摘要
运用量子理论和MATLAB软件,从理论推导和数值仿真两方面系统地研究了直角坐标系和极坐标系下量子力学中二维线性谐振子的基本特性.结果表明:除特殊情况外,直角坐标系下的二维线性谐振子的简并度、能量本征值和概率密度分布的极大值个数分别为n_(x)+n_(y)+1,hω(n_(x)+n_(y)+1)和(n_(x)+1)(n_(y)+1),且波函数与平面Ψ=0的交线数为n_(x)+n_(y).极坐标系下的简并度和能量本征值分别为2n_(r)+|m|+1和hω(2n_(r)+|m|+1),且波函数与平面Ψ=0的交线数为2n_(r)+|m|;在n_(r)=0的情况下,概率密度分布的极大值个数为2|m|.这种可视化的结果与理论推导结果完全一致.
Using quantum theory and MATLAB software,the basic properties of two-dimensional linear harmonic oscillators in quantum mechanics are systematically studied in rectangular coordinate system and polar coordinate systems from theoretical derivation and numerical simulation.The results show that,with the exception of special case,the degeneracy,energy eigenvalue and the maximum number of probability density distributions of two-dimensional linear harmonic oscillator in the rectangular coordinate system are n_(x)+n_(y)+1,hω(n_(x)+n_(y)+1)and(n_(x)+1)(n_(y)+1),respectively.And the number of intersection line between wave function and the plane withΨ=0 is n_(x)+n_(y).In polar coordinate system,the degeneracy and energy eigenvalues of two-dimensional linear harmonic oscillator are 2n_(r)+|m|+1 and hω(2n_(r)+|m|+1),and the number of intersection line between wave function and the plane withΨ=0 is 2n_(r)+|m|.In the case of n_(r)=0,the maximum number of probability density distributions is 2|m|.The results of this visualization are in complete agreement with the theoretical results.
作者
杨伟
张杰
郑兴荣
高晓红
张郃
Yang Wei;Zhang Jie;Zheng Xingrong;Gao Xiaohong;Zhang He(College of Electrical Engineering, Longdong University, Qingyang 745000, China)
出处
《宁夏大学学报（自然科学版）》
CAS
2022年第2期164-168,176,共6页
Journal of Ningxia University(Natural Science Edition)
基金
甘肃省自然科学基金资助项目(21JR7RM188)
庆阳市自然科学基金资助项目(QY2021A-F014)。
关键词
二维线性谐振子
量子理论
直角坐标系
极坐标系
本征函数
概率密度
仿真
two-dimensional linear harmonic oscillator
quantum theory
rectangular coordinate system
polar coordinate system
eigenfunction
probability density
simulation
