隧道效应下的Schrödinger方程吸收边界条件  

Absorbing Boundary Conditions for Schrödinger Equation under Tunneling Effect

作　　者：张阔[1,2] ZHANG Kuo(China Center for Information Industry Development,Beijing 100048,China;Beijing CCID Publishing&Media Co.,Ltd.,Beijing 100048,China)

机构地区：[1]中国电子信息产业发展研究院,北京100048 [2]北京赛迪出版传媒有限公司,北京100048

出　　处：《新一代信息技术》2019年第17期47-52,共6页New Generation of Information Technology

基　　金：国家科技重大专项(项目编号:2011ZX05020)。

摘　　要：Schrödinger方程是揭示量子微观体系中隧道效应等特殊科学现象的理论依据。为了采用数值方法准确模拟微观粒子在无限大区域的运动特征,需要在有限区域外围施加吸收边界条件,然而在隧道效应下,Schrödinger方程性态发生变化,经典吸收边界条件失去完备性。立足于对现有理论框架的回顾和理解,以Clayton-Engquist和PML两种经典吸收边界条件的构造思想为指引,提出隧道效应下Schrödinger方程吸收边界条件的构造构想。对Schrödinger方程两种性态下的吸收边界进行试算验证,进而将两种性态相结合,构建半无限深方势阱,实现隧道效应模拟。两种吸收边界下的模拟结果均展现出优秀的一致有效性。顺带地,虽然隧道效应在宏观层面是不可解释的,但将Schrödinger方程与声波方程等经典物理方程相类比得知,其不可解释的原因是可解释的。Schrödinger equation is the theoretical basis for revealing the special scientific phenomena such as the tunneling effect in the quantum micro-system.In order to accurately simulate the motion characteristics of micro-particles in an infinite region using the numerical methods,absorbing boundary conditions need to be imposed on the periphery of a finite region.However,under the tunneling effect,the behavior of Schrödinger equation changes so that the classical absorbing boundary conditions lose completeness.Based on the review and understanding of the existing theoretical framework,and under the guidance of the construction ideas of classical Clayton-Engquist and PML absorbing boundary conditions,a conception of constructing the absorbing boundary conditions for Schrödinger equation under the tunneling effect is put forward.The absorbing boundaries under two behaviors of Schrödinger equation are verified by trial calculation,and then a semi-infinite deep square potential well is constructed by combining the two behaviors to simulate the tunneling effect.The simulation results under both absorbing boundaries show excellent uniformly validity.Incidentally,despite that the tunneling effect is not interpretable at a macro level,the reason why it is not interpretable,is interpretable,by comparing the Schrödinger equation with classical physical equations such as the acoustic wave equation.

关 键 词：Schrödinger方程 隧道效应 吸收边界条件 Clayton-Engquist PML 半无限深方势阱本文著录格式:张阔.隧道效应下的Schrödinger方程吸收边界条件[J].新一代信息技术 2019 2(17):47-52 

分 类 号：O431.2[机械工程—光学工程]