Implementation details of a variational method to solve the time independent Schrödinger equation

Abstract

The time independent Schrödinger equation is a differential equation of great interest in computational physics. In many cases, it is impossible to reach an analytical solution for it, due to the potential function complexity, therefore numerical methods play an important role in its solution for practical cases. By means of numerical methods it is possible to solve the stationary Schrödinger equation for arbitrary potentials, allowing the study of interesting potentials that exhibit fascinating phenomena. Some of these potentials are the Kronig-Penney and pseudo-Coulomb potential functions, or more single like barrier potential function. Nevertheless, in many cases, the implementation of the sequence of steps needed to solve the differential equation is not straightforward. In this work we present and explain a sequence of steps to solving the time independent Schrödinger equation by means of the variational method, and apply it to solve non-periodic potential functions, like the harmonic oscillator potential well and rectangular potential barrier, and periodic potential functions like the Kronig-Penney and pseudo-Coulomb. Our main purpose is for this work to be an introduction to the computational quantum mechanics field.