线性谐振子量子特性的可视化研究

A research on visualization of quantum properties of linear harmonic oscillator

基于量子理论和数值计算,重点研究了量子力学中二、三维线性谐振子的基本特性,包括波函数及其能量、能级、简并度和几率密度,得到了三维谐振子的四维图形,同时得到了二维线性谐振子的三维图形.结果表明,线性谐振子能量只能取分立值,能量是量子化的;谐振子的能级是均匀分布的,且相邻两能级间隔为ΔE=ħω.二维线性谐振子的简并度为N+1,但N=0时,对应的基态波函数无简并;波函数与平面Ψ=0的交线数为N;几率密度图能更直观地显示出几率密度的峰值个数与大小,且几率密度分布的极大值个数为(n_(x)+1)(n_(y)+1).三维情况下,以基态波函数为准,在一定范围内([−3,3]),x,y,z轴上的切片数分别为n_(x)+1,n_(y)+1,n_(z)+1,总的切片数为n_(x)+n_(y)+n_(z)+3.另外,通过MATLAB软件得到了三维线性谐振子的四维空间切片图,可视化的结果与理论结果完全吻合.

Based on quantum theory and numerical calculation,we have mainly studied the basic characteristic of n-dimensional linear harmonic oscillator,including wave function,energy,energy levels,degeneracy and probability density,and obtain four-dimensional figures of three-dimensional harmonic oscillator and 3D figures of two-dimensional harmonic oscillator.The results show that the energy of a linear harmonic oscillator is discrete and the energy is quantized.The energy of a harmonic oscillator is evenly distributed,two adjacent energy level spacing isΔE=ħω.The degeneracy of two-dimensional linear harmonic oscillator is N+1,but the corresponding ground-state wave function has no degeneracy when N=0.The intersection line between wave function and the plane withΨ=0 is N.The probability density graph can more intuitively show the number and magnitude of the peak value of probability density,and the number of maximal value of probability density distribution is(n_(x)+1)(n_(y)+1).For the three-dimensional linear harmonic oscillator,based on the ground state wave function,within a certain range([−3,3]),the slices numbers of the x,y and z axis are n_(x)+1,n_(y)+1,n_(z)+1,respectively.The total number of slices is n_(x)+n_(y)+n_(z)+3.In addition,this research has obtained the four-dimensional space slice diagrams of three-dimensional linear harmonic oscillator by MATLAB software.The results of this visualization have established consistence with the theoretical results.