Modified Finite Difference Method on the 1-D Schrodinger Equation

The time-dependent Schrödinger equation (TDSE) is a fundamental law in understanding the states of many microscopic systems. Such systems occur in nearly all branches of physics and engineering, including atomic, high-energy, and solid-state physics just to name a few. A robust and efficient algorithm to solve the TDSE would be an essential goal in these respective fields. In this study we use the well-known method for solving the TDSE, the finite difference method (FDM) but with an important modification: formulating the TDSE in a form suitable for symplectic integration to conserve flux. In this presentation we discuss case studies and present results on stability, flux conservation, as well as scattering probabilities in 1D potentials. We show that the modified method is a stable, promising, and accurate method for linear domains over lower dimensions with arbitrary potentials. We also discuss challenging problems of scaling to higher dimensions and more refined grids.