Fourth order real space solver for the time-dependent Schr\"odinger equation with singular Coulomb potential

TL;DR

This paper introduces a high-order numerical method combining split-operator and Crank-Nicolson schemes to solve the 3D axially symmetric Schrödinger equation with Coulomb potential, achieving high accuracy and efficiency.

Contribution

A novel hybrid splitting scheme that merges split-operator and Crank-Nicolson methods for improved accuracy and robustness in solving the Schrödinger equation with Coulomb potential.

Findings

Achieves fourth order accuracy in space and time discretization.

Demonstrates high precision in simulating optical tunneling in hydrogen.

Validates the method's efficiency and robustness through numerical experiments.

Abstract

We present a novel numerical method and algorithm for the solution of the 3D axially symmetric time-dependent Schr\"odinger equation in cylindrical coordinates, involving singular Coulomb potential terms besides a smooth time-dependent potential. We use fourth order finite difference real space discretization, with special formulae for the arising Neumann and Robin boundary conditions along the symmetry axis. Our propagation algorithm is based on merging the method of the split-operator approximation of the exponential operator with the implicit equations of second order cylindrical 2D Crank-Nicolson scheme. We call this method hybrid splitting scheme because it inherits both the speed of the split step finite difference schemes and the robustness of the full Crank-Nicolson scheme. Based on a thorough error analysis, we verified both the fourth order accuracy of the spatial discretization in the optimal spatial step size range, and the fourth order scaling with the time step in the case of proper high order expressions of the split-operator. We demonstrate the performance and high accuracy of our hybrid splitting scheme by simulating optical tunneling from a hydrogen atom due to a few-cycle laser pulse with linear polarization.