The Schrödinger eigenvalue march

Abstract

A simple numerical method for the determination of Schrödinger equation eigenvalues is introduced. It is based on a marching process that starts from an arbitrary point, proceeds in two opposite directions simultaneously and stops after a tolerance criterion is met. The method is applied to solving several 1D potential problems including symmetric double-well (ammonia inversion problem) and Johnson asymmetric potentials, 3D hydrogen atom and Morse potential. Band structure calculation can equally be tackled by extending marching to the complex plane as illustrated with the Kronig–Penney problem.