The method of single expression (MSE) as a prospective modeling tool for boundary value problems: an extension from nano-optics to quantum mechanics

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 counter-propagating 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.