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American Journal of Physics

The contracting square quantum well

D. N. Pinder

Published Jan 1, 1990• 60 citations
Publisher

Abstract

The one-dimensional square quantum well of varying width is shown to be an instructive example for testing the appropriateness of approximate solutions in quantum mechanics. The range of wall speeds over which the adiabatic approximation is applicable is investigated and related to the initial energy of the quantum system for both expanding and contracting wells. However, of more interest is the applicability of the sudden approximation to these systems. It is shown that this approximation is appropriate for expanding wells at quite modest wall speeds, whereas the approximation cannot be applied for contracting wells no matter how rapidly the well width contracts. The physical reasons for the inapplicability of the sudden approximation for contracting wells is investigated.

DOI
10.1119/1.16319

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The one-dimensional square quantum well of varying width is shown to be an instructive example for testing the appropriateness of approximate solutions in quantum mechanics. The range of wall speeds over which the adiabatic approximation is applicable is investigated and related to the initial energy of the quantum system for both expanding and contracting wells. However, of more interest is the applicability of the sudden approximation to these systems. It is shown that this approximation is appropriate for expanding wells at quite modest wall speeds, whereas the approximation cannot be applied for contracting wells no matter how rapidly the well width contracts. The physical reasons for the inapplicability of the sudden approximation for contracting wells is investigated.
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.
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.
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.
We analyze the validity of the adiabatic approximation, and in particular the reliability of what has been called the ``standard criterion'' for validity of this approximation. Recently, this criterion has been found to be insufficient. We will argue that the criterion is sufficient only when it agrees with the intuitive notion of slowness of evolution of the Hamiltonian. However, it can be insufficient in cases where the Hamiltonian varies rapidly but only by a small amount. We also emphasize the distinction between the adiabatic theorem and the adiabatic approximation, two quite different, although closely related, ideas.