QuantumFDTD -- A computational framework for the relativistic Schr\"odinger equation

TL;DR

QuantumFDTD version 3 extends a computational framework to solve both relativistic and non-relativistic three-dimensional Schrödinger equations using FFT-based methods, supporting arbitrary potentials and parity projections for diverse quantum physics applications.

Contribution

The paper introduces the inclusion of the relativistic Schrödinger equation and optimized FFT-based kinetic energy terms into the QuantumFDTD code, enhancing its capabilities.

Findings

Successfully extended QuantumFDTD to relativistic cases.

Implemented FFT-based kinetic energy calculations.

Supports arbitrary external potentials and parity projection.

Abstract

We extend the publicly available quantumfdtd code. It was originally intended for solving the time-independent three-dimensional Schr\"odinger equation via the finite-difference time-domain (FDTD) method and for extracting the ground, first, and second excited states. We (a) include the case of the relativistic Schr\"odinger equation and (b) add two optimized FFT-based kinetic energy terms for the non-relativistic case. All the three new kinetic terms are computed using Fast Fourier Transform (FFT). We release the resulting code as version 3 of quantumfdtd. Finally, the code now supports arbitrary external file-based potentials and the option to project out distinct parity eigenstates from the solutions. Our goal is quark models used for phenomenological descriptions of QCD bound states, described by the three-dimensional Schr\"odinger equation. However, we target any field where solving either the non-relativistic or the relativistic three-dimensional Schr\"odinger equation is required.