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

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.本研究提出一款简易的一维原子模型，该模型处于含时强短脉冲电磁场（electromagnetic field，EM field）中，旨在帮助物理专业本科生学习如何运用多种数值方法与时间传播方案，探究该系统的含时演化行为。在此模型中，我们采用软核相互作用（soft-core interaction）替代精确库仑势（Coulomb potential），以规避原点处的奇点问题。尽管该模型存在一定局限性，但已有研究表明，它能够合理复现全三维氢原子中的物理过程。本模型可作为教学工具，助力物理专业本科生掌握计算与软件开发的实践技能。