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European Journal of Physics

Quantum bouncer on a spring

John Lekner, Huyen Nguyen

Published Jul 13, 2009• 7 citations
Publisher

Abstract

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 .

DOI
10.1088/0143-0807/30/5/l03

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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 discuss the time development of Gaussian wave packet solutions of the “quantum bouncer” (a quantum mechanical particle subject to a uniform downward force, above an impermeable flat surface). We focus on the evaluation and visualization of the expectation values and uncertainties of position and momentum variables during a single quasi-classical period as well as during the long-term collapsed phase and several revivals. This approach complements existing analytic and numerical analyses of this system, as well as being useful for comparison with similar results for the harmonic oscillator and infinite well cases.
The coherent states of the simple harmonic oscillator with an impenetrable barrier at its center are studied. This half oscillator is the equivalent of a pendulum that bounces elastically off a vertical wall directly below the point of suspension with the angle of swing sufficiently small. The system can also be considered as a paddle ball, where the paddle is fixed and the ball is constrained by a spring attached to the paddle. The coherent states are almost the same as the familiar Gaussian coherent states of the full oscillator, except when they overlap the barrier. The solutions can be easily extended to two and three dimensions and gravity can be included if the impenetrable barrier is vertical. To better understand the form of the expectation values of the position and momentum, we investigate some general aspects of the effect of impenetrable barriers on the dynamics of wavepackets.
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.
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.
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.