Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and Theoretical

A time-dependent exact solution for wave-packet scattering at a rectangular barrier

Victor F. Los, Mykola ‘Nicholas’ V Los

Published Feb 20, 2012• 11 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8113/45/9/095302

Related papers

An Exact Solution of the Time-Dependent Schrödinger Equation with a Rectangular Potential for Real and Imaginary Times

Victor F. Los, Mykola ‘Nicholas’ V Los

A propagator for the one-dimensional time-dependent Schr¨odinger equation with an asymmetric rectangular potential is obtained, by using the multiple-scattering theory. It allows the consideration of the reflection and transmission processes as the scattering of a particle at the potential jump (in contrast to the conventional wave-like picture) and the account for the non-classical counterintuitive contribution of the backward-moving component of the wave packet attributed to a particle. This propagator completely resolves the corresponding time-dependent Schr¨odinger equation (defines the wave function w(x, t)) and allows the consideration of the quantum mechanical effects of a particle reflection from the potential downward step/well and a particle tunneling through the potential barrier as a function of the time. These results are related to fundamental issues such as measuring the time in quantum mechanics (tunneling time, time of arrival, dwell time). For the imaginary time, which represents an inverse temperature (t → −ih/kBT), the obtained propagator is equivalent to the density matrix for a particle that is in a heat bath and is subject to the action of a rectangular potential. This density matrix provides information about particles’ density in the different spatial areas relative to the potential location and on the quantum coherence of different particle spatial states. If one passes to the imaginary time (t → −it), the matrix element of the calculated propagator in the spatial basis provides a solution to the diffusion-like equation with a rectangular potential. The obtained exact results are presented as the integrals of elementary functions and thus allow a numerical visualization of the probability density |w(x, t)|2, the density matrix, and the solution of the diffusion-like equation. The results obtained may also be applied to spintronics due to the fact that the asymmetric (spin-dependent) rectangular potential can model the potential profile in layered magnetic nanostructures.

Ukrainian Journal of Physics•Apr 1, 2016•0 citations

Time-Dependent Scattering by an Asymmetric Spin-Dependent Rectangular Potential in Nanostructures

Victor F. Los, Luxoft Eastern Europe, 14-b Vasylkivska Str., 03040 Kyiv, Ukraine, N. V. Los

An exact time-dependent solution for the wave function (r, t) of a particle moving in the presence of an asymmetric rectangular well/barrier potential varying in one dimension is obtained by applying a novel for this problem approach using multiple scattering theory (MST) for the calculation of the space-time propagator.This approach, based on the found effective potentials localized at the potential jumps and responsible for transmission through and reflection from the considered rectangular potential, enables considering these processes from standpoint of a particle (rather than a wave).The solution describes these quantum phenomena as time dependent and is related to both the fundamental issues (such as time measurement) of quantum mechanics and the kinetic theory of nanostructures due to the fact that the considered potential can model the spin-dependent potential profile of the magnetic multilayers used in spintronics devices.It is presented in terms of integrals of elementary functions and is a sum of the forward-and backward-moving components of the wave packet.The relative contribution of these components and their interference as well as of the potential asymmetry to both the probability density (x, t) 2 and the particle dwell time is considered and numerically visualized for the narrow and broad energy (momentum) distributions of the initial Gaussian wave packet.As shown, in the case of a broad initial wave packet, the quantum-mechanical counterintuitive

METALLOFIZIKA I NOVEISHIE TEKHNOLOGII•Mar 23, 2016•0 citations

An exact solution of the time-dependent Schrödinger equation with a rectangular potential for real and imaginary time

Victor F. Los, Mykola ‘Nicholas’ V Los

A propagator for the one dimensional time-dependent Schrödinger equation with an asymmetric rectangular potential is obtained using the multiple-scattering theory approach. It allows for the consideration of the reflection and transmission processes as the particle scattering at the potential jump (in contrast to the conventional wave-like picture) and for accounting for the nonclassical counterintuitive contribution of the backward-moving component of the wave packet attributed to the particle. This propagator completely resolves the corresponding time-dependent Schrödinger equation (defines the wave function $ψ(x,t)$) and allows for considering the quantum mechanical effects of a particle reflection from the potential downward step/well and a particle tunneling through the potential barrier as a function of time. These results are related to fundamental issues such as measuring time in quantum mechanics (tunneling time, time of arrival, dwell time). For imaginary time, which represents an inverse temperature $(t\rightarrow-i{\hbar}/k_{B}T)$, the obtained propagator is equivalent to the density matrix for a particle that is in a heat bath and is subject to a rectangular potential. If one shifts to imaginary time $(t\rightarrow-it)$, the matrix element of the calculated propagator in the spatial basis provides a solution to the diffusion-like equation with a rectangular potential. The obtained exact results are presented as the integrals from elementary functions and thus allow for a numerical visualization of the probability density $|ψ(x,t)|^2$, the density matrix and the solution of the diffusion-like equation. The results obtained may also be useful for spintronics applications due to the fact that the asymmetric (spin-dependent) rectangular potential can model the potential profile in layered magnetic nanostructures.

arXiv:1509.02809•Sep 9, 2015•0 citations

Wave packet scattering from time-varying potential barriers in one dimension

R. M. Dimeo

We discuss a solution of the time-dependent Schrödinger equation that incorporates absorbing boundary conditions and a method for extracting the reflection and transmission probabilities for wave packets interacting with time-dependent potential barriers. We apply the method to a rectangular barrier that moves with constant velocity, an oscillating rectangular barrier, a locally periodic barrier with an amplitude modulated by a traveling wave, and a locally periodic potential with an amplitude modulated by a standing wave. Visualizations of the reflection phenomena are presented with an emphasis on understanding these systems from their dynamics. Applications to non-stationary neutron optics experiments are discussed briefly.

American Journal of Physics•Jan 23, 2014•11 citations

Reflections on a Wave Packet Approach to Quantum Mechanical Barrier Penetration

Mark H. Bramhall, Barry M. Casper

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.

American Journal of Physics•Sep 1, 1970•29 citations

Dynamic transmission and reflection phenomena for a time-dependent rectangular potential

D. L. Haavig, Ronald G. Reifenberger

The time-dependent Schr\"odinger equation is solved for a one-dimensional rectangular potential whose height changes sinusoidally in time. By using standard wave-matching techniques, dynamic transmission and reflection probabilities are found. It is shown that the flux transmitted through the barrier region is modulated in time at integer multiples of ${\ensuremath{\omega}}_{b}$, the angular frequency at which the potential oscillates. The results obtained for a delocalized plane wave incident on the barrier are extended by considering the interaction of a Gaussian wave packet with the oscillating potential barrier. The modulation of the transmitted and reflected probability density by the oscillating barrier is clearly demonstrated by a series of calculations performed to illustrate this interaction. These calculations also show that the probability density leaving the barrier region is composed of individual wave packets having different average velocities, a result which is a direct consequence of the transitions induced by the time-dependent potential barrier under consideration. Calculations are also performed which show that the flux transmitted across the potential barrier depends on the phase of the oscillating barrier at the time the incident Gaussian wave packet begins to interact with it. The results of these model calculations suggest that experiments properly designed to measure the transmission probability of a quantum particle as it escapes from a metal surface are likely to provide information about the time formation of the dynamic image potential.

Physical review. B, Condensed matter•Dec 15, 1982•47 citations
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.
A propagator for the one-dimensional time-dependent Schr¨odinger equation with an asymmetric rectangular potential is obtained, by using the multiple-scattering theory. It allows the consideration of the reflection and transmission processes as the scattering of a particle at the potential jump (in contrast to the conventional wave-like picture) and the account for the non-classical counterintuitive contribution of the backward-moving component of the wave packet attributed to a particle. This propagator completely resolves the corresponding time-dependent Schr¨odinger equation (defines the wave function w(x, t)) and allows the consideration of the quantum mechanical effects of a particle reflection from the potential downward step/well and a particle tunneling through the potential barrier as a function of the time. These results are related to fundamental issues such as measuring the time in quantum mechanics (tunneling time, time of arrival, dwell time). For the imaginary time, which represents an inverse temperature (t → −ih/kBT), the obtained propagator is equivalent to the density matrix for a particle that is in a heat bath and is subject to the action of a rectangular potential. This density matrix provides information about particles’ density in the different spatial areas relative to the potential location and on the quantum coherence of different particle spatial states. If one passes to the imaginary time (t → −it), the matrix element of the calculated propagator in the spatial basis provides a solution to the diffusion-like equation with a rectangular potential. The obtained exact results are presented as the integrals of elementary functions and thus allow a numerical visualization of the probability density |w(x, t)|2, the density matrix, and the solution of the diffusion-like equation. The results obtained may also be applied to spintronics due to the fact that the asymmetric (spin-dependent) rectangular potential can model the potential profile in layered magnetic nanostructures.
An exact time-dependent solution for the wave function (r, t) of a particle moving in the presence of an asymmetric rectangular well/barrier potential varying in one dimension is obtained by applying a novel for this problem approach using multiple scattering theory (MST) for the calculation of the space-time propagator.This approach, based on the found effective potentials localized at the potential jumps and responsible for transmission through and reflection from the considered rectangular potential, enables considering these processes from standpoint of a particle (rather than a wave).The solution describes these quantum phenomena as time dependent and is related to both the fundamental issues (such as time measurement) of quantum mechanics and the kinetic theory of nanostructures due to the fact that the considered potential can model the spin-dependent potential profile of the magnetic multilayers used in spintronics devices.It is presented in terms of integrals of elementary functions and is a sum of the forward-and backward-moving components of the wave packet.The relative contribution of these components and their interference as well as of the potential asymmetry to both the probability density (x, t) 2 and the particle dwell time is considered and numerically visualized for the narrow and broad energy (momentum) distributions of the initial Gaussian wave packet.As shown, in the case of a broad initial wave packet, the quantum-mechanical counterintuitive
A propagator for the one dimensional time-dependent Schrödinger equation with an asymmetric rectangular potential is obtained using the multiple-scattering theory approach. It allows for the consideration of the reflection and transmission processes as the particle scattering at the potential jump (in contrast to the conventional wave-like picture) and for accounting for the nonclassical counterintuitive contribution of the backward-moving component of the wave packet attributed to the particle. This propagator completely resolves the corresponding time-dependent Schrödinger equation (defines the wave function ψ(x,t)ψ(x,t)ψ(x,t)) and allows for considering the quantum mechanical effects of a particle reflection from the potential downward step/well and a particle tunneling through the potential barrier as a function of time. These results are related to fundamental issues such as measuring time in quantum mechanics (tunneling time, time of arrival, dwell time). For imaginary time, which represents an inverse temperature (t→−iℏ/kBT)(t\rightarrow-i{\hbar}/k_{B}T)(t→−iℏ/kB​T), the obtained propagator is equivalent to the density matrix for a particle that is in a heat bath and is subject to a rectangular potential. If one shifts to imaginary time (t→−it)(t\rightarrow-it)(t→−it), the matrix element of the calculated propagator in the spatial basis provides a solution to the diffusion-like equation with a rectangular potential. The obtained exact results are presented as the integrals from elementary functions and thus allow for a numerical visualization of the probability density ∣ψ(x,t)∣2|ψ(x,t)|^2∣ψ(x,t)∣2, the density matrix and the solution of the diffusion-like equation. The results obtained may also be useful for spintronics applications due to the fact that the asymmetric (spin-dependent) rectangular potential can model the potential profile in layered magnetic nanostructures.
We discuss a solution of the time-dependent Schrödinger equation that incorporates absorbing boundary conditions and a method for extracting the reflection and transmission probabilities for wave packets interacting with time-dependent potential barriers. We apply the method to a rectangular barrier that moves with constant velocity, an oscillating rectangular barrier, a locally periodic barrier with an amplitude modulated by a traveling wave, and a locally periodic potential with an amplitude modulated by a standing wave. Visualizations of the reflection phenomena are presented with an emphasis on understanding these systems from their dynamics. Applications to non-stationary neutron optics experiments are discussed briefly.
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.
The time-dependent Schr\"odinger equation is solved for a one-dimensional rectangular potential whose height changes sinusoidally in time. By using standard wave-matching techniques, dynamic transmission and reflection probabilities are found. It is shown that the flux transmitted through the barrier region is modulated in time at integer multiples of ${\ensuremath{\omega}}_{b}$, the angular frequency at which the potential oscillates. The results obtained for a delocalized plane wave incident on the barrier are extended by considering the interaction of a Gaussian wave packet with the oscillating potential barrier. The modulation of the transmitted and reflected probability density by the oscillating barrier is clearly demonstrated by a series of calculations performed to illustrate this interaction. These calculations also show that the probability density leaving the barrier region is composed of individual wave packets having different average velocities, a result which is a direct consequence of the transitions induced by the time-dependent potential barrier under consideration. Calculations are also performed which show that the flux transmitted across the potential barrier depends on the phase of the oscillating barrier at the time the incident Gaussian wave packet begins to interact with it. The results of these model calculations suggest that experiments properly designed to measure the transmission probability of a quantum particle as it escapes from a metal surface are likely to provide information about the time formation of the dynamic image potential.