# Nonstandard Finite Difference Time Domain (NSFDTD) Method for Solving the Schr\"odinger Equation

## TL;DR

This paper introduces a nonstandard finite difference time domain (NSFDTD) method that improves the accuracy of solving the Schrödinger equation by iteratively refining eigen-energy estimates using a modified numerical scheme.

## Contribution

The paper presents a novel NSFDTD approach that enhances eigen-energy calculations for quantum systems by replacing standard derivatives with non-standard schemes and iterative refinement.

## Key findings

- Validated on infinite square well, harmonic oscillator, and Morse potentials.
- Achieves more accurate eigen-energies compared to standard FDTD.
- Demonstrates iterative improvement in numerical solutions.

## Abstract

In this paper, an improvement of the finite difference time domain (FDTD) method using a non-standard finite difference scheme is presented. The standard numerical scheme for the second derivative in the spatial domain is replaced by a non-standard numerical scheme. In order to apply the non-standard FDTD (NSFDTD), first estimates of eigen-energies of a system are needed and computed by the standard FDTD method. These first eigen-energies are then used by the NSFDTD method to obtain improved eigen-energies. The NSFDTD method can be performed iteratively using the resulting eigen-energies to obtain accurate results. In this paper, the NS-FDTD method is validated for infinite square well, harmonic oscillator and Morse potentials.