On numerical solutions of the time-dependent Schr\"odinger equation

TL;DR

This paper reviews an explicit numerical method for solving the time-dependent Schrödinger equation, emphasizing its simplicity, accuracy, efficiency, and adaptability to higher dimensions and separate wave function components.

Contribution

It introduces a straightforward, explicit algorithm that can be extended to multi-dimensional systems and allows separate calculation of wave function parts, improving computational flexibility.

Findings

Method achieves significant accuracy and efficiency.

Algorithm extends to higher spatial dimensions.

Enables separate calculation of real and imaginary wave function parts.

Abstract

We review an explicit approach to obtaining numerical solutions of the Schr\"odinger equation that is conceptionally straightforward and capable of significant accuracy and efficiency. The method and its efficacy are illustrated with several examples. Because of its explicit nature, the algorithm can be readily extended to systems with a higher number of spatial dimensions. We show that the method also generalizes the staggered-time approach of Visscher and allows for the accurate calculation of the real and imaginary parts of the wave function separately.