一维无限深势阱中粒子的定态波函数的展开问题  

On the Expansion Problem of the Wavefunctions of a Stationary State for a Particle in One-Dimensinal Potentional Well with Infinitely High Walls

作　　者：刘登云[1] 

机构地区：[1]山西师范大学物理系,山西临汾041004

出　　处：《山西师范大学学报（自然科学版）》2000年第2期30-35,共6页Journal of Shanxi Normal University(Natural Science Edition)

基　　金：山西省留学回国人员基金资助课题

摘　　要：对于一维无限深势阱中粒子的定态函数的展开问题 ,我们发现 ,在区间 ( 0≤ x≤ a)上 ,当能量量子数 n为偶数时 ,定态波函数可以表示为两列反向行进的平面波的迭加 .而当 n为奇数时 ,由于定态波函数不是该区间上的周期函数 ,不能表示为两列反向行进的平面波的迭加 .仅当势阱宽度趋于无穷大 ,或当 n很大 ,在连续谱基底上的展开系数变为两个中心位置对称的 δ函数的迭加时 ,定态波函数才能表示为两列反向行进的平面波的迭加 .The expansion problem of the wavefunctions of a stationary state for a particle in one dimensional potential well with infinitely high walls in studied. We found, in the interval ( 0≤x≤a ), that the wavefunctions of a stationary state can be expressed as the superposition of two plane waves, which traveling in the opposite directions, when the energy quantum number n is even. However, it can not be done when n is odd, because they are not the period functions in this interval. If only if the width of the well tends to infinity, or n is very large, and the expanding coefficients on the base of continuous spectrum become to the superposition of two symmetrical delta functions, all of the stationary wavefunctions can be expanded as the superposition of two plane waves.

关 键 词：定态波函数 周期函数 量子力学 无限深势阱 粒子 

分 类 号：O413.1[理学—理论物理]