The method of single expression (MSE) as a prospective modeling tool for boundary value problems: an extension from nano
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The method of single expression (MSE) as a prospective modeling tool for boundary value problems: an extension from nano‑optics to quantum mechanics H. V. Baghdasaryan1 · T. M. Knyazyan1 · T. Baghdasaryan2 · T. T. Hovhannisyan1 · M. Marciniak3 Received: 15 June 2020 / Accepted: 24 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Mathematical description of the wave phenomena in nano-optics and quantum mechanics is similar and requires wavelength-scale analysis of wave interaction with nano-layers in optics and micro-particle interaction with potential barriers or wells in quantum mechanics. Traditionally, when dealing with boundary problems in nano-optics and quantum mechanics, the same fundamental approach of counter-propagating waves is often being used, when general solutions of the wave equations are presented as a sum of counterpropagating waves. This type of solution presentation relies on the superposition principle restricting correct description of strong intensity-dependent nonlinear wave-matter interaction. The non-traditional method of single expression (MSE) does not exploit the superposition principle, but rather uses resulting field representation and backward-propagation algorithm allowing to obtain correct steady-state solutions of boundary value problems without approximations and at any value of wave intensity by taking into account correctly intensity-dependent nonlinearity, loss or gain in a medium. In the present work a detailed description of the MSE approach extended for one dimensional quantum mechanical boundary value problems is presented. Results of numerical simulations by the MSE of electron tunneling through rectangular single and double potential barriers are presented and discussed. Keywords Schrödinger equation · Boundary value problem · Quantum tunneling · Resonant tunneling · Method of single expression · Numerical simulation
* H. V. Baghdasaryan [email protected] M. Marciniak [email protected] 1
Fiber Optics Communication Laboratory, National Polytechnic University of Armenia, 105, Terian Str., 0009 Yerevan, Armenia
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Department of Applied Physics and Photonics (TONA), Brussels Photonics (B‑PHOT), Vrije Universiteit Brussel (VUB), Pleinlaan 2, 1050 Brussels, Belgium
3
National Institute of Telecommunications, 1. Szachowa Str., 04‑894 Warsaw, Poland
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1 Introduction The wave phenomena is in the core of many models in physics and examples include, but are not limited to electromagnetic waves, acoustic and mechanical waves in solids, liquids and gases. Waves also feature when describing micro-particles in non-relativistic and relativistic quantum mechanics (Fitzpatrick 2013, 2005; Towne 1988). Mathematical description of wave phenomena in different fields is usually based on identical types of secondorder differential equations (Fitzpatrick 2013; Griffiths and Steinke 2001; Pain 2005; Towne 1988). In case of monochromatic waves and frequency-domain analys
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