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Chinese Journal of Quantum Electronics

Reflection and transmission of material wave through an arbitrary potential barrier

Deng Man-lan

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

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.

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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.
In this paper, use is made of the tools of analytical mechanics and the concept of operators to obtain the time-independent and time-dependent Schrodinger wave equations for quantum mechanical systems. Derivations are embarked upon of expressions for reflection and transmission coefficients for a particle of mass m as well as of energy E moving under different potential set-ups across step functions, barriers and well functions. The tunneling effect is then discussed. The transmission probability equation obtained in this research has been observed to be more accurate than the transmission probability expression deduced by some researchers in 2014 for a tunneling barrier. This research work finds applications in nuclear magnetic resonance imaging systems, synchrotrons, gyrators, accelerators, and in electrodynamics.
Abstract The reflection and transmission coefficients describe the behavior of the matter wave incident on a potential barrier. They can be expressed in terms of the probability with which the matter wave can be reflected or transmitted. The central equation accounting for the behavior of the matter wave is the Schrödinger equation. The Schrödinger equation is the second order partial differential equation. However, in a stationary state, the Schrödinger equation is reduced to the time independent Schrödinger equation. This time independent Schrödinger equation is the second order linear ordinary differential equation. Since the time independent Schrödinger equation is linear, superposition of any of the two solutions to the time independent Schrödinger equation is also a solution. In this paper, we focus on the superposition of various potentials. The reflection and transmission coefficients from the superposition of various potentials are obtained. A comparison between the exact coefficients and those obtained by the 2 × 2 transfer matrix is made. The relationship between the transmission coefficient of the superposed potential and that of each individual potential is found. The results show that the transmission coefficient obtained from the 2 × 2 transfer matrix is of a lower bound on the exact transmission coefficient.
A simple, yet accurate, method of solving the Schrödinger equation across an arbitrary potential barrier is described, based on the analytic transfer-matrix technique. This paper gives the final analytical expression for the transmission probability. Numerical calculations for four typical potential barrier structures are compared with the traditional WKB, modified WKB and the modified Airy function (MAF) methods. Excellent agreement with an exact example is demonstrated. Most importantly, in cases where other potential-barrier methods fail, our method performs well.
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.