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American Journal of Physics

Reflections on a Wave Packet Approach to Quantum Mechanical Barrier Penetration

Mark H. Bramhall, Barry M. Casper

Published Sep 1, 1970• 29 citations
Publisher

Abstract

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.

DOI
10.1119/1.1976566

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Routledge eBooks•May 11, 2018•0 citations
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.
In this paper, we discuss the tunneling time of a quantum particle through a rectangular barrier. The reflection and transmission times associated with the wave packets representing the particle are discussed. By using an initial Gaussian momentum distribution, we carry out a comparative analysis of the stationary phases of the incident, reflected, and transmitted wave packets leading to the reflection and transmission times $\ensuremath{\Delta}{t}_{R}$ and $\ensuremath{\Delta}{t}_{T},$ respectively. In the present treatment of this old and very known problem we take into account the deformations of the reflected and transmitted momentum distributions. These deformations produce a dependence of the reflection and transmission times on the location of the initial wave packet. In a parallel calculation, by numerically monitoring the time evolution of the system, we characterize a reflection and a transmission time. Such times agree with the ones obtained via the stationary phase method.
The reflection of wave packets from a quantum well bounded by rectangular potential barriers, and which has a tunneling transmission resonance, is discussed. Numeric solutions to Schrödinger’s time-dependent equation show that the shape of the probability function ψψ* for a reflected wave packet can be quite different from that of the incident packet. Either a symmetric double hump or an asymmetric shape are possible for the reflected ψψ*, depending on the sharpness of the resonance. These shapes are explained analytically in terms of the reflection coefficient.
We develop a new variant of the wave-packet analysis and solve the tunneling time problem for one particle. Our approach suggests an individual asymptotic description of the quantum subensembles of transmitted and reflected particles both at the final and initial stage of tunneling. We find the initial states of both subensembles, which are non-orthogonal. The latter reflects ultimately the fact that at the initial stage of tunneling it is impossible to predict whether a particle will be transmitted through or reflected off the barrier. At the same time, in this case, one can say about the to-be-transmitted and to-be-reflected subensembles of particles. We show that before the interaction of the incident packet with the barrier both the number of particles and the expectation value of the particle's momentum are constant for each subensemble. Besides, these asymptotic quantities coincide with those of the corresponding scattered subensemble. On the basis of this formalism we define individual delay times for both scattering channels. Taking into account the spreading of wave packets, we define the low bound of the scattering time to describe the whole quantum ensemble of particles. All three characteristic times are derived in terms of the expectation values of the position and momentum operators. The condition which must be fulfilled for completed scattering events is derived. We propose also the scheme of a {\it gedanken} experiment that allows one to verify our approach.
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.
A simple exact solution for the spacetime particle propagator in the presence of a one-dimensional rectangular potential is obtained. This propagator solves the Cauchy problem for the corresponding time-dependent Schrodinger equation in terms of the forward-moving (positive momentum) and backward-moving (negative momentum) components of the wavefunction. The exact formulae for the time-dependent wavefunction in different spatial regions (before, inside and after the barrier) are found. These results demonstrate that generally the forward- and backward-moving components of the wavefunction as well as their interference contribute to the probability density of finding a particle in different spacetime regions. The time of arrival to and the dwell time in the above-mentioned spatial regions are also obtained. The relative contribution of the positive- and negative-momentum components of the wavefunction as well as those of the below- and above-barrier energy regions are discussed for the case of an initial Gaussian wave packet. Numerical visualization of the probability density to find a particle in the different spacetime regions is performed and discussed in the tunneling regime.
The interaction between incident and target particles is inferred from the numbers and angular distribution of scattered particles and from their energy distribution. It is important to recognize that a scattering region, as represented for example by a rectangular barrier, is a region where, in classical terms, forces are applied to the incident particles. A full analysis of scattering events must, in principle, embody the description of a localized free particle that travels toward a center of force and is scattered by it. Quantum-mechanically, that implies a theoretical analysis in terms of packet states, with an explicit account of their change of position with time. One of the most striking effects described by quantum mechanics is the transmission of incident particles across a potential barrier whose height is greater than the particle energy. Classically, these particles lack the energy to get over the barrier, yet for quantum-mechanical systems some fraction of incident particles are transmitted.