Four numerical approaches to solve the time-independent Schrödinger equation

Abstract

Schrödinger’s equation was a milestone in the development of Quantum Mechanics and continues to be essential in the study of non-relativistic systems to this day. Quantum Mechanics textbooks generally address problems in which it is possible to find analytical expressions for the eigenenergies and eigenstates of the system. However, we do not know analytical solutions for numerous physical systems of interest. Thus, numerical methods are essential to complement analytical approaches. In this work, we present numerical methods for solving Quantum Mechanics problems. We begin by discussing aspects of the theory we use to motivate the different numerical approaches. Next, we introduce discretization and dimensionless approaches, aiming to use them in numerical simulations. We present the shooting, matching, matrix and variational Monte Carlo methods. We apply the techniques to different physical systems and discuss the advantages and disadvantages of each one. We hope students can reproduce the results presented and extend them to their applications of interest.

Keywords
Time-independent Schrödinger’s equation; Variational Monte Carlo; Shooting method; Matching method