Analysis of Two Dimensional Potential Well Quantum Systems by D₄ Group

The wave function of quantum systems is generally solved by the Schrödinger equation.but for some systems with high symmetry, the adoption of group theory methods can make the problem more convenient.In this paper, the D₄ group is used to analyze Energy Levels in two-dimensional potential well quantum systems.Firstly, the steady-state Schrödinger equation, eigenvectors, eigenenergy levels, and perturbation effects of a two-dimensional infinite deep square potential well quantum mechanical system are analyzed; Then, through symmetry analysis, it is concluded that the group of the system belongs to D₄ group, and the characteristics of D₄ group associated with two-dimensional square potential well quantum system are explained; Finally, the intrinsic wave function and energy level structure of a two-dimensional potential well quantum system are studied using the D₄ group.At the time, the energy level characteristics of regular degeneracy and accidental degeneracy under symmetric perturbation are solved as the same time.The main innovation of this article is to analyze energy levels and to solve the splitting problem of degenerate energy levels under symmetric perturbations by group theory methods.