Spectral And Boundary Value Methods For Two-dimensional Schrodinger Equation

The Schr dinger equation is the basic equation of quantum mechanics inthe physical system. It can clearly describe the regular of how the quantum evolvesover time. By solving the Schr dinger equation which the micro systemcorrespond to, we can get the wave function and energy, and thus calculate theprobability distribution of the particles, further understand the nature ofit. In chemistry, physics and other fields of scientific research, the results ofsolving the Schrodinger equation are basically consistent with the actual. In recentyears, many scholars investigated the Schr dinger equation with complex potentialfunction based on various methods, and explained a series of importantphenomena. Therefore, solving the Schr dinger equation is significant.The main purpose of this paper is to solve two dimensional Schr dingerequation by using the Galerkin-Chebyshev spectral method and the boundaryvalue method. Firstly we use the spectral method to approximate the spatialderivation, discretize the two dimensional Schr dinger equation，and transform theoriginal problem into a set of linear ordinary differential equations in thecomplex field. Then we use the boundary value method to solve the equations, thenumerical solutions is the solutions of the original problem, and then we analyzethe error. Finally we use Matlab to conduct the numerical simulations, and give theimages of the numerical solutions and errors, which show that the methods havehigh precision and good stability.