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Quantum Electronics

Evolution of a wave packet scattered by a one-dimensional potential

А. Ж. Хачатрян, Al.G. Alexanyan, V. A. Khoetsyan, N.A. Alexanyan

Published Jun 30, 2013• 1 citation

Abstract

We consider the evolution of a wave packet that is made up of a group of the wave functions describing the stationary scattering process and tunnels through a one-dimensional potential of arbitrary form. As the main characteristics of the time difference of the tunnelling process, use is made of the propagation speed of the wave-packet maximum. We show that the known Hartman formula for the tunnelling time corresponds to the wave packet with a wavenumber-uniform spectral composition in the case, when the phase and transmission coefficient modulus dispersions are taken into account only in the linear approximation. The amplitude of the main peak of the transmitted wave intensity is proven to be independent of the tunnelling time and is determined by the transmission coefficient of the spectral component at the carrier frequency and the spectral width of the wave packet. In the limit of an infinitely wide potential barrier the amplitude of the wave-packet maximum is shown to tend to zero slower than the tunnelling time tends to its asymptotic value, i.e., indeed we deal with the paradox of an infinitely large propagation speed of a wave disturbance through the barrier.

DOI: 10.1070/qe2013v043n06abeh015002

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I show that the MacColl-Hartman effect, namely, the saturation of the group delay time of sub-barrier quantum tunneling as a function of the barrier width, comes from the saturating behavior of a more fundamental concept - the phase of the stationary wave function. The explanation of saturation is given based on the decomposition of the stationary wave function into the spectrum of wave numbers and formulation of the initial condition for the direction of propagation of the incident matter wave. It is also shown that the saturation plateau of MacColl and Hartman actually doesn't continue indefinitely, but has a finite length. After the plateau, the sub-barrier tunneling time monotonically increases with increasing width of the potential, and this applies both to the maximum's of the wave packet and to the average tunneling time.

In this work a wave process evolution problem in a field of an arbitrary one-dimensional potential is considered. We investigate the wave packets derived based on linear combinations of scattering wave functions taken with magnitudes of state indexes near to resonance tunneling case. It is shown that the wave processes with different parities of asymptotes describe a wave process where the group and phase velocities have opposite directions. The conducted consideration allows to state that the bound states can be interpreted as wave packets of the scattering wave functions to be derived under a certain way.

In this work a wave process evolution problem in a field of an arbitrary one-dimensional potential is considered. We investigate the wave packets derived based on linear combinations of scattering wave functions taken with magnitudes of state indexes near to resonance tunneling case. It is shown that the wave processes with different parities of asymptotes describe a wave process where the group and phase velocities have opposite directions. The conducted consideration allows to state that the bound states can be interpreted as wave packets of the scattering wave functions to be derived under a certain way.

A new variant of the packet analysis to solve the tunneling time problem for the so-called completed elastic scattering (a potential barrier of an arbitrary form is located in the finite spatial interval) is presented. In particular, we propose an explanation of the "paradoxes" of wave packet tunneling, and elaborate a rule enabling one to extract the tunneling times for a particle from the analysis of tunneling the corresponding wave packet. It is shown that the quantum ensembles of transmitted and reflected particles are statistically independent not only after the scattering but also before this event. We find the corresponding counterparts for the transmitted and reflected packets, which describe separately both scattering channels at early times, and define the expectation values of delay times for each channel. Besides, to describe jointly all particles at the stage when they cannot be divided into two independent subensembles, we find the scattering time taking into account broadening the packets. We derive the condition to be satisfied for the completed scattering. The times above are defined in terms of the well-known quantum-mechanical averages for the position and wave-number operators.

Abstract We examine the tunnelling process of incident Hermite-Gaussian wavepackets which feature several nodes and peaks. An analytic formula for the transmitted wave function is developed which applies to arbitrary incident wave functions and potential barriers with known transmission amplitudes. This formula permits the accurate numerical calculation of the transmitted wave packet, even when the transmission probability is extremely small. For square potential barriers, the shape of the transmitted packet, its relationship to the shape of the incident packet, the momentum of the transmitted packet, and the duration of the tunnelling process are investigated. The shape of the reconstructed transmitted wave packet is also studied for the smooth repulsive Pöschl-Teller potential.

It is shown that in the case of the one-particle one-dimensional scattering problem for a given time-independent potential, for each state of the whole quantum ensemble of identically prepared particles, there is an unique pair of (subensemble's) solutions to the Schrödinger equation, which, as we postulate, describe separately transmission and reflection: in the case of nonstationary states, for any instant of time, these functions are orthogonal and their sum describes the state of all particles; evolving with constant norms, one of them approaches at late times the transmitted wave packet and another approaches the reflected packet. Both for transmission and reflection, 1) well before and after the scattering event, the average kinetic energy of particles is the same, 2) the average starting point differs, in the general case, from that for all particles. It is shown that for reflection, in the case of symmetric potential barriers, the domain of the motion of particles is bounded by the midpoint of the barrier region. We define (exact and asymptotic) transmission and reflection times and show that the basic results of our formalism can be, in principle, checked experimentally.