Meshless basis set for solving one-dimensional time independent Schrodinger Equation

Abstract

The ability of Schrödinger equation in representing the energies and wave function of the molecular system has attracted many scientists to find the best solution for it. Due to the restriction of this equation that can only be solved analytically for the simple molecular models; numerous of numerical methods were introduced to solve it. The Fourier grid Hamiltonian (FGH) method introduced by Marston and Balint-Kurti in 1989 solved the one-dimensional time independent Schrödinger equation for H2 molecule by using the plane wave basis set and coupling with the Fast Fourier Transform (FFT) technique to reduce the computational time. In this study, we implement the Meshless Element Free Galerkin (MEFG) method to solve the same problem. The localized basis sets adopted in this method and the compactness properties of the weight function lead us to generate a sparse Hamiltonian matrix in finding the eigenvalues and eigenfunctions. The aim of this study is to develop a new numerical approach to solve the one-dimensional time independent Schrödinger equation and as a preliminary research work for us to investigating into the computational quantum mechanics studies.