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

American Journal of Physics

Tunneling between Two Moving Wells in the Adiabatic Approximation

Philip A. Deutchman, David C. Koelsch

Published Sep 1, 1974• 1 citation
Publisher

Abstract

A solution in the adiabatic approximation of the Born-Oppenheimer type to Schrödinger's time-dependent equation in one dimension for a model three-body system is presented. The model problem for this system is a particle of mass m interacting with two attractive square wells that are receding at constant relative speed. Graphs of the time development of the probability distribution for the tunneling particle are presented for various initial conditions. The series of graphical solutions is useful as a visual aid in understanding the time development of a simplified three-body problem and the results of the model problem are used as a basis for understanding a nuclear reaction leading to three bodies in the final state.

DOI
10.1119/1.1987828

Related papers

Quantum dynamics using the time-dependent Schrödinger equation

Abraham Nitzan

Abstract This chapter the reviews essential concepts and methods of time-dependent quantum mechanics. (1) Formal solutions of the time-dependent Schrödinger equation. (2) The two-level system. (3) Interacting systems: The time-dependent self-consistent field approximation and the Ehrenfest approximation. (4) The Born–Oppenheimer approximation and nuclear potential surfaces. (5) Time-dependent Hamiltonians: A two-level system in a time-dependent field. (6) Non-adiabatic dynamics: The surface-hopping method. (7) Expressing the time evolution in terms of the Green operator. (8) Representations (the Schrödinger and Heisenberg representations, the interaction representation). (9) Time-dependent perturbation theory. (10) Quantum dynamics of the free particles. (11) Quantum dynamics of the harmonic oscillator. (12) The raising/lowering operators formalism. (13) Tunneling.

•Jun 14, 2024•1 citation

Infinite Square-Well Potential with a Moving Wall

S.W. Doescher, Melvin H. Rice

The problem of a particle in a one-dimensional infinite square-well potential with one wall moving at constant velocity is treated by means of a complete set of functions which are exact solutions of the time-dependent Schrödinger equation. Comparison is made with a first-order perturbation treatment, and numerical results are presented for a particle initially in the ground state.

American Journal of Physics•Dec 1, 1969•121 citations

Approximations of time-dependent phenomena in quantum mechanics: adiabatic versus sudden processes

S V Melnichuk, W. van Dijk, Yukihisa Nogami

By means of a one-dimensional model of a particle in an infinite square-well potential with one wall moving at a constant speed, we examine aspects of time-dependent phenomena in quantum mechanics such as adiabatic and sudden processes. The particle is assumed to be initially in the ground state of the potential with its initial width. The time dependence of the wavefunction of the particle in the well is generally more complicated when the potential well is compressed than when it is expanded. We are particularly interested in the case in which the potential well is suddenly compressed. The so-called sudden approximation is not applicable in this case. We also study the energy of the particle in the changing well as a function of time for expansion and contraction as well as for expansion followed by contraction and vice versa.

European Journal of Physics•Apr 5, 2005•3 citations

Time dependence and line shape of spontaneous quantum tunneling

B. U. Felderhof

The time dependence of spontaneous quantum tunneling of a particle from a potential well through a barrier in one or three dimensions is studied on the basis of the solution of the time-dependent Schrödinger equation as an initial-value problem by the method of Laplace transform. The wavefunction in the outer space is expressed in terms of special functions known from the theory of classical diffusion. A concise expression is derived for the probability current into the outer space. At long times the current oscillates about zero before decaying with a t −4 long-time tail. A comparison is made with the solution obtained by an expansion of the initial wavefunction in terms of the complete set of scattering states and bound states.

Journal of Physics A Mathematical and Theoretical•Oct 7, 2008•2 citations

Approximations of time-dependent phenomena in quantum mechanics: adiabatic versus sudden processes

S V Melnichuk, W. van Dijk, Yukihisa Nogami

By means of a one-dimensional model of a particle in an infinite square-well potential with one wall moving at a constant speed, we examine aspects of time-dependent phenomena in quantum mechanics such as adiabatic and sudden processes. The particle is assumed to be initially in the ground state of the potential with its initial width. The time dependence of the wavefunction of the particle in the well is generally more complicated when the potential well is compressed than when it is expanded. We are particularly interested in the case in which the potential well is suddenly compressed. The so-called sudden approximation is not applicable in this case. We also study the energy of the particle in the changing well as a function of time for expansion and contraction as well as for expansion followed by contraction and vice versa.

European Journal of Physics•Dec 15, 2004•6 citations

Applications of Schrödinger Equation: Potential (Quantum) Wells

Z. Ficek

The wave nature of particles plays an important role in their physical properties that, for example, particles confined into a small bounded area can have only particular discrete energies. This chapter explains how to solve the time-independent Schrodinger equation. It considers four cases of a one-dimensional motion of particles confined in potentials rapidly changing with x : infinite potential quantum well; a potential step; square-well potential; and tunneling through a potential barrier. The chapter helps to learn the characteristic properties of solutions to this equation. It shows the differences between the predictions of quantum mechanics and classical physics. Quantum tunneling is important in the understanding of a number of physical phenomena such as thermonuclear reactions and conduction in metals and semiconductors. The most advanced application of quantum tunneling is the scanning tunneling microscope. The details of the calculations are left for the readers as a tutorial problem.

•Mar 3, 2017•0 citations
A solution in the adiabatic approximation of the Born-Oppenheimer type to Schrödinger's time-dependent equation in one dimension for a model three-body system is presented. The model problem for this system is a particle of mass m interacting with two attractive square wells that are receding at constant relative speed. Graphs of the time development of the probability distribution for the tunneling particle are presented for various initial conditions. The series of graphical solutions is useful as a visual aid in understanding the time development of a simplified three-body problem and the results of the model problem are used as a basis for understanding a nuclear reaction leading to three bodies in the final state.
Abstract This chapter the reviews essential concepts and methods of time-dependent quantum mechanics. (1) Formal solutions of the time-dependent Schrödinger equation. (2) The two-level system. (3) Interacting systems: The time-dependent self-consistent field approximation and the Ehrenfest approximation. (4) The Born–Oppenheimer approximation and nuclear potential surfaces. (5) Time-dependent Hamiltonians: A two-level system in a time-dependent field. (6) Non-adiabatic dynamics: The surface-hopping method. (7) Expressing the time evolution in terms of the Green operator. (8) Representations (the Schrödinger and Heisenberg representations, the interaction representation). (9) Time-dependent perturbation theory. (10) Quantum dynamics of the free particles. (11) Quantum dynamics of the harmonic oscillator. (12) The raising/lowering operators formalism. (13) Tunneling.
By means of a one-dimensional model of a particle in an infinite square-well potential with one wall moving at a constant speed, we examine aspects of time-dependent phenomena in quantum mechanics such as adiabatic and sudden processes. The particle is assumed to be initially in the ground state of the potential with its initial width. The time dependence of the wavefunction of the particle in the well is generally more complicated when the potential well is compressed than when it is expanded. We are particularly interested in the case in which the potential well is suddenly compressed. The so-called sudden approximation is not applicable in this case. We also study the energy of the particle in the changing well as a function of time for expansion and contraction as well as for expansion followed by contraction and vice versa.
The time dependence of spontaneous quantum tunneling of a particle from a potential well through a barrier in one or three dimensions is studied on the basis of the solution of the time-dependent Schrödinger equation as an initial-value problem by the method of Laplace transform. The wavefunction in the outer space is expressed in terms of special functions known from the theory of classical diffusion. A concise expression is derived for the probability current into the outer space. At long times the current oscillates about zero before decaying with a t −4 long-time tail. A comparison is made with the solution obtained by an expansion of the initial wavefunction in terms of the complete set of scattering states and bound states.
By means of a one-dimensional model of a particle in an infinite square-well potential with one wall moving at a constant speed, we examine aspects of time-dependent phenomena in quantum mechanics such as adiabatic and sudden processes. The particle is assumed to be initially in the ground state of the potential with its initial width. The time dependence of the wavefunction of the particle in the well is generally more complicated when the potential well is compressed than when it is expanded. We are particularly interested in the case in which the potential well is suddenly compressed. The so-called sudden approximation is not applicable in this case. We also study the energy of the particle in the changing well as a function of time for expansion and contraction as well as for expansion followed by contraction and vice versa.
The wave nature of particles plays an important role in their physical properties that, for example, particles confined into a small bounded area can have only particular discrete energies. This chapter explains how to solve the time-independent Schrodinger equation. It considers four cases of a one-dimensional motion of particles confined in potentials rapidly changing with x : infinite potential quantum well; a potential step; square-well potential; and tunneling through a potential barrier. The chapter helps to learn the characteristic properties of solutions to this equation. It shows the differences between the predictions of quantum mechanics and classical physics. Quantum tunneling is important in the understanding of a number of physical phenomena such as thermonuclear reactions and conduction in metals and semiconductors. The most advanced application of quantum tunneling is the scanning tunneling microscope. The details of the calculations are left for the readers as a tutorial problem.