev. E 50, 3116 (1994)] is supplied using entirely straightforward methods, i.e., without appealing to operator formalism. This reexamination confirms the consistency of the work in that paper by extracting a term proportional to time that Dembrinski et al. [Phys. Rev. E 53, 4243 (1996)] claim is not present. The arguments of Dembinski et al. are not refuted, but it is pointed out that they are based upon methods within which one readily encounters paradoxes such [p${\mathrm{^}}^{2}$,p^]\ensuremath{\ne}0 and [x^,p^]\ensuremath{\ne}i^. The present exposition avoids such methods. Unfortunately I am not prepared to address their numerical results. This will have to await further investigation. However, an example of numerical results of my own is presented in order to illustrate the richness of the dynamics of the model. \textcopyright{} 1996 The American Physical Society.
Abstract We apply the time-dependent supersymmetry methods of Bagrov, Samsonov and Shekoyan to the Schrödinger equation for a quantum bouncer. A new family of potentials, that correspond to the linear gravitational potential with a small oscillatory component superimposed upon it, is produced. Both the frequency and amplitude of the oscillatory part can be controlled and the corresponding eigenfunctions are found. These solutions are explored and basic observables are calculated. In particular we examine how motion in such a potential can be distinguished from motion in a linear gravitational field. We also point out some pedagogical aspects of this project.
It is shown here that the non-stationary Schrodinger equation can be solved exactly for two quantum models subject to Dirichlet boundary conditions. One of them is a modified problem of a quantum bouncer, i.e. the problem of a particle falling down in the gravitational field on a moving (oscillating) platform such as a loudspeaker. The second model is a 'cuff-off oscillator' with a moving infinite potential wall and a time-dependent frequency. In both cases exact solutions are given in closed forms, easy to use. Their possible applications are also indicated. In each of the models extra coordinate- and time-dependent phase factors are generated by moving boundaries in the former case giving rise to a non-local effect in quantum mechanics.