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arXiv:1811.06054

On Scattering of 1-D Wave Packets

N. F. Berk

Published Nov 14, 2018• 1 citation
arXivHTMLPDF

Abstract

We present an exact solution to the one-dimensional (1-D) scattering-from-a-barrier problem for an incident neutron described by a wave packet. As an aid to presenting our approach, we spend some time on a basic review of how wave packets appear in standard quantum mechanics (SQM), paying attention to interpretive aspects of a working theory both familiar on the one hand while historically subject to debates, confusion, and misunderstandings on the other. Several appendices also are included to address various mathematical issues that may be helpful to some readers.

DOI
10.48550/arxiv.1811.06054

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We present an exact solution to the one-dimensional (1-D) scattering-from-a-barrier problem for an incident neutron described by a wave packet. As an aid to presenting our approach, we spend some time on a basic review of how wave packets appear in standard quantum mechanics (SQM), paying attention to interpretive aspects of a working theory both familiar on the one hand while historically subject to debates, confusion, and misunderstandings on the other. Several appendices also are included to address various mathematical issues that may be helpful to some readers.
We discuss "the plane wave approximation" to quantum mechanical scattering using simple one-dimensional examples. The central points of the paper are that (a) plane waves should be thought of as infinitely wide wave packets, and (b) the calculations of reflection and transmission probabilities RRR and TTT in standard textbook presentations involve an approximation which is almost never discussed. We argue that it should be discussed explicitly, and that doing so provides a simple and intuitively revealing alternative way to derive and understand certain formulas. Using an under-appreciated exact expression for wave packet scattering probabilities, we calculate, for two standard examples, expressions for RRR and TTT for an incident Gaussian wave packet. Comparing these results to the corresponding ones calculated using the plane wave approximation helps illuminate the domain of applicability of that approximation.
The book contains four chapters. The first one deals with elementary problems of one dimensional quantum mechanics of scalar particles, and their solution is given by much more simpler method than in standard textbooks. Attention should be paid to problems with periodic potentials. In the second chapter methods of one-dimensional quantum mechanics are applied to three-dimensional scattering problems. Here a new approach to the dynamic diffraction theory based on multiple wave scattering method is proposed. Diffraction on a crystalline plane is solved exactly, and is shown to vanish in contradiction with experimental observations. In the third chapter, all the problems of the first one are generalized to spinor particles. The most interesting questions are related to polarization of particles and rotation of polarization by means of radio-frequency fields, which create superpositions of states with different energies and speeds. The fourth chapter is devoted to fundamental questions of quantum mechanics. The theory of weak measurements is criticized. Some experiments related to wave packets and to determination of quantization axis of non polarized neutron beams are proposed.
In previous studies, by the methods of wave mechanics, of one-dimensional motion of particles in cases in which there are intervals in which the value of the potential energy function V(x)V(x)V(x) exceeds the value of the total energy EEE, attention has been confined to wave functions of the form $f(x, E)\mathrm{exp}(\ensuremath{-}\frac{2\ensuremath{\pi}\mathrm{iEt}}{h})$. In the present note wave packets are considered, instead of these trains of waves.The function V(x)V(x)V(x) is taken as follows: V(x),=0forx<0andforx>a,=V0>0for0<x<a.{V(x),=0 \mathrm{for} x<0 \mathrm{and}\mathrm{for} x>a,}{={V}_{0}>0 \mathrm{for} 0<x<a.}V(x),=0forx<0andforx>a,=V0​>0for0<x<a. A wave function is set up which initially represents a wave packet moving toward the point x=0x=0x=0 from the left. The separation of the incident packet into a reflected packet and a transmitted packet is studied. It is found that the transmitted packet appears at the point x=ax=ax=a at about the time at which the incident packet reaches the point x=0x=0x=0, so that there is no appreciable delay in the transmission of the packet through the barrier.
According to Standard Quantum Mechanics (SQM), known as the Copenhagen Interpretation, the complete description of a system of particles is provided by its wave function. However, in the de Broglie-Bohm theory of Bohmian Quantum Mechanics (BQM), the additional element which is introduced apart from the wave function is the particle position, conceived in the classical sense as pursuing a definite continuous track in space-time. In BQM formulation, depending on the configuration of the potential barrier and the energy of the packet, the particle trajectories have been shown to take distinct paths. We will consider several barrier heights and show that in a Bohmian interpretation of the problem, there is no such thing as Quantum Tunnelling.
We consider the time development of a one-dimensional Gaussian wave packet. Noting that the quantum mechanical predictions concerning position and momentum measurements of a free particle described by a Gaussian packet are consistent with the predictions one would make on the basis of a classical “Naive Model,” we investigate whether this model can account for the scattering of a particle by a rectangular barrier. Obtaining numerical solutions of the time dependent Schrödinger equation for a series of packets with varying incident momenta, we find that the Naive Model is inadequate to explain the quantum mechanical transmission. We study the relative speeds of the transmitted and reflected packets, discussing a number of possible interpretations and indicating the quantum mechanical result. Finally, we note that for certain choices of the parameters describing the packet and the barrier, the reflected packet is composed of two distinct humps.