Solution and numerical analysis of two-dimensional time-independent Schrdinger equation based on finite difference method

Abstract

With the continuous development and widespread use of quantum mechanics, solving the Schrdinger equation has become a hot research topic. The finite difference method has the advantages of simple calculation and high accuracy, which means that it has high potential in solving the numerical solutions of the Schrdinger equation. In this paper, we deeply explore the problem of using the finite difference method to solve the numerical solution of the time-independent Schrdinger equation, propose a solution method based on the finite difference method, and evaluate its performance under different conditions. Firstly, by analyzing the principles and characteristics of the finite difference method, we construct a difference format for the time-independent Schrdinger equation. Then, by converting the difference format of the numerical solutions of the equation into a matrix, the numerical calculation problem is transformed into a matrix eigenvalue and eigenvector problem. Finally, for different physical scenarios, the established model is numerically solved and its performance is analyzed. This study found that the constructed numerical solution method exhibits high accuracy and stability in solving the numerical solutions of the time-independent Schrdinger equation. In different physical scenarios, this method can provide satisfactory results, thus verifying the feasibility of applying the finite difference method to this problem.