Finite-difference approach to solving Heisenberg’s operator equations of motion: Application to one-dimensional time dependent Hamiltonians

Abstract

We reviewed and expanded on the finite difference approach introduced by Moncrief [Phys. Rev. D 28, 2485 (1983)] and Bender [Phys. Rev. Lett. 55, 901 (1985)] for solving Heisenberg’s operator equations of motion. In this approach, finite-difference recurrence relations are used to evolve the matrix representation of the operators in time. The advantages and disadvantages of this approach for the study of quantum processes in real time are discussed. The approach performed very well as illustrated by examples of harmonic and Morse oscillators interacting with continuous wave and pulsed laser fields.