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International Journal of Mathematical Education in Science and Technology

Bouncing Wave Packets

J. V. Greenman

Published Jan 1, 1972• 0 citations
Publisher

Abstract

Summary It is argued that quantum mechanics is a better theory than classical mechanics because it gives all the classical results and in addition explains new phenomena that classical mechanics cannot convincingly explain. However, little time seems to be spent in introductory courses on quantum mechanics to show that one can indeed obtain even the simplest classical results. In this paper we discuss two simple classical phenomena, a ball bouncing against a wall and a ball bouncing backwards and forwards inside a box. By explicitly finding the square integrable solutions of the time‐dependent Schrodinger equation we show that there is an analogous quantum mechanical bouncing process. It is possible to find explicit solutions near the classical limit since there is negligible dispersion of the wave functions, a fact that enables the integrals over the improper eigenfunctions to be evaluated with little error.

DOI
10.1080/0020739720030103

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Summary It is argued that quantum mechanics is a better theory than classical mechanics because it gives all the classical results and in addition explains new phenomena that classical mechanics cannot convincingly explain. However, little time seems to be spent in introductory courses on quantum mechanics to show that one can indeed obtain even the simplest classical results. In this paper we discuss two simple classical phenomena, a ball bouncing against a wall and a ball bouncing backwards and forwards inside a box. By explicitly finding the square integrable solutions of the time‐dependent Schrodinger equation we show that there is an analogous quantum mechanical bouncing process. It is possible to find explicit solutions near the classical limit since there is negligible dispersion of the wave functions, a fact that enables the integrals over the improper eigenfunctions to be evaluated with little error.
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.
When the solution of the time independent Schrödinger equation is a real function, the velocity of the associated quantum particle is vanishing in Bohmian Mechanics. We can obtain a moving particle if we consider that its velocity can be related not to the phase of the wave function but to its amplitude. By using the rules of standard quantum mechanics, we have constructed a model generalizing the momentum operator and modifying the corresponding Schrödinger equation. We obtain a new nonlinear equation of the form studied by Ermakov and Pinney and we apply the model to the cases of the particle in a box and of a bouncing ball. The new formula of quantum velocity and the corresponding motion of quantum particles have an immediate correspondence with the behavior of the bodies in classical mechanics.
In this chapter, the Schrödinger equation is “derived” for particles that can be described by de Broglie waves. The Schrödinger equation is very different from the corresponding equation of motion in classical mechanics. In order to illustrate the fundamental differences between the two theories, one of the simplest problems of particle dynamics is solved in both Newtonian and quantum mechanics. This simple example also helps to show that quantum mechanics is the fundamental theory and classical mechanics is an approximation, a remarkably good approximation, when considering macroscopic objects. The solution of the Schrödinger equation is presented for a particle inside a box and the quantization condition is derived. The amazing possibility of quantum tunneling when a particle is incident on a barrier of height larger than the energy of the incident particle is also discussed. Finally the three-dimensional Schrödinger equation is solved for the hydrogen atom.
Classical mechanics traces a particle’s movement through space and time through x(t). For example, one may consider the case of a particle moving at constant speed (momentum p) in one direction or a particle bouncing back and forth at constant speed in a box. These problems are described in time and have direction and position associated with time etc. It is possible to define a probability P(x,t) in the classical picture based on delta time, the amount of time a particle spends in a little region of space dx. P(x,t), by itself, loses much relevant information present in the x(t) scenario because a particle moving to the right, or to the left, or bouncing back and forth all have the same probability, 1/L, where L is length. Furthermore, this probability is the same for all p (momentum) values. Thus a probability description, without extra information, is a subset of the relevant information in the problem. As a result, to describe the classical picture fully, one might say that a particle has a probability of 1/L, but also a mass mo and velocity +v (i.e. momentum mov nonrelativistically). For the particle bouncing back and forth in a box with constant p, one needs to give the initial position at time=0 and then a subsequent time (as well as the initial velocity and mo) in order to fully describe the problem, even though probability = 1/L. In the literature (see (1)), there have been many attempts to derive the quantum Schrodinger equation from a classical Fockker-Planck equation or a continuity equation (which follows from the Fokker-Planck equation as shown in (1)). The continuity equation contains a probability P(x,t) and a velocity(x,t) and is generally argued to represent classical physics. It does, if one retains appropriate classical conditions such as the discontinuity of density for a particle in a box (i.e. density=0 outside the box, but 1/L inside) and the notion of differing directions within the box. In fact, one may write density=W*(x,t)W(x,t) where W(x,t) is an ad hoc complex function and still retain compatibility with classical physics. If one does not, however, retain the classical conditions completely, nonclassical physics is introduced, even if density(x,t)=1/L. For example, in quantum mechanics density(x) is considered to be continuous in all x. Thus for a particle moving in a box, density outside the box is 0 so at the box wall it must also be 0. In the classical case, the density is 1 inside the box and 0 outside. There is a discontinuity classically and this cannot be changed and this contradicts the quantum mechanical picture. Thus quantum mechanics does not follow form classical Fokker-Planck or continuity equations which retain classical conditions.. Furthermore, because time is completely removed in the FP-continuity approach as applied to quantum mechanics, one cannot distinguish between forwards and backwards motion. As seen by the velocity field given in (1), v is zero for a bound particle which completely removes the sense of time and violates the classical physics conditions. It is, however, not zero for a particle moving in one direction. Thus there seems to be no issue in describing a Fokker-Planck-continuity equation as describing classical physics. One may even introduce density(x,t)= W*(x,t)W(x,t) provided one retains classical conditions allowing for discontinuity in density and different W(x,t) values at different times i.e. a discontinuity in W(x,t) at a box wall. If these conditions, however, are not met, then one no longer describes classical physics, but a new kind of physics (quantum physics). Thus one cannot argue that quantum mechanics follows from classical statistical Fokker-Planck-continuity equations. It may follow from these if one adds a set of extra conditions. Given these extra conditions, however, one could derive quantum mechanics without starting with classical density.
The quantum mechanical problem of a particle bouncing between two walls is formulated both in terms of the Heisenberg equations of motion and the Schrödinger equation. The reason for considering the Heisenberg equations is to understand the quantal nature of the impulsive forces. It is shown that these two formulations are compatible if, in addition to the classical impulsive forces, there are singular forces proportional to [Formula: see text] and [Formula: see text], i.e., forces of quantum origin. When these forces are added to the free-particle Hamiltonian then the total Hamiltonian is self-adjoint, and from it one can derive the boundary condition that must be imposed on the wave function. As an application of this formulation one can study the quantum mechanics of classical dynamical systems expressible as difference equations, e.g., the problem of a particle trapped between two walls moving relative to each other. The total Hamiltonian can also be used to study the question of the separability of the wave equation for such a motion.