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Journal of Physics Conference Series

Time delay in the reflection of particles by semi-harmonic wells

Óscar Rosas-Ortiz, Sara Cruz y Cruz, Nicolás Fernández-García

Published Aug 24, 2012• 3 citations
Publisher

Abstract

The phase time of wave reflection from a one-dimensional semi-harmonic well includes a delay term which can be negative according to the incoming energy and the well parameters in a quantum scattering process. The model is a rectangular well which is embedded in an environment composed by a zero potential energy to its right and a harmonic-like potential to its left.

DOI
10.1088/1742-6596/380/1/012018

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The phase time of wave reflection from a one-dimensional semi-harmonic well includes a delay term which can be negative according to the incoming energy and the well parameters in a quantum scattering process. The model is a rectangular well which is embedded in an environment composed by a zero potential energy to its right and a harmonic-like potential to its left.
We study the scattering of a particle by a square barrier potential that has a parabolic hole in between. The barrier is parameterized in such a form that the hole can be partially or completely removed. This can be also chosen to be symmetric or asymmetric with respect to the hole. Some expressions for the phase time are given and the time spent by the particle in the interaction zone of the barrier is calculated. It is shown that the related time delay depends on the symmetry operations that one can do on the potential.
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.
The problem of time delay in one-dimensional quantum-mechanical scattering by a potential barrier is studied in the framework of time-dependent scattering theory. It is shown how the concept of sojourn time can be used to define three separate time delays: the time delay for transmission through the barrier, the time delay for reflection, and the total time delay (i.e., averaged over transmission and reflection). In addition to the anticipated dependence on the S matrix and its energy derivative, the time delays depend on certain parameters which can be interpreted as describing positions of detectors in a gedanken-experiment measurement. The problem of defining separate time delays for transmission and reflection appears to be rooted in the foundations of quantum theory and its interpretation.
We present a step by step introduction to the notion of time-delay in classical and quantum mechanics, with the aim of clarifying its foundation at a conceptual level. In doing so, we motivate the introduction of the concepts of “fuzzy” and “free-flight” sojourn times that we use to provide the most general possible definition for the quantum time-delay, valid for simple and multichannel scattering systems, with or without conditions on the observation of the scattering particle, and for incoming wave packets whose energy can be smeared out or sharply peaked (fixed energy). We conclude our conceptual analysis by presenting what we think is the right interpretation of the concepts of sojourn and delay times in quantum mechanics, explaining why, in ultimate analysis, they should not be called “times.”
Abstract We present a step by step introduction to the notion of time-delay in classical and quantum mechanics, with the aim of clarifying its foundation at a conceptual level. In doing so, we motivate the introduction of the concepts of “fuzzy” and “free-flight” sojourn times that we use to provide the most general possible definition for the quantum time-delay, valid for simple and multichannel scattering systems, with or without conditions on the observation of the scattering particle, and for incoming wave packets whose energy can be smeared out or sharply peaked (fixed energy). We conclude our conceptual analysis by presenting what we think is the right interpretation of the concepts of sojourn and delay times in quantum mechanics, explaining why, in ultimate analysis, they should not be called “times.”