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

AIP conference proceedings

Quantum wave package propagation through a potential step

Maxim Belov, George G. Krylov

Published Jan 1, 2012• 0 citations
Publisher

Abstract

One-dimensional wave package propagation through the potential step is considered. Several approaches are tested for the ability to obtain transmission coefficients. Consideration includes direct numeric solution of the time-dependent Schrödinger Equation with the step potential, usage of the spectral decomposition of the propagator. An asymptotic formula is proposed for calculation of the transmission coefficient, which produces accurate results and is much easier to apply then the others considered approaches.

DOI
10.1063/1.4745571

Related papers

Transmission Coefficients for One-stepped and Double-stepped Potential Barriers

Hao Yu-hua

The transfer matrix and the formulae of transmission coefficients of particle tunneling through one-dimensional multi-stepped potential barriers are studied by Erwin schrodinger equation.The results of transmission coefficient and the curves of transmission coefficients varying with the energy of incident particles for one-stepped potential barriers and double-stepped potential barriers are calculated.

Journal of Changzhou Institute of Technology•Jan 1, 2006•0 citations

Numerical Solution Of The Time-Dependent Schrodinger equation

Bernardine Renaldo Wong, Swee-Ping Chia, Kurunathan Ratnavelu +1 more

We implement a method to numerically solve the Time‐Dependent Schrödinger Equation (TDSE) in one dimension. The time‐dependence is modeled using the Crank‐Nicholson formula. The discrete transparent boundary conditions are employed using a Numerov approximation formula. A normalized Gaussian wavefunction is propagated across a potential step and across a potential well of modified Poschl‐Teller form, respectively. Preliminary results will be presented.

AIP conference proceedings•Jan 1, 2009•2 citations

Analyses of reflection and transmission at moving potential step

Samura, Toshiharu, Masato Ohmukai

The reflection and transmission of wave functions at a potential step is a well-known issue in a textbook of quantum mechanics. We studied the reflection and transmission characteristics analytically when the potential step is moving at a constant velocity $v$ in the same direction as an incident wave function by means of solving the time-dependent Schrödinger equation. As for an infinite potential step, it is known that group velocity is the same as the moving velocity of the potential step. We found two interesting results when the potential step has a finite height of $V_0$. The transmission occurs when the kinetic energy of incident wave function is larger than the effective potential hight of $(\sqrt{\frac{m}{2}}v + \sqrt{V_0})^2$. The other result is that the reflectivity depends on $x$, which derives from the interference between the incident and the reflected wave functions.

arXiv:quant-ph/0411113•Nov 16, 2004•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

Quantum Mechanical Potential Step Functions, Barriers, Wells and the Tunneling Effect

Gilbert A. Ibitola, Olanrewaju Ajanaku

In this paper, use is made of the tools of analytical mechanics and the concept of operators to obtain the time-independent and time-dependent Schrodinger wave equations for quantum mechanical systems. Derivations are embarked upon of expressions for reflection and transmission coefficients for a particle of mass m as well as of energy E moving under different potential set-ups across step functions, barriers and well functions. The tunneling effect is then discussed. The transmission probability equation obtained in this research has been observed to be more accurate than the transmission probability expression deduced by some researchers in 2014 for a tunneling barrier. This research work finds applications in nuclear magnetic resonance imaging systems, synchrotrons, gyrators, accelerators, and in electrodynamics.

World Journal of Applied Physics•Dec 22, 2016•0 citations

Transmission probability and traversal time in scattering by a one-dimensional potential of finite range

John E. G. Farina

Scattering by a one-dimensional potential V=V(x) which is constant outside (-a, a) (a>0) is investigated. Exact expressions for the transmission and reflection coefficients are obtained in terms of f 0 (a), f 1 (a), f' O (a) and f' 1 (a), where f 0 and f 1 are the solutions in (-a, a) satisfying the boundary conditions f 0 (-a)=1, f 0 '(-a)=0, f 1 (-a)=0 and f 1 '(-a)=1. These expressions are used to show that the first-order WKB approximation conserves particles. Numerical results are obtained from these expressions for the transmission and reflection probabilities by (i) using the WKB formulae and (ii) solving the time-independent Schrodinger equation numerically in (-a, a). The time T for the centre of a wavepacket to traverse the interval (-a, a) is also obtained in terms of f 0 (a), f 1 (a), f 0 '(a) and f 1 '(a), and some of its properties studied. In particular, it is shown that T takes the classical form in the classical limit.

Journal of Physics A Mathematical and General•Jun 7, 1988•5 citations
One-dimensional wave package propagation through the potential step is considered. Several approaches are tested for the ability to obtain transmission coefficients. Consideration includes direct numeric solution of the time-dependent Schrödinger Equation with the step potential, usage of the spectral decomposition of the propagator. An asymptotic formula is proposed for calculation of the transmission coefficient, which produces accurate results and is much easier to apply then the others considered approaches.
The reflection and transmission of wave functions at a potential step is a well-known issue in a textbook of quantum mechanics. We studied the reflection and transmission characteristics analytically when the potential step is moving at a constant velocity vvv in the same direction as an incident wave function by means of solving the time-dependent Schrödinger equation. As for an infinite potential step, it is known that group velocity is the same as the moving velocity of the potential step. We found two interesting results when the potential step has a finite height of V0V_0V0​. The transmission occurs when the kinetic energy of incident wave function is larger than the effective potential hight of (m2v+V0)2(\sqrt{\frac{m}{2}}v + \sqrt{V_0})^2(2m​​v+V0​​)2. The other result is that the reflectivity depends on xxx, which derives from the interference between the incident and the reflected wave functions.
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.
In this paper, use is made of the tools of analytical mechanics and the concept of operators to obtain the time-independent and time-dependent Schrodinger wave equations for quantum mechanical systems. Derivations are embarked upon of expressions for reflection and transmission coefficients for a particle of mass m as well as of energy E moving under different potential set-ups across step functions, barriers and well functions. The tunneling effect is then discussed. The transmission probability equation obtained in this research has been observed to be more accurate than the transmission probability expression deduced by some researchers in 2014 for a tunneling barrier. This research work finds applications in nuclear magnetic resonance imaging systems, synchrotrons, gyrators, accelerators, and in electrodynamics.
Scattering by a one-dimensional potential V=V(x) which is constant outside (-a, a) (a>0) is investigated. Exact expressions for the transmission and reflection coefficients are obtained in terms of f 0 (a), f 1 (a), f' O (a) and f' 1 (a), where f 0 and f 1 are the solutions in (-a, a) satisfying the boundary conditions f 0 (-a)=1, f 0 '(-a)=0, f 1 (-a)=0 and f 1 '(-a)=1. These expressions are used to show that the first-order WKB approximation conserves particles. Numerical results are obtained from these expressions for the transmission and reflection probabilities by (i) using the WKB formulae and (ii) solving the time-independent Schrodinger equation numerically in (-a, a). The time T for the centre of a wavepacket to traverse the interval (-a, a) is also obtained in terms of f 0 (a), f 1 (a), f 0 '(a) and f 1 '(a), and some of its properties studied. In particular, it is shown that T takes the classical form in the classical limit.