n维线性谐振子特性的MATLAB数值仿真  

MATLAB numerical simulation of characteristics of n-dimensional linear harmonic oscillator

作　　者：唐歡 郑兴荣 李小明 李小龙 李娜 谢凯强 TANG Huan;ZHENG Xingrong;LI Xiaoming;LI Xiaolong;LI Na;XIE Kaiqiang(College of Electrical Engineering,Longdong University,Qingyang 745000,China;Huanxian Secondary Vocational School,Qingyang 745700,China)

机构地区：[1]陇东学院电气工程学院,甘肃庆阳745000 [2]环县职业中等专业学校,甘肃庆阳745000

出　　处：《青海大学学报》2020年第6期68-76,共9页Journal of Qinghai University

基　　金：甘肃省教育厅高等学校创新能力项目(2019A-112);国家自然科学基金项目(11565018)。

摘　　要：基于量子理论和数值计算,本文系统地研究量子力学中n维线性谐振子的基本特性。重点分析n维谐振子的波函数和几率密度,并借助MATLAB软件计算一维和二维一些能级的波函数及其几率密度,得到可视化的结果。并运用MATLAB软件对经典理论和量子理论中线性谐振子的几率分布进行图像对比。结果表明:一维情况下,波函数与Ψ=0的直线的交点个数为n,波函数有n个节点,且在节点处找到粒子的几率为零;几率密度分布满足归一性。二维情况下,谐振子的简并度为N+1,但N=0时,对应的基态波函数无简并;波函数与Ψ=0平面的交线数为N;一般情况下几率密度分布的极大值个数为(nx+1)(ny+1)。当量子数n很大时,几率密度在量子结果和经典结果上越接近,两种情况在平均上已相当符合,差别只在于|Ψ|2的迅速振荡上。Based on the quantum theory and numerical calculation,this paper systematically studies the basic characteristics of n-dimensional linear harmonic oscillator in quantum mechanics and focuses on the analysis of the wave function and probability density of n-dimensional harmonic oscillator. It also calculates the wave function and probability density of some energy levels in one and two dimensions with the help of MATLAB software to obtain the visualization results. The results show that in one-dimensional case,the number of intersections between the wave function and the straight line with ψ = 0 is n and the number of the nodes of wave function is n. Besides,the probability of finding particles at the nodes is zero;the distribution of probability density satisfies the normalization. In the case of two dimensions,the degeneracy of harmonic oscillator is N + 1,while the corresponding ground state wave function has no degeneracy when N = 0;the number of intersection lines between wave function and ψ = 0 plane is N. In general,the maximum number of probability density distribution is( nx+ 1)( ny+ 1). The probability density is much closer to the classical result when the quantum number is extremely large. The two cases are quite consistent on average because their difference only lies in the rapid oscillation of | Ψ |2.

关 键 词：n维线性谐振子 量子理论 波函数 几率密度 仿真 

分 类 号：O413.1[理学—理论物理] O413.3[理学—物理]