谱方法和边界值法求解二维薛定谔方程硕士学位专业论文.docVIP

摘 要 薛定谔方程是物理系统中量子力学的基础方程，它可以清楚地说明量子在系统中随时间变化的规律。通过求解微观系统所对应的薛定谔方程，我们能够得到其波函数以及对应的能量，从而计算粒子的分布概率，进一步来了解其性质。在化学和物理等诸多科学研究领域当中，薛定谔方程求解的结果都与实际很相符。近年来，很多学者通过各种方法研究具有复杂势函数的薛定谔方程，解释了很多重要的物理现象，因此对薛定谔方程的求解具有相当重要的意义。 本文主要是用Galerkin-Chebyshev谱方法和边界值法求解二维薛定谔方程。首先运用Galerkin-Chebyshev谱方法来对空间导数进行近似，离散二维薛定谔方程，从而将原问题转化为复数域上的线性常微分方程组。然后用边界值法求解该方程组，所求得的数值解即为原问题的解，之后进行误差分析。最后利用Matlab进行数值模拟，给出数值解的图像以及误差曲面图像，结果显示此方法精度高且具有很好的稳定性。 关键词：Galerkin-Chebyshev谱方法；边界值法；数值解； 精度高；稳定 Abstract The Schr?dinger equation is the basic equations of quantum mechanics?in the?physical system. It can clearly describe the regular of the quantum evolves over time. By solving the Schr?dinger equation which the micro?system correspond, we can get?the wave function?and energy, and thus calculate the probability distribution of the particles, further understand the nature of it.?In?chemistry, physics?and other?fields of scientific research,?the ?results of solving the Schrodinger?equation are basically consistent with the actual.?In recent years,??many researchers used a variety of methods to?investigate the Schr?dinger equation with complex?potential function,?and explained a lot of?important phenomena.?Thus?solving the Schr?dinger?equation?has very important significance. The main purpose of this paper is to solve the two dimensional?Schr?dinger equation through the Galerkin-Chebyshev?spectral method?and the?boundary value?method. First we use the?spectral method?to approximate the spatial derivation, discretize the?two dimensional Schr?dinger equation， original?problem into a set of linear ordinary differential equations in? the complex number field.?Then by using the boundary value?method to solve the equations,?that the numerical?solutions is the solutions of the original problem, and then analyze the error. Finally we use?Matlab to conduct the numerical simulation, and give the images of?the numerical solutions and?errors, which show that the methods have high precision and good stability. Keywords: Sc