基于狄拉克方程推导求解一维势垒问题  

Resolving problems of one-dimensional potential barriers based on Dirac equation

作　　者：王怀玉[1] WANG Huaiyu(Department of Physics,Tsinghua University,Beijing,100084,China)

机构地区：[1]清华大学物理系,北京100084

出　　处：《华北科技学院学报》2022年第1期97-107,共11页Journal of North China Institute of Science and Technology

基　　金：国家重点研发计划资助项目(2018YFB0704304)。

摘　　要：量子力学教科书中,粒子做非相对论运动遇到势垒时,按照薛定谔方程来求解。作者之前的研究表明,在粒子能量小于势能的势垒区域,应该运用负动能薛定谔方程。本文重新处理了六个量子力学上常见的例子,一维有限深势阱,三维有限深球形势阱和有限高球形势垒,线性势,Kronig-Penney模型,WKB近似方法。结果表明:在有限高无限宽的势垒内部,波函数为零;在有限高有限宽势垒内部,波函数是平面波的叠加;线性势中的波函数具有对称或者反对称性质;势垒穿透的透射系数是随着势垒宽度有周期性变化的。这些结果都与量子力学教科书有所区别。相比之下,本文的结果更为合理。原因是负动能薛定谔是从相对论量子力学方程做低动量近似得到的结果,具有坚实的理论基础。In quantum mechanics(QM)textbooks,the non-relativistic motion of a particle is treated with Schr dinger equation.The author’s previous study showed that when a particle is in a region where the potential is larger than its energy,negative kinetic energy(NKE)Schr dinger equation ought to be used.In the present work,six problems that often appear in usual QM textbooks are re-treated.They are one-dimensional potential well with finite depth,three-dimensional potential well with finite depth,potential barrier with finite height,linear potential,Kronig-Penney model and WKB approximation method.The following conclusions are obtained.Inside a potential barrier with finite height but infinite width,the wave function is zero.In a potential barrier with finite height and width,the wave function is the superposition of plane waves.The wave functions of a particle subject to a linear potential are of some symmetry.The transmission coefficient of a barrier varies with the barrier width periodically.It is argued that compared to those in QM textbooks,our results are more reasonable.This is because the NKE Schr dinger equation was derived rigorously by taking low momentum approximation of the relativistic QM equation,so that this equation has a solid foundation.

关 键 词：薛定谔方程 负动能薛定谔方程 一维有限深方势阱 线性势 KRONIG-PENNEY模型 WKB近似方法 透射系数O413 

分 类 号：O413[理学—理论物理]