Non-standard FDTD implementation of the Schr\"odinger equation

TL;DR

This paper applies a non-standard finite-difference time-domain method to solve the time-dependent Schrödinger equation, incorporating potential variations and implementing Perfectly Matched Layers for improved simulation accuracy.

Contribution

It introduces a novel non-standard FDTD approach for the Schrödinger equation, accounting for different potential energies and deriving specific equations for these scenarios.

Findings

Effective handling of potential energy differences in Schrödinger equation simulations

Derivation of equations involving hyperbolic sine functions for higher potentials

Implementation of Perfectly Matched Layers using the non-standard FDTD method

Abstract

In this work, we apply the Cole's non-standard form of the FDTD to solve the time dependent Schr\"odinger equation. We deduce the equations for the non-standard FDTD considering an electronic wave function in the presence of potentials which can be higher or lower in comparison with the energy of the electron. The non-standard term is found to be almost the same, except for a sine functions which is transformed to a hyperbolic sine function,as the argument is imaginary when the potential has higher energy than the electron. Perfectly Matched Layers using this methodology are also presented.