A comparison of numerical approaches to the solution of the time-dependent Schrödinger equation in one dimension

Abstract

We present a simple, one-dimensional model of an atom exposed to a time-dependent intense, short-pulse EM field with the objective of teaching undergraduates how to apply various numerical methods to study the behavior of this system as it evolves in time using several time propagation schemes.In this model, the exact Coulomb potential is replaced by a soft-core interaction to avoid the singularity at the origin. While the model has some drawbacks, it has been shown to be a reasonable representation of what occurs in the fully three-dimensional hydrogen atom.The model can be used as a tool to train undergraduate physics majors in the art of computation and software development. Program summaryProgram Title: 1d hydrogen light interactionProgram Files doi:http://dx.doi.org/10.17632/2275fmvdzc.1Code Ocean Capsule:https://doi.org/10.24433/CO.1476487.v1Licensing provisions: MIT licenseProgramming language: FORTRAN90Nature of problem: The one dimensional time dependent Schrödinger equation has been shown to be quite useful as a model to study the Hydrogen atom exposed to an intense, short pulse, electromagnetic field. We use a model potential that is cut-off near x=0 and avoids the singularity of the true 1-D potential, but retains the characteristic Rydberg series and continuum to study excitation and ionization of the true H atom. The code employs a number of numerical methods to understand and compare the efficacy and accuracy when applied to this model problem.Solution method: The program uses and contrasts a number of approaches; the Crank–Nicolson, Short Iterative Lanczos, various incarnations of the split-operator and the Chebychev method have been programmed. These methods have been compared using a 3-point finite difference (FD) discretization of the space coordinate. For completeness, some attention has also been given to using 5–9 FD formulas in order to show how higher order discretization affects the accuracy and efficiency of the methods but the primary focus of the method is the time propagation.Additional comments including restrictions and unusual features: The main purpose of this code is as a teaching tool for undergraduates interested in acquiring knowledge of numerical methods and programming skills useful to a practicing computational physicist.