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

European Journal of Physics

Two-dimensional wave packets through a one-dimensional quantum barrier

Ph. Grossel, Françoise Depasse

Published Dec 30, 2004• 1 citation
Publisher

Abstract

We describe a two-dimensional wave packet going through a one-dimensional quantum barrier by using the calculation of a rich basis of stationary states, evaluated in the frame of the powerful transfer matrix method. Application to a specific case—a double resonant barrier—enlightens the simplicity and the clarity of the method, furnishing at once each stationary state and the movie of the wave packet.

DOI
10.1088/0143-0807/26/1/017

Related papers

A time-dependent study of resonant tunneling through a double barrier

Andrew P. Stamp, Gregory C. McIntosh

The resonant tunneling of a wave packet is studied, for a one-dimensional double barrier, by solving the time-dependent Schrödinger equation. The transmission is determined as a function of the spatial width of the incident wave packet. It is shown that if the double barrier acts as an energy filter, then the spatial width of the transmitted wave packet is increased, due to the trapping of the particle between the two barriers for a time, ℏ/Γ, where Γ is the width of the resonance. The reflected wave packet consists of two components: one corresponding to reflection from the first barrier and the other due to the particle being trapped between the barriers for a time ℏ/Γ.

American Journal of Physics•Mar 1, 1996•19 citations

Reflection and transmission behaviour of a particle in a resonant tunnelling barrier

Ph. Grossel, Françoise Depasse, J. M. Vigoureux

Stationary states are used to describe the transmission and the reflection of a wave packet through a resonant tunnelling barrier. The transmitted and the reflected spectra are studied and different links between numerical evaluations and the important characteristics of the resonant regime are underlined. The differences between the behaviours of wide and narrow near-resonant wave packets are delineated: two time scales appear in the case of quasi-bound states.

Journal of Physics A Mathematical and General•Nov 7, 2002•11 citations

Time delay in tunnelling through a potential barrier

J R Fletcher

A simple method is proposed for calculation of the time taken for a particle to tunnel through a potential barrier. The method is applicable to long or short wave packets. For short wave packets the time is reduced below the results given by the stationary phase method.

Journal of Physics C Solid State Physics•Jan 20, 1985•61 citations

Towards time-dependent wave packet tunnelling via complex Riccati equations

Moise Bonilla–Licea, Dieter Schuch

Abstract Since the discovery of tunnelling in the early days of quantum mechanics, there has been considerable interest in finding an effective method to determine the transmission of wave packets through a potential barrier. In this work, we combine the transfer matrix method, known from stationary tunnelling problems, with the construction of Gaussian wave packets via superposition of stationary states with the knowledge about the dynamics of generalized Gaussian-shaped coherent states in terms of complex Riccati equations. In the low-energy regime, this leads to an approximate analytical expression for the transmission probability and its dynamics. Extension to more general potential shapes appears straightforward.

Journal of Physics Conference Series•Aug 1, 2020•1 citation

Transfer-Matrix Formalism of One-Dimensional Potential Barrier Problems

Tao Cai

The paper introduces a simplified and exact transfer matrix formalism.That can be used to study a series of one dimensional potential barrier problems with only two general purpose matrices one to propagate a wave function over a region of constant potential,the other to take a wave function over a discontinuity in a potential.Not only are standard results recovered for one potential barrier problem with this technique,but it is easy to extend this formalism into two,even more many potential barier cases,which facilitates wide ranging explorations of one dimensional quantum mechanics.

Journal of Sichuan Teachers College•Jan 1, 2000•0 citations

2重障壁による台形波束の共鳴トンネル現象

Teruaki Minagawa, Yuko TATSUMI, Miki INOUE

2重障壁ポテンシャルに障壁よりも低いエネルギーをもつ台形波束を入射して共鳴トンネル現象が観測されるか調べた。時間を含む1次元シュレディンガー方程式を数値的に解き,パソコン画面上で波束の運動を観察した。共鳴は,2重障壁の間に波束が長時間にわたって多量にたまることによって識別できた。また,波束をフーリエ変換して得た波数分布から透過波と反射波の割合をみることによって,波束が2重障壁をトンネル透過する程度が判別できた。2重障壁の固有の量子エネルギー準位に対応して台形波束が2重障壁を共鳴的にトンネル透過することが,いろいろな条件の場合について,鮮明に観測された。その結果,2重障壁は特定のエネルギーを持つ波束すなわち粒子のみを選択的に透過させる優れたフィルターの役割をすることがわかった。|A resonance tunnel effect was investigated by leading a wave-packet of trapezoid shape into double potential barriers. The time-dependent Schrödinger equation was numerically solved, and the motion of a wave packet was observed on the screen of a personal computer. The state that a wave packet is resonant with double potential barriers showed a distinct phenomenon that a large part of a wave packet is stored between two barriers for a long time. Tunneling of a wave packet was clearly recognized by comparing an amount of the transmitted wave with that of the reflected one in the distribution of wave vectors that was obtained by the Fourier transformation of a wave packet. A resonance tunnel effect was demonstrated for various kinds of double potential barriers. It was showed that a couple of double potential barriers functions as an excellent filter that allows particles with specified energies to pass through.

•Aug 31, 1999•0 citations
We describe a two-dimensional wave packet going through a one-dimensional quantum barrier by using the calculation of a rich basis of stationary states, evaluated in the frame of the powerful transfer matrix method. Application to a specific case—a double resonant barrier—enlightens the simplicity and the clarity of the method, furnishing at once each stationary state and the movie of the wave packet.
The resonant tunneling of a wave packet is studied, for a one-dimensional double barrier, by solving the time-dependent Schrödinger equation. The transmission is determined as a function of the spatial width of the incident wave packet. It is shown that if the double barrier acts as an energy filter, then the spatial width of the transmitted wave packet is increased, due to the trapping of the particle between the two barriers for a time, ℏ/Γ, where Γ is the width of the resonance. The reflected wave packet consists of two components: one corresponding to reflection from the first barrier and the other due to the particle being trapped between the barriers for a time ℏ/Γ.
Abstract Since the discovery of tunnelling in the early days of quantum mechanics, there has been considerable interest in finding an effective method to determine the transmission of wave packets through a potential barrier. In this work, we combine the transfer matrix method, known from stationary tunnelling problems, with the construction of Gaussian wave packets via superposition of stationary states with the knowledge about the dynamics of generalized Gaussian-shaped coherent states in terms of complex Riccati equations. In the low-energy regime, this leads to an approximate analytical expression for the transmission probability and its dynamics. Extension to more general potential shapes appears straightforward.
The paper introduces a simplified and exact transfer matrix formalism.That can be used to study a series of one dimensional potential barrier problems with only two general purpose matrices one to propagate a wave function over a region of constant potential,the other to take a wave function over a discontinuity in a potential.Not only are standard results recovered for one potential barrier problem with this technique,but it is easy to extend this formalism into two,even more many potential barier cases,which facilitates wide ranging explorations of one dimensional quantum mechanics.
2重障壁ポテンシャルに障壁よりも低いエネルギーをもつ台形波束を入射して共鳴トンネル現象が観測されるか調べた。時間を含む1次元シュレディンガー方程式を数値的に解き,パソコン画面上で波束の運動を観察した。共鳴は,2重障壁の間に波束が長時間にわたって多量にたまることによって識別できた。また,波束をフーリエ変換して得た波数分布から透過波と反射波の割合をみることによって,波束が2重障壁をトンネル透過する程度が判別できた。2重障壁の固有の量子エネルギー準位に対応して台形波束が2重障壁を共鳴的にトンネル透過することが,いろいろな条件の場合について,鮮明に観測された。その結果,2重障壁は特定のエネルギーを持つ波束すなわち粒子のみを選択的に透過させる優れたフィルターの役割をすることがわかった。|A resonance tunnel effect was investigated by leading a wave-packet of trapezoid shape into double potential barriers. The time-dependent Schrödinger equation was numerically solved, and the motion of a wave packet was observed on the screen of a personal computer. The state that a wave packet is resonant with double potential barriers showed a distinct phenomenon that a large part of a wave packet is stored between two barriers for a long time. Tunneling of a wave packet was clearly recognized by comparing an amount of the transmitted wave with that of the reflected one in the distribution of wave vectors that was obtained by the Fourier transformation of a wave packet. A resonance tunnel effect was demonstrated for various kinds of double potential barriers. It was showed that a couple of double potential barriers functions as an excellent filter that allows particles with specified energies to pass through.