Solving the time-dependent Schr\"odinger equation with absorbing boundary conditions and source terms in Mathematica 6.0

TL;DR

This paper introduces a simplified numerical method for solving the time-dependent Schrödinger equation with absorbing boundary conditions and source terms, using a Crank-Nicholson scheme with analytical matrix inversion suitable for Mathematica and similar software.

Contribution

It presents an analytically inverted matrix approach that simplifies implementing the Crank-Nicholson method with source terms in common scientific computing software.

Findings

Efficient implementation of Schrödinger equation solutions in Mathematica.

Ability to include source terms naturally in the numerical scheme.

Simplification of the numerical method through analytical matrix inversion.

Abstract

In recent decades a lot of research has been done on the numerical solution of the time-dependent Schr\"odinger equation. On the one hand, some of the proposed numerical methods do not need any kind of matrix inversion, but source terms cannot be easily implemented into this schemes; on the other, some methods involving matrix inversion can implement source terms in a natural way, but are not easy to implement into some computational software programs widely used by non-experts in programming (e.g. Mathematica). We present a simple method to solve the time-dependent Schr\"odinger equation by using a standard Crank-Nicholson method together with a Cayley's form for the finite-difference representation of evolution operator. Here, such standard numerical scheme has been simplified by inverting analytically the matrix of the evolution operator in position representation. The analytical inversion of the N x N matrix let us easily and fully implement the numerical method, with or without source terms, into Mathematica or even into any numerical computing language or computational software used for scientific computing.