Higher Order Unitary Integrators for the Schrödinger Equation

Abstract

For classical Hamiltonian dynamics, much progress has recently been made in manifestly symplectic integrators. In this work some of these integrators will be generalized to the Schrodinger equation. There they correspond to algorithms which preserve unitarity and time reversal invariance exactly. In particular, we apply them to the one dimensional harmonic and anharmonic oscillators. The accuracies of three different algorithms are compared. 1 Introduction: Symplectic integrators In the present paper we shall discuss some new integration algorithms for Schrodinger equations with coordinate independent kinetic energies. These algorithms are rather straightforward modifications of recently proposed algorithms for classical mechanical systems. Thus we shall first discuss these integrators which are known as "symplectic integrators" since they preserve the symplectic structure of classical mechanics. Symplectic integrators were introduced by De Vogelaere in 1956, in a series of unpublis..