Physical Review C

Symmetry relationship of quantum tunneling and its applications

Peiwei Wen, C. J. Lin, H. M. Jia, Liu Yang, N. R. Ma, F. Yang

Published Jun 3, 2024• 2 citations

Abstract

Quantum mechanics serves as the fundamental cornerstone for numerous areas of research. Finding simple and analytical quantum transmission coefficients for potentials is a rare occurrence. Notably, we have discovered a readily provable yet unreported symmetry tunneling law. For an infinite parabolic potential, it can be proved analytically that the sum of the one-dimensional tunneling probabilities at incident energies equal to the barrier height plus or minus any energy deviation is consistently 1. Based on this relationship, the original WKB approximation and the Kemble's formula can be generalized to the above-barrier energy region in a simplified but more accurate manner, referred to as the symmetric WKB approach in this context. For the realistic potential between two nuclei, numerical results demonstrate that the symmetry relationship at the above-barrier energy region and the sub-barrier energy is also well satisfied, as well as in the cases of the multichannel WKB method. The symmetry tunneling law has been effectively extended to the Eckart potential as well. Such symmetry in tunneling may have universal characteristics but its underlying reason remains to be uncovered, which may help us to further understand the quantum mechanisms.

DOI: 10.1103/physrevc.109.064602

Related papers

Calculation a New Transmission Coefficient of Tunneling for an Arbitrary Potential Barrier and Application to Alpha Decay

Hasan Hüseyin Erbil

According to classical physics theories, a moving particle cannot move to an environment with greater potential energy than its total energy during movement. But according to quantum theories, this event is known to be. This event is called tunneling. Tunneling is a probability, and it is measured by a transition coefficient. Correct calculation of this coefficient is very important because very sensitive and important instruments have been developed based on this event, and many events in nature can be explained by tunneling. This coefficient is generally calculated by semi-classical approaches (WKB) and the known formula is an approximate formula. In this paper, the general transmission coefficient of a potential barrier with arbitrary form is calculated by a simple method without any approximation. The results are applied to calculate the half-life values of the nuclei that emit alpha particles. The half-life values obtained from our calculations and the classical method (WKB) have been compared, and it has been found that the new half-life values are exactly consistent with the experimental values.

Journal of Photonic Materials and TechnologyJan 1, 20193 citations

A Study Of Quantum Tunneling Through Potential Barriers

Akshay Bhatia

This research paper explores the theoretical foundations of quantum tunneling, explaining how particles can pass through potential barriers despite lacking the classical energy to do so. It develops the mathematical framework using the Schrödinger equation, derives the transmission coefficient for a rectangular potential barrier, introduces the WKB approximation for more realistic barriers, and examines key applications such as alpha decay, scanning tunneling microscopy (STM), and the Josephson effect. The paper also reviews the historical development, experimental verification, and limitations of quantum tunneling, highlighting its significance as one of the most fundamental and experimentally validated predictions of quantum mechanics.

Jul 24, 20260 citations

Some important aspects of the tunneling phenomenon through quantum structures

E. Polatdemir

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.

Jan 1, 20020 citations

Smooth double barriers in quantum mechanics

Avik Dutt, Sayan Kar

Quantum mechanical tunneling across smooth double barrier potentials is modeled using Gaussian functions and is analyzed numerically and by using the WKB approximation. The transmission probability, resonances as a function of incident particle energy, and their dependence on the barrier parameters are obtained for various cases. We also discuss the tunneling time, for which we obtain generalizations of the known results for rectangular barriers.

arXiv:1008.1640American Journal of PhysicsNov 18, 201027 citations

Transmission coefficient in double barrier quantum tunnelling effect

Zhongzheng Duan, Wenxi Luo, Xiaohan Xu

At the turn of the twentieth century, the establishment of quantum theory propelled rapid advancements, particularly in the understanding of quantum tunnelling—a fundamental phenomenon in quantum mechanics crucial for various physical processes. The quantum phenomenon of particle passes through potential barriers is of great importance. In classical physics, when the energy of a particle is less than the height of a double barrier structure, it is impossible for it to pass through. However, quantum mechanics allows a particle to penetrate the barrier and emerge on the other side. This paper explores the quantum tunnelling effect, focusing on the single potential barrier model in one dimension and subsequently extending to the double potential barrier model. The Schrödinger equation provides the foundational framework for elucidating the motion of microscopic particles, emphasizing wave-particle duality inherent in quantum mechanics. The analysis of the single potential barrier model involves solving the Schrödinger equation in different regions, determining wave functions and coefficients through boundary conditions. The transmission coefficient is derived, representing the probability of a particle passing through a barrier. In the case of a “thick” barrier, an approximate form for transmission coefficient is provided, demonstrating the exponential decrease in transmission probability with increasing barrier thickness.

Theoretical and Natural ScienceDec 19, 20230 citations

Reflection and transmission probabilities from various potentials using quasilinearization and WKB approximation techniques

Kittinut Singhanat

Quantum tunneling is a phenomenon in quantum mechanics. In this phenomenon, Quantum particle can go through potential barrier that potential energy is higher than total energy of quantum particle. Physicists are interested in transmitted and reflected value of Quantum particle that is transmission and reflection probabilities. Then, Schrödinger equation is used to find transmission and reflection probabilities by directly solving. However, some potential barriers are too complex to solve directly. Then, we use another method to solve more complex problem. That method is called "quasilinearization''. Quasilinearization method (QLM) is a numerical method to solve nonlinear differential equations. In this thesis, we compare the quasilinearization method with the well-known WKB approximation method. The results show that the transmission and reflection probabilities calculated by the QLM and WKB approximation have the same trend as the transmission and reflection probabilities calculated from the exact solution.

Jan 1, 20250 citations