Non-Standard and Numerov Finite Difference Schemes for Finite Difference Time Domain Method to Solve One- Dimensional Schrödinger Equation

Abstract

<span>The purpose of this paper is to show some improvements of the finite-difference time domain (FDTD) method using Numerov and non-standard finite difference (NSFD) schemes for solving the one-dimensional Schr</span><span>ö</span><span>dinger equation. Starting with results of the unmodified FDTD method, Numerov-FD and NSFD are applied iteratively to produce more accurate results for eigen energies and wavefunctios. Three potential wells, infinite square well, harmonic oscillator and Poschl-Teller, are used to compare results of FDTD calculations. Significant improvements in the results for the infinite square potential and the harmonic oscillator potential are found using Numerov-NSFD scheme, and for Poschl-Teller potential are found using Numerov scheme.</span>

An alternating-direction hybrid implicit-explicit finite-difference time-domain method for the Schrödinger equation

“…A multitude of different implementations exist, depending on the spatial discretization and the time propagation, each having their strengths and weaknesses. Moreover, the FDTD method has also been applied to various related equations such as the non-linear Schrödinger equation [20][21][22][23][24][25][26], the Gross-Pitaevskii equations [27], and the stationary Schrödinger equation by performing a Wick rotation [28][29][30][31].…”