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.

中文翻译：

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单表达式方法 (MSE) 作为边界值问题的前瞻性建模工具：从纳米光学到量子力学的扩展

纳米光学和量子力学中波动现象的数学描述是相似的，需要对波与光学中纳米层的相互作用以及微米粒子与量子力学中的势垒或阱的相互作用进行波长尺度分析。传统上，在处理纳米光学和量子力学中的边界问题时，经常使用与反向传播波相同的基本方法，当波动方程的一般解表示为反向传播波的总和时。这种类型的解决方案依赖于叠加原理，限制了对强强度相关非线性波-物质相互作用的正确描述。单表达式（MSE）的非传统方法没有利用叠加原理，而是使用结果场表示和反向传播算法，通过正确考虑介质中与强度相关的非线性、损失或增益，允许在没有近似值和任何波强度值的情况下获得边界值问题的正确稳态解。在目前的工作中，详细描述了针对一维量子力学边界值问题扩展的 MSE 方法。介绍并讨论了电子隧道通过矩形单势垒和双势垒的 MSE 数值模拟结果。介质中的损失或收益。在目前的工作中，详细描述了针对一维量子力学边界值问题扩展的 MSE 方法。介绍并讨论了电子隧道通过矩形单势垒和双势垒的 MSE 数值模拟结果。介质中的损失或收益。在目前的工作中，详细描述了针对一维量子力学边界值问题扩展的 MSE 方法。介绍并讨论了电子隧道通过矩形单势垒和双势垒的 MSE 数值模拟结果。