Solution of the Schrödinger Equation

Abstract

This chapter deals with the application of group theory to the solution of the Schrödinger equation. The Schrödinger equation is a partial differential equation with second-order derivatives in space and a first-order derivative in time and represents the central equation of non-relativistic quantum mechanics. The Schrödinger equation contains the Hamilton operator whose eigenvalues represent the energy levels of the system. The symmetry of the Hamilton operator leads to the group of the Schrödinger equation. The irreducible representations of this group fully determine the symmetry of the solutions of the Schrödinger equation as well as the degeneracy of the energy levels. Furthermore, the concept of perturbation theory is introduced. Crystal field theory is a semi-empirical theory build in the framework of linear time-independent perturbation theory. Group theory is used to determine the splitting of energy levels due to a crystal field as well as the terms within the effective crystal field Hamiltonian. Finally, time-dependent perturbation theory is discussed and transition probabilities and selection rules are formulated.