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Journal of Physics A Mathematical and General

Two classes of exactly solvable quantum models with moving boundaries

Adam J. Makowski

Published Jun 7, 1992• 46 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/25/11/040

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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.
By applying the standard analytical techniques of solving partial differential equations, we have obtained the exact solution in terms of the Fourier sine series to the time-dependent Schrödinger equation describing a quantum one-dimensional harmonic oscillator of time-dependent frequency confined in an infinite square well with the two walls moving along some parametric trajectories. Based upon the orthonormal basis of quasi-stationary wave functions, the exact propagator of the system has also been analytically derived. Special cases like (i) a confined free particle, (ii) a confined time-independent harmonic oscillator, and (iii) an aging oscillator are examined, and the corresponding time-dependent wave functions are explicitly determined. Besides, the approach has been extended to solve the case of a confined generalized time-dependent harmonic oscillator for some parametric moving boundaries as well.
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 method of the effective Schrodinger equation applied to the one-dimensional problem of a moving particle in a potential box with a potential function and high-frequency time-dependent boundary conditions is investigated. The method of converting the initial equation written for a system with time-dependent boundary conditions to an equation describing a system in a constant region is briefly presented. The resulting system can be considered as a particle in a potential box, located in the field of some effective potential, different from the initial one. Next we consider a particular case of wall motion, with the subsequent construction of the effective Schrodinger equation. Rapid oscillations of the potential box as a whole (i.e., without changing its width) lead to an additional term in the effective equation proportional to the second derivative of the potential. This, in particular, can lead to the formation of potential wells at potential peaks in which bound states of a particle are possible. In the considered case, it was assumed that the oscillation amplitudes are small in comparison with the characteristic dimensions of the system.
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 .