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arXiv:0709.1101 · Journal of Physics A Mathematical and Theoretical

Surprises in the suddenly-expanded infinite well

Claude Aslangul

Published Feb 5, 2008• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8113/41/7/075301

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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.
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.
The time evolution and its implications with regard to the approach to equilibrium of a single-particle system in a suddenly expanded box with perfectly reflecting walls is investigated using density-matrix techniques. It is shown that the time-dependent quantities exhibit a periodic behavior which is independent of the initial density matrix as long as the dimensions of the system are finite. The period is found to be $\frac{2\ensuremath{\pi}\ensuremath{\hbar}}{{E}_{1}}$ where E1{E}_{1}E1​ is the ground-state energy of the expanded system. The continuum approximation is employed to determine the explicit time dependence of the particle density of a system initially in thermal equilibrium. It is found that the density for an expansion into a finite volume decays essentially as 1t\frac{1}{t}t1​ with small oscillations superposed on this behavior for times ranging from shortly after expansion up to one-fourth the characteristic period.
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 Gedanken experiment is described to explore a counter-intuitive property of quantum mechanics. A particle is placed in a one-dimensional infinite well. The barrier on one side of the well is suddenly removed and the chamber dramatically enlarged. At specific, periodically recurring, times the particle can be found with probability one at the opposite end of the enlarged chamber in an interval of the same size as the initial well. With the help of symmetry considerations these times are calculated and shown to be dependent on the mass of the particle and the size of the enlarged chamber. Parameter ranges are given, where the non-relativistic nature of standard quantum mechanics becomes particularly apparent.