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IEEE International Conference on Plasma Science (ICOPS)

A Theoretical Comparison of Quantum Mechanical Tunneling and Electromagnetic Wave Propagation Through a Barrier

N. Riga, Rebecca Seviour, Kevin L. Jensen, Joel L. Lebowitz, Donald A. Shiffler

Published May 22, 2022• 0 citations

Abstract

We present a theoretical study of the correspondence between quantum mechanics (QM) and electromagnetism (EM) using the transfer matrix approach (TMA). This analogy has been studied previously [1] , [2] , [3] ; here we adapt it to electromagnetic wave propagation through an isotropic metamaterial for comparison with quantum tunneling through a one-dimensional barrier. Wigner trajectory calculations for quantum mechanical tunneling have been developed previously [4] , [5] to present a method for modeling electron emission at an atomistic level and to allow for coupling to larger-scale particle-based simulations. To provide a means for computational verification and macroscopic experimental validation of these microscale simulations, we use the TMA to calculate both the QM and EM transmission and reflection coefficients, which enables direct comparison of the respective outputs.

DOI: 10.1109/icops45751.2022.9813100

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We present a theoretical study of the correspondence between quantum mechanics (QM) and electromagnetism (EM) using the transfer matrix approach (TMA). This analogy has been studied previously [1] , [2] , [3] ; here we adapt it to electromagnetic wave propagation through an isotropic metamaterial for comparison with quantum tunneling through a one-dimensional barrier. Wigner trajectory calculations for quantum mechanical tunneling have been developed previously [4] , [5] to present a method for modeling electron emission at an atomistic level and to allow for coupling to larger-scale particle-based simulations. To provide a means for computational verification and macroscopic experimental validation of these microscale simulations, we use the TMA to calculate both the QM and EM transmission and reflection coefficients, which enables direct comparison of the respective outputs.

In this paper, we introduce the theoretical framework underlying our proposed methodology of verification and validation (V&V) for quantum mechanical emission models using analogous macroscopic electromagnetic systems. We derive the correspondence between quantum mechanics and electromagnetism using the transfer matrix approach and describe the electromagnetic analog that will be used to anchor the atomistic quantum tunneling simulations. Finally, we illustrate this correspondence by comparing the quantum mechanical and electromagnetic systems for some simple, analytically soluble examples and outline future V&V work based on the framework presented here.

Expressions are derived for the electromagnetic fields of guided waves which are analogous to the quantum-mechanical equations representing barrier tunneling. This analogy is achieved by comparing the propagation constant of the Schrödinger time-independent wave equation with that of the electromagnetic wave equation in waveguide and by comparing the de Broglie wavelength of a particle with the wavelength of the waves that propagate the energy. This results in an expression relating the form of an arbitrary one-dimensional energy barrier to the physical dimensions of a section of waveguide. The analogy is tested by the propagation of energy in the TE10 mode at both the 3- and 6-cm bands for the cases of rectangular and hyperbolic barriers. Although evanescent modes are present at the discontinuous regions, the analog for the rectangular barrier, which is considered to be the worse case, is verified when an effective barrier length leff of about 1.2l is used. This experimental verification demonstrates the possibility of waveguide simulation of quantum-mechanical energy barriers and the practicality of utilizing an electromagnetic analog for demonstrating the tunneling phenomenon and provides a method for measurement of the transmission coefficient through an arbitrarily shaped barrier.

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.

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.

The Generalized Method of Transfer Matrix (MMT) appears as an alternative to quantum phenomena research in semiconductor materials and optoelectronic components. In this paper the MMT is applied to the propagation phenomena of a particle through a well or a potential barrier to the quantum case. The relations scattering amplitudes are deduced using the coefficients of the transfer matrix (transmission and reflection). The results are compared with related literature, to determine the accuracy and scope of the MMT, namely which can be applied to study many other problems of wave propagation and electrons, even in layered media as photonic crystals.