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American Journal of Physics

Effect of discontinuities in the chopped wave state upon momentum distribution

Cathleen A. Brenner, Orville L. Brill

Published Jun 1, 1988• 0 citations
Publisher

Abstract

The effects of discontinuities in the zeroth and first derivatives of the Schrödinger wavefunction upon the momentum distribution of a particle are illustrated for the chopped wave state. Smoothing the discontinuities in the chopped wave state by use of a modulated Gaussian results in a single continuous, differentiable wavefunction which, in the corresponding limits, permits proper mathematical treatment of the uncertainty in momentum for four common cases: the chopped wave, the free particle, the Gaussian packet, and the localized particle.

DOI
10.1119/1.15768

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The effects of discontinuities in the zeroth and first derivatives of the Schrödinger wavefunction upon the momentum distribution of a particle are illustrated for the chopped wave state. Smoothing the discontinuities in the chopped wave state by use of a modulated Gaussian results in a single continuous, differentiable wavefunction which, in the corresponding limits, permits proper mathematical treatment of the uncertainty in momentum for four common cases: the chopped wave, the free particle, the Gaussian packet, and the localized particle.
The method of convolutions is used to form smoothed functions from approximate wave functions which are discontinuous or have discontinuous first derivatives. As a parameter ε in the smoothed function decreases, the smoothed function approaches the approximate wave function. The expectation values for physical properties corresponding to the approximate wave function are defined to be the limit as ε approaches zero of the expectation values corresponding to the smoothed function. It is found that if the approximate wave function is discontinuous, the corresponding expectation value for the kinetic energy is infinite. Therefore, it seems unlikely that discontinuous approximate wave functions can ever be useful. However, if the approximate wave function is continuous but has a discontinuity in its first derivative, then, as a result of the discontinuity, there is a contribution δT̄12 to the expectation value of the kinetic energy. For a one-dimensional problem δT̄12=−(ℏ2/2m)ψ*(0) [ψ2′(0)−ψ1′(0)].Here ψ(0) is the value of the approximate wave function at the point zero where the discontinuity in its first derivative occurs, and ψ1′(0) and ψ2′(0) are the first derivatives of ψ taken from the left and from the right, respectively, at this point. Similarly, for an N-dimensional problem having a surface S12=0 over which the first derivatives of the approximate wave function are discontinuous, δT̄12= ∫ surfaceψ*[(∂/∂n) (ψ2−ψ1)]dS12.Here the ∂/∂n is the normal derivative with the normal pointed from region 1 towards region 2.
In quantum theory particles are represented as wave packets.Shock wave analysis of quantum equations of motion shows that wave function representation in general and wave packet description in particular contain discontinuities due to a non-zero quantum force corresponding to quantum potential.The quantum force causes wave packet dispersion which results in the intersection of characteristic curves developing a shock discontinuity.Since quantum force vanishes for localized quantum density waves [1], it is thus established that localized quantum density waves form the only class of continuous wave representation of particles in quantum theory.
We analyze the time-behaviour of the Gaussian wave packet of a quantum particle moving in the field of a linear force, undergoing a continuous nondemolition observation of its position. In particular, the time-dependence of position and momentum dispersions, as well as position-momentum uncertainty is investigated. The discussion includes the case of a free particle and initially coherent state of a harmonic oscillator. It is found that in the beginning of observation the momentum dispersion and position-momentum uncertainty always increase in contrast to the position dispersion. It is also proved that all considered functions asymptotically take the form of harmonic oscillations, decaying to finite values. For initially coherent states, the asymptotic bounds for all considered functions are given.
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.