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American Journal of Physics

On the sudden expansion of an infinite square well

J. G. Cordes, David Kiang, Hideto Nakajima

Published Feb 1, 1984• 3 citations
Publisher

Abstract

It is shown that most initial states, upon sudden expansion of the wall of an infinite square well, lead to an infinite dispersion in the energy. Circumstances under which ΔE remains finite and constant are discussed. The situation with respect to a macroscopic system is examined.
DOI
10.1119/1.13727

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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.
I study the time-evolution of a particle prepared in the ground state of an infinite well after the latter is suddenly expanded. It turns out that the probability density |Ψ(x, t) | 2 shows up quite a surprising behaviour: for definite times, plateaux appear for which |Ψ(x, t) | 2 is constant on finite intervals for x. Elements of theoretical explanation are given by analyzing the singular component of the second derivative ∂xxΨ(x, t). Analytical closed expressions are obtained for some specific times, which easily allow to show that, at these times, the density organizes itself into regular patterns provided the size of the box in large enough; more, above some critical time-dependent size, the density patterns are independent of the expansion parameter. It is seen how the density at these times simply results from a construction game with definite rules acting on the pieces of the initial density.
I study the time evolution of a particle prepared in the ground state of an infinite well after the latter is suddenly expanded. It turns out that the probability density |Ψ( x , t )| 2 shows up quite a surprising behaviour: for definite times, plateaux appear for which |Ψ( x , t )| 2 is constant on finite intervals for x . Elements of theoretical explanation are given by analysing the singular component of the second derivative ∂ xx Ψ( x , t ). Analytical closed expressions are obtained for some specific times, which easily allow us to show that, at these times, the density organizes itself into regular patterns provided the size of the box is large enough; more, above some critical size depending on the specific time, the density patterns are independent of the expansion parameter. It is seen how the density at these times simply results from a construction game with definite rules acting on the pieces of the initial density.