Numerical solutions of the Schrodinger equation with source terms or time-dependent potentials

TL;DR

This paper introduces a numerical method combining the generalized Crank-Nicolson approach with Euler-MacLaurin expansion to accurately solve the time-dependent Schrödinger equation with source terms and time-dependent potentials.

Contribution

The paper presents a novel numerical scheme that improves precision in solving the Schrödinger equation with nonhomogeneous terms, extending existing methods.

Findings

Method achieves accuracy comparable to traditional approaches.

Systematic precision increase allows error estimation.

Validated against analytically solvable models.

Abstract

We develop an approach to solving numerically the time-dependent Schrodinger equation when it includes source terms and time-dependent potentials. The approach is based on the generalized Crank-Nicolson method supplemented with an Euler-MacLaurin expansion for the time-integrated nonhomogeneous term. By comparing the numerical results with exact solutions of analytically solvable models, we find that the method leads to precision comparable to that of the generalized Crank-Nicolson method applied to homogeneous equations. Furthermore, the systematic increase in precision generally permits making estimates of the error.