Comparison of Numerical Approaches to the Time-Dependent Schr\"odinger Solutions in One Dimension

TL;DR

This paper compares various numerical time propagation schemes for solving the 1D Schrödinger equation, showing that more accurate methods can be more efficient overall despite higher per-step costs.

Contribution

It provides a comparative analysis of different numerical schemes for the time-dependent Schrödinger equation, highlighting the efficiency of higher-order methods in one-dimensional models.

Findings

Higher-order methods allow larger time-steps, reducing overall computation time.

Simple low-order schemes require very small time-steps, increasing total computation.

Some accurate methods are computationally cheaper overall despite higher per-step costs.

Abstract

We examine the performance of various time propagation schemes using a one-dimensional model of the hydrogen atom. In this model the exact Coulomb potential is replaced by a soft-core interaction. The model has been shown to be a reasonable representation of what occurs in the fully three-dimensional hydrogen atom. Our results show that while many numerically simple (low order) propagation schemes work, they often require quite small time-steps. Comparing them against more accurate methods, which may require more work per time-step but allow much larger time-steps, can be illuminating. We show that at least in this problem, the compute time for a number of the more accurate methods is actually less than lower order schemes. Finally, we make some remarks on what to expect in generalizing our findings to more than one dimension.