An Accurate Pentadiagonal Matrix Solution for the Time-Dependent Schr\"{o}dinger Equation

TL;DR

This paper introduces a highly accurate pentadiagonal scheme for solving the time-dependent Schrödinger equation, improving numerical precision and enabling efficient analysis of bipartite wavepacket dynamics.

Contribution

It develops a novel pentadiagonal Crank-Nicolson method using a five-point stencil, enhancing accuracy over standard approaches for quantum evolution simulations.

Findings

Solutions are significantly more accurate than standard methods.

The method enables decoupling bipartite dynamics into independent single-particle problems.

Conditions for product state preservation are derived.

Abstract

One of the unitary forms of the quantum mechanical time evolution operator is given by Cayley's approximation. A numerical implementation of the same involves the replacement of second derivatives in Hamiltonian with the three-point formula, which leads to a tridiagonal system of linear equations. In this work, we invoke the highly accurate five-point stencil to discretize the wave function onto an Implicit-Explicit pentadiagonal Crank-Nicolson scheme. It is demonstrated that the resultant solutions are significantly more accurate than the standard ones. We also discuss the resolution of bipartite wavepacket dynamics and derive conditions under which a product state from the laboratory perspective remains a product state from the center-of-mass point of view. This has profound applications for decoupling complicated bipartite dynamics into two independent single-particle problems.