Simulation of Time-Dependent Schrödinger Equation in the Position and Momentum Domains

Abstract

The paper outlines the development of a new, spectral method of simulating the Schr ö dinger equation in the momentum domain. The kinetic energy operator is approximated in the momentum domain by exploiting the derivative property of the Fourier transform. These results are compared to a position-domain simulation generated by a fourth-order, centered-space, finite-difference formula. The time derivative is approximated by a four-step predictor-corrector in both domains. Free-particle and square-well simulations of the time-dependent Schr ö dinger equation are run in both domains to demonstrate agreement between the new, spectral methods and established, finite-difference methods. The spectral methods are shown to be accurate and precise.