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Physical Review A

Generalized reflection time for one-dimensional structures

Cheng Yin, Zhuangqi Cao

Published Dec 9, 2009• 7 citations
Publisher

Abstract

A generalized reflection time which follows directly from the Schr\"odinger equation without approximation is obtained. By the Winger-Eisenbud approach, this reflection time is conceptually well defined and can be applied to arbitrary continuous potentials including both quantum tunneling and quantum reflection issue. A simple explanation for the quantum reflection phenomenon is proposed and the explicit expression of the difference between the quantum reflection time and the classical reflection time is also derived.

DOI
10.1103/physreva.80.064102

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A generalized reflection time which follows directly from the Schr"odinger equation without approximation is obtained. By the Winger-Eisenbud approach, this reflection time is conceptually well defined and can be applied to arbitrary continuous potentials including both quantum tunneling and quantum reflection issue. A simple explanation for the quantum reflection phenomenon is proposed and the explicit expression of the difference between the quantum reflection time and the classical reflection time is also derived.
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The proper basis for the calculation of transmission and reflection times for wave packets scattered off arbitrary tunneling structures in one dimension is considered. With packets narrow in wave-number space, we demonstrate that the classic phase times are indeed correct to lowest order. Explicit, general expressions for the leading correction terms for finite wave packets are given. The physics associated with these corrections is discussed. We also consider the dwell time, as it is currently defined, and derive a general relation between this dwell time and the phase times. This relation shows when the dwell time can and cannot be used. Finally, we discuss wave packets transmitted from narrow resonances, and derive an explicit, exact formula for the tunneling time with resonance transmission from a symmetric double barrier. Comparison with earlier approximate results is made.
Abstract A distinguishing feature of quantum mechanics, as compared with classical mechanics, is the phenomenon of matter exhibiting wave-like properties. A direct consequence of this wave property of particles is the tunneling of particles through potential energy barriers.-3 The classical particle is forbidden from penetrating the barrier region. In this chapter, the problem of one-dimensional (1D) quantum mechanical tunneling of particles through potential barriers of arbitrary shape is treated. We show that tunneling is a mixed boundary value problem, and obtain the action integral that leads to Schrodinger’s equation and at the same time gives the equations defining the boundary conditions (BCs).
Based on the similarity between time-independent Schrodinger equation and Helmholtz equation, the analytical transfer matrix method was utilized to analyze macroscopic quantum tunneling effect, and the potential filed and eigenvalue were corresponded to the refractive index and propagation constant, respectively.Without solving the Schrodinger equation and any approximation,the exact analytic expressions of reflection coefficient and reflection probability of the particle were obtained.Initial results show the equations obtained here are exact and have clear physical meanings.It is expected to be applied in the researches of many basic quantum phenomena,such as quantum tunneling effect,quantum reflection, quantum particles and tunneling resonance.