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arXiv:1705.02543

Particle in infinite potential well with variable walls

Bernhard K. Meister

Published May 7, 2017• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1705.02543

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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.
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.
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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.
The general problem is studied on a simple example. A quantum particle in an infinite one-dimensional well potential is considered. Let the boundaries of well changes in a finite time TTT. The standard methods for calculating probability of transition from an initial to the final state are in general inapplicable since the states of different wells belong to different Hilbert spaces. If the final well covers only a part of the initial well (and, possibly, some outer part of the configuration space), the total probability of the transition from any stationary state of the initial well into {\bf all} possible states of the final well is less than 1 at T→0T\to 0T→0. If the problem is regularized with a finite-height potential well, this missing probability can be understood as a non-zero probability of transitions into the continuous spectrum, despite the fact that this spectrum disappears at the removal of regularization. This phenomenon ("transition to nowhere'') can result new phenomena in some fundamental problems, in particular at description of earlier Universe. We discuss also how to calculate the probabilities of discussed transitions at final TTT for some ranges of parameters.
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.
In this work, the motion of a particle in one dimension under the influence of the linearly downward potential well is studied within the context of the non-relativistic quantum mechanics. The attention is paid on the paradoxical phenomenon of the reflection of a particle that is in contrast between classical and quantum physics. Classically, the reflection effect occurs only at a potential barrier. To demonstrate such counter-intuitive phenomenon, the Schrödinger equation is solved to obtain the reflection coefficient in the scattering state by considering an incident particle that is represented by a monochromatic plane wave having an energy E > 0, propagates freely from left to right, pass through the potential well. The continuity conditions at boundaries give the desired result that is expressed in terms of the Airy functions which depends on the incident energy E, the strength jV 0 j and the range L of the well. The value of the reflection coefficient R lies in the interval 0 < R < 1, and its behavior is the decreasing function with respect to the range L.