Boundary Conditions And Numerical Solution Of Schrodinger Equation

The Schrdinger equation is a very important partial differential equation in modern mathematics and quantum physics and quantum chemistry study, it has great significance in many fields. Such as quantum semiconductor, electromagnetic wave propagation, seismic migration and so on, many practical problems need to be solved numerically of Schrdinger equation and various forms of solution.However, given a initial value problems of Schrdinger equation, there are still many problems related to how to achieve its value. Firstly, to construct an proper boundary conditions of initial value problem is the beginning of studying the numerical solution. In many eyes of scholars of experimental physics and engineering technology, assigned to the zero boundary condition is still appropriate in the larger area. But the drawbacks of this approach is not possible for a long time span to describe, because the waveform can be transferred to the original area on the border. If we want to describe a longer span of time evolved the question, it is necessary to the original computational domain to expand again, the inevitable to spend a lot of human and financial resources. Since the last century, many partial differential scholars in an attempt to construct an appropriate boundary conditions to depicting the Schrdinger equation for a longer span of time derivative at the small area. This thesis launches the research work on this problem on the basis of previous. This thesis is mainly divided into the following four parts:The first part begins with a short introduction to the Schrdinger equation of background knowledge, and provides several examples of analysis. Finally, brief summarizes the research status of Schrdinger equation at home and abroad.The second part, starting from the one-dimensional linear Schrdinger equation, we construct its boundary conditions, including accurate boundary conditions and artificial boundary conditions. Describes the discrete method of them,and thus provides discrete method for the initial boundary value problem of derivative from it. Finally, some numerical examples is given at the end.The third part, one-dimensional extended to high dimension, which does not directly on boundary conditions of constructing for high dimensional problems,but transform and processing for the equation. This approach aims to make the high dimensional problem into one dimension problem. On the other hand, we also use more direct splitting theory gives a numerical method for high dimensional problems.The fourth part, we discuss some special Schrdinger equation, including nonlinear and linear example, when we discuss them and distinguish the second part.