Solution Of Schrodinger-Poisson Equations Based On Finite Element Method

Schrodinger’s equation is the basic equation of the quantum mechanics,it also called the Schrodinger wave equation.It describes the transform of the wave function with the time in physic systems. Meanwhile,it also applys to the electromagnetic wave propagation, underwater acoustics,optical fiber and some design of the optoelectronic devices.It simulates a electromagnetic wave problem in the Weak structures.Especially,it have applied greatly to the computation of the large scale quantum mechanics.Poisson’s equation is a common partial differential equation(PDE) in electrostatics,mechanical engineering and theoretical physics.It describles the relationship between the potential function of electrostatic field and the source charges. In the computer simulation of the modern optoelectronic devices, it is the key point to solve the Schrodinger’s equation and Poisson’s equation. So it becomes a very important subject that we set up a efficient numerical method to solve the Schrodinger’s equation and Poisson’s equation.Compared to the most numerical methods in this paper, we find the finite element method is a very efficient numerical method(FEM) to solve the Schrodinger’s equation and Poisson’s equation with a given boundary condition. Hence, we have analysised the finite element method in detail.Besides, we have used the finite element method to solve the general Schrodinger’s equation and Poisson’s equation.At last, we have delevoped the solver in FEM of the Schrodinger’s equation and Poisson’s equation.We have finished these works below in this paper:First of all, we have analysised how to solve the partial differential equation(PDE) using the finite element method in detail. Then, we have solved the general Schrodinger’s equation and Poisson’s equation using the finite element method and we have given the solving ideas and steps in detail.Thirdly, after compared to those most numerical computational tools, we decide to use the the open source C++library libMesh to solve a two dimension general Poisson’s equation and a three dimension general Schrodinger’s equation. We have also checked and visualized the solutions. We have verified the accuracy and flexibility of using the finite element method to solve partial differential equation, especially for Schrodinger’s equation and Poisson’s equation. At last, we have applied these source code based on libMesh of solving the Schrodinger’s equation and Poisson’s equation to our research group of the computer simulation of terahertz antenna.