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European Journal of Physics

Transmittance and reflectance formulations: a typical isomorphism between the rectangular potential-barrier in quantum mechanics and the dielectric slab in electromagnetic-optics

Belkacem Meziane

Published Feb 9, 2018• 1 citation
Publisher

Abstract

Abstract The main intent of this paper is to demonstrate that the theoretical formulations of the reflectance and transmittance characteristics of an initially free quantum particle, heading towards some restricted region of space, inside which it undergoes the effect of an electric-potential-barrier, are isomorphic to those of an incident plane- and linearly-polarized electromagnetic wave, which propagates towards and through a homogeneous dielectric slab.

DOI
10.1088/1361-6404/aa9cbf

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Abstract The main intent of this paper is to demonstrate that the theoretical formulations of the reflectance and transmittance characteristics of an initially free quantum particle, heading towards some restricted region of space, inside which it undergoes the effect of an electric-potential-barrier, are isomorphic to those of an incident plane- and linearly-polarized electromagnetic wave, which propagates towards and through a homogeneous dielectric slab.
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In this work, the motion of a particle in one dimension under the influence of the linearly downward potential well is studied within the context of the non-relativistic quantum mechanics. The attention is paid on the paradoxical phenomenon of the reflection of a particle that is in contrast between classical and quantum physics. Classically, the reflection effect occurs only at a potential barrier. To demonstrate such counter-intuitive phenomenon, the Schrödinger equation is solved to obtain the reflection coefficient in the scattering state by considering an incident particle that is represented by a monochromatic plane wave having an energy E > 0, propagates freely from left to right, pass through the potential well. The continuity conditions at boundaries give the desired result that is expressed in terms of the Airy functions which depends on the incident energy E, the strength jV 0 j and the range L of the well. The value of the reflection coefficient R lies in the interval 0 < R < 1, and its behavior is the decreasing function with respect to the range L.
The interaction between incident and target particles is inferred from the numbers and angular distribution of scattered particles and from their energy distribution. It is important to recognize that a scattering region, as represented for example by a rectangular barrier, is a region where, in classical terms, forces are applied to the incident particles. A full analysis of scattering events must, in principle, embody the description of a localized free particle that travels toward a center of force and is scattered by it. Quantum-mechanically, that implies a theoretical analysis in terms of packet states, with an explicit account of their change of position with time. One of the most striking effects described by quantum mechanics is the transmission of incident particles across a potential barrier whose height is greater than the particle energy. Classically, these particles lack the energy to get over the barrier, yet for quantum-mechanical systems some fraction of incident particles are transmitted.
We calculate the effects of normal transmission through an amplifying dielectric slab on the properties of an incident pulse of light. The transmitted pulse shows shifts in peak position and additional lengthening or shortening with respect to the unamplified incident pulse. The magnitudes of these effects are generally larger than those of the corresponding changes that occur in transparent or attenuating slabs. They are interpreted in terms of the interference of multiply reflected contributions to the transmitted pulse. The theory is valid for pulses of nonclassical light, but the same reshaping occurs for appropriate pulses of classical light.
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.
We apply a recently developed scheme for electromagnetic field quantization in dispersive and absorbing dielectrics to calculate the effects of perpendicular propagation through a finite-temperature dielectric slab on the properties of an incident pulse of light. The theory applies to incident pulses of nonclassical light and also reproduces the results for classical pulses in the appropriate limit. The transmitted pulse is assumed to be measured by a detector that receives radiation only from the direction normal to the slab surfaces. The Poynting vector of the transmitted light includes contributions from the incident pulse, whose shift in peak position and additional broadening or narrowing are determined, and from the thermal emission of the slab, which tends to a black-body form in the appropriate limits.