Time-dependent integral equation approach to quantum dynamics of systems with time-dependent potentials

Abstract

The solution of the time-dependent Schrödinger equation in the form exp(−i Ht/ℏ)Ψ(0) is valid only for Hamiltonian systems for which H is independent of time. This fact precludes the solution in this form for any time interval, t, during which H changes significantly. We present a formal solution, by means of integral equations, which is valid for time-dependent potentials and arbitrarily long times, as well as for time-dependent stochastic potentials. An important feature of the approach is the explicit inclusion of time ordering of noncommuting operators even when the formal solution is converted into a computationally viable form. The resulting time-ordered expression provides a basis for numerical algorithms for treating such systems to arbitrary accuracy. Several schemes for implementing the approach are presented. Some problems for which the present approach is the appropriate one include: (1) mixed quantum and classical systems (e.g., systems containing both light and heavy particles); (2) systems in which the interaction with radiation is treated semiclassically and (3) nonisolated systems. In addition we show briefly how the same techniques can be applied to treat the von Neumann equation for the density matrix, time-dependent Hohenberg—Kohn—Sham density functional equations (where the time dependence arises from a mixed quantum-classical treatment of electron-ion interactions in solids), and to an example of a nonisolated system, namely a driven oscillator.