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Journal of Physics Conference Series

Tunneling Time of the Triangular Barrier and the Rectangular Barrier

Qingling Li, Shuang Zheng

Published Mar 1, 2023• 2 citations
Publisher

Abstract

Abstract In this paper, we calculate the phase time and dwell time of the triangular barrier and the rectangular barrier. The phase time and dwell time firstly increase and then decrease with wave number. In the tunneling region, the phase time decreases with the increase of barrier height, and firstly decreases and then increases with the increase of barrier width. Dwell time decreases with the increase of barrier height and increases with the increase of barrier width.

DOI
10.1088/1742-6596/2470/1/012032

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•Jan 1, 2002•0 citations
Abstract In this paper, we calculate the phase time and dwell time of the triangular barrier and the rectangular barrier. The phase time and dwell time firstly increase and then decrease with wave number. In the tunneling region, the phase time decreases with the increase of barrier height, and firstly decreases and then increases with the increase of barrier width. Dwell time decreases with the increase of barrier height and increases with the increase of barrier width.
量子隧穿作为没有经典对应的纯粹量子力学现象，其发生是否是瞬时的，即是否需要有限的时间，自量子力学建立到现在一直是人们悬而未决饱受争论的研究课题。本文通过计算粒子经过三角势垒的Wigner相位时间以及驻留时间可发现：随着波数k的变化，透射相位与反射相位呈两种相反变化趋势，而驻留时间则与透射相位时间变化相同；但相位时间和驻留时间均会出现峰值，且有振荡行为。 Quantum tunneling is a pure quantum phenomenon without any classical analogy. Whether it is instantaneous process or a process with finite time is an intensively debated issue since the early days of quantum mechanics. In this paper, we calculate the Wigner phase time and dwell time for a particle tunneling through a triangle barrier. We find that with the change of wave number k, the transmission phase time and the reflection phase show two opposite trends, while the dwell time changes the same as the transmission phase time. However, the phase time and dwell time both appear peak and oscillate.
In the discussion of temporary behaviors of quantum tunneling, people usually like to focus their attention on rectangular barrier with steep edges, or to deal with smooth barrier with semi-classical or even numerical calculations. Very few discussions on analytic solutions of tunneling through smooth barrier appear in the literature. In this paper, we provide two such examples, a semi-infinite long barrier V(x)=A2[1+tanh(x/a)] and a finite barrier V(x) = A sech2(x/a). To each barrier, we calculate the associated phase time and dwell time after obtaining the analytic solution. The results show that, different from rectangular barrier, phase time or dwell time does increase with the length parameter a controlling the effective extension of the barrier. More interestingly, for the finite barrier, phase time or dwell time exhibits a peak in k-space. A detailed analysis shows that this interesting behavior can be attributed to the strange tunneling probability Ts(k), i.e., Ts(k) displays a unit step function-like profile Θ(k − k0), especially when a is large, say, a ≫ 1/κ, 1/k. And k0≡mA/ħ is exactly where the peak appears in phase or dwell time k-spectrum. Thus only those particles with k in a very narrow interval around k0 are capable to dwell in the central region of the barrier sufficiently long.
For a one-dimensional barrier of arbitrary shape, an expression of the dwell time of tunneling particles has been obtained by subdividing it into infinitesimal rectangular barrier elements and then summing the dwell times spent by particles inside the barrier elements, and the dwell times in a parabolic potential barrier were calculated. The results show that the dwell times are almost independent of the barrier width as it is greater than a characteristic width W c . This can be ascribed that in the width range beyond W c , the tunneling probabilities through the barrier elements are very small, thus the times taken to tunnel through them make little contribution to the summation of the dwell times in all the barrier elements.
The electron tunnelling phase time τp and dwell time τD through an associated delta potential barrier U(x) = ξδ(x) are calculated and both are in the order of 10−17−10−16s. The results show that the dependence of the phase time on the delta barrier parameter ξ can be described by the characteristic length lc = 2/meξ and the characteristic energy Ec = meξ2/2 of the delta barrier, where me is the electron mass, lc and Ec are assumed to be the effective width and height of the delta barrier with lcEc = ξ, respectively. It is found that τD reaches its maximum and τD = τp as the energy of the tunnelling electron is equal to Ec/2, i.e. as lc = DB, DB is de Broglie wave length of the electron.
As was shown in quant-ph/0405028, the state of a tunneling particle can be uniquely presented as a coherent superposition of two states to describe alternative sub-processes, transmission and reflection. In this paper, on the basis of the stationary wave functions for these sub-processes, we give new definitions of the dwell times for transmission and reflection. In the case of rectangular potential barriers the dwell times are obtained explicitly. In contrast with the well-known Büttiker's dwell-time, our dwell time for transmission increases exponentially, for the under-barrier tunneling, with increasing the barrier's width. By our approach the well-known Hartman effect is rather an artifact resulted from an improper interpretation of the wave-packet tunneling and experimental data, but not a real physical effect accompanying the tunneling phenomenon.
Tunneling phenomenon is one of the wondrous events that take place particularly in quantum mechanics and it has no counterpart in classical physics. We focus on its two important implications : tunneling time and tunneling probability. There has been no consensus on tunneling times among the tunneling time community for last seven decades. So far, most of the attempts to calculate tunneling times have apparently been centered on a simple rectangular barrier example. We implement the tunneling times with the most widely accepted approach, which is using physical Larmor clock, into a nonsymmetrical barrier and a double barrier. It results in different and unique insights into the usual understanding of tunneling time. Three characteristics tunneling time expressions are thoroughly studied in these quantum structures. The difference between dwell time and phase-delay time is once again approved. The unique characteristics of the phase-delay time, which was thought to be exactly similar to dwell time when we assign higher energies to the particles, is also revealed in the nonsymmetrical barrier that it should actually be regarded as a distinct feature from the dwell time. The singularity of reflection time in rectangular barrier is no longer valid in nonsymmetrical barrier. The finiteness of reflection time fairly shows that different quantum structures might bear some taboo breaking results. Since a double barrier involves much more complicated features and calculations, the calculation of tunneling times under this barrier is praiseworthy. The dwell time is the most crucial, troublesome and distinctive tunneling time element to look into for a double barrier. Besides the dwell and phase-delay time, we also calculate the three characteristic times as well as the reflection time for a double barrier. Another area that we study within the context of tunneling phenomenon is the tunneling probability. Recently, there has been ground shaking attempts in modifying the conventional WKB formulae. We follow one of the modified versions of WKB formulae in order to be able to calculate the tunneling probability through different quantum structures where conventional WKB formulae fail. These generalized WKB connection formulae are applied to some complex quantum potentials by seeking aid from the Green's function and the path integral approach. One of the biggest problems in quantum mechanics (visualization) is partially overcome through implementing the method of diagrammatic representations that make the life easier in understanding the events happening in the quantum world when we attempt to calculate the transmission and reflection probabilities. This part therefore has a pedagogical sense too in terms of interpreting the results in a diagrammatic way with the help of the path integral approach.