Gray and Verosky have recently studied the reformulation of the N-state matrix representation of the time-dependent Schrödinger equation as an N-degrees of freedom classical Hamiltonian system. This opens the possibility of using in quantum dynamics numerical integrators originally devised for classical mechanics. When the Hamiltonian matrix is time-dependent, Gray and Verosky suggest the use of a Magnus approximation before reducing the quantum system to its classical format. We show that Magnus approximations are not necessary and suggest an alternative technique. With the new technique it is possible to obtain simple integrators of arbitrarily high orders of accuracy that can be applied to all matrix Schrödinger problems with a, possibly time-dependent, real Hamiltonian matrix. The connection between the new approach and high-order split-operator methods is studied.
Gray and Verosky have recently studied the reformulation of the N-state matrix representation of the time-dependent Schrödinger equation as an N-degrees of freedom classical Hamiltonian system. This opens the possibility of using in quantum dynamics numerical integrators originally devised for classical mechanics. When the Hamiltonian matrix is time-dependent, Gray and Verosky suggest the use of a Magnus approximation before reducing the quantum system to its classical format. We show that Magnus approximations are not necessary and suggest an alternative technique. With the new technique it is possible to obtain simple integrators of arbitrarily high orders of accuracy that can be applied to all matrix Schrödinger problems with a, possibly time-dependent, real Hamiltonian matrix. The connection between the new approach and high-order split-operator methods is studied.
The general, N state matrix representation of the time-dependent Schrödinger equation is equivalent to an N degree of freedom classical Hamiltonian system. We describe how classical mechanical methods and ideas can be applied towards understanding and modeling exact quantum dynamics. Two applications are presented. First, we illustrate how qualitative insights may be gained by treating the two state problem with a time-dependent coupling. In the case of periodic coupling, Poincaré surfaces of section are used to view the quantum dynamics, and features such as the Floquet modes take on interesting interpretations. The second application illustrates computational implications by showing how Liouville’s theorem, or more generally the symplectic nature of classical Hamiltonian dynamics, provides a new perspective for carrying out numerical wave packet propagation. We show how certain simple and explicit symplectic integrators can be used to numerically propagate wave packets. The approach is illustrated with an application to the problem of a diatomic molecule interacting with a laser, although it and related approaches may be useful for describing a variety of problems.
We consider the numerical propagation of models that combine both quantum and classical degrees of freedom, usually, electrons and nuclei, respectively. We focus, in our computational examples, on the case in which the quantum electrons are modeled with time-dependent density-functional theory, although the methods discussed below can be used with any other level of theory. Often, for these so-called quantum-classical molecular dynamics models, one uses some propagation technique to deal with the quantum part and a different one for the classical equations. While the resulting procedure may, in principle, be consistent, it can however spoil some of the properties of the methods, such as the accuracy order with respect to the time step or the preservation of the geometrical structure of the equations. Few methods have been developed specifically for hybrid quantum-classical models. We propose using the same method for both the quantum and classical particles, in particular, one family of techniques that proves to be very efficient for the propagation of Schrödinger-like equations: the (quasi)-commutator free Magnus expansions. These have been developed, however, for linear systems, yet our problem is nonlinear: formally, the full quantum-classical system can be rewritten as a nonlinear Schrödinger equation, i.e., one in which the Hamiltonian depends on the system itself. The Magnus expansion algorithms for linear systems require the application of the Hamiltonian at intermediate points in a given propagating interval. For nonlinear systems, this poses a problem as this Hamiltonian is unknown due to its dependence on the state. We approximate it by employing a higher order extrapolation using previous steps as input. The resulting technique can then be regarded as a multistep technique or, alternatively, as a predictor corrector formula.
Hybrid quantum-classical systems combine both classical and quantum degrees of freedom. Typically, in chemistry, molecular physics, or saterials science, the classical degrees of freedom describe atomic nuclei (or cations with frozen core electrons), whereas the quantum particles are the electrons. Although many possible hybrid dynamical models exist, the essential one is that the so-called Ehrenfest dynamics that results from the simple partial classical limit applied to the complete quantum Schrodinger equation. Few numerical methods are developed specifically for the mixing of this sort of systems. Here we present a preliminary study of the performance of a family of recently developed propagators: the (quasi) commutator-free Magnus expansions. These methods, however, were initially designed for nonautonomous linear equations. We employ them for the nonlinear Ehrenfest system, by approximating the state value at whenever step within the propagation, using an extrapolation from previous time steps.
Magnus expansion (ME) provides a general way to expand the real-time propagator of a time-dependent Hamiltonian within the exponential such that the unitarity is satisfied at any order. We use this property and explicit integration of Lagrange interpolation formulas for the time-dependent Hamiltonian within each time interval and derive approximations that preserve unitarity for the differential time evolution operators of general time-dependent Hamiltonians. The resulting second-order approximation is the same as using the average of Hamiltonians for two end points of time. We identify three fourth-order approximations involving commutators of Hamiltonians at different times and also derive a sixth-order expression. A test of these approximations along with other available expressions for a two-state time-dependent Hamiltonian with sinusoidal time dependences provides information on the relative performance of these approximations and suggests that the derived expressions can serve as useful numerical tools for time evolution in time-resolved spectroscopy, quantum control, quantum sensing, real-time ab initio quantum dynamics, and open system quantum dynamics.
Abstract Two methods for the numerical integration of the time‐dependent Schrödinger equation with given initial conditions (initial wave packet) are presented. The first method (method A) is based on the Schrödinger representation of the quantum‐dynamical system while the second one (method B) is based upon the intermediate representation. In both cases the quantum dynamical equation is transformed into a system of Hamilton‐Jacobi type equations of motion as occurring in multi particle classical dynamics, i.e. standard molecular dynamics techniques can be applied for the integration. The dynamics of a minimum uncertainty Gaussian wave packet in a strongly anharmonic oscillator is taken as an example.
This talk is about a method for solving the numerical initial value problem of the time-dependent Schrodinger equation with a Hamiltonian which may depend on time. It is only for time independent Hamiltonians that I was able to describe the accuracy and stability properties in the form of proven theorems. All practical experience shows, that the time-dependent case behaves equally well. In this method, the Hamiltonian comes in only as a hermitian linear operator and not as a mathematical expression that can be reordered or otherwise transformed in order to speed up the necessary calculations. In all cases where it is a feasible computational task to apply the Hamilton operator to a generic state vector, my suggestion is to try this method as a workhorse. It can be considered a ‘Simpson’s rule’ for quantum dynamics. There is no way for me to explain the matter from a technical point of view in 10 minutes, so I’ll try to convey the basic idea of the method. This idea is to treat quantum dynamics in closer analogy to classical dynamics as one usually does. One may argue, that there is a fundamental difference between classical and quantum dynamics since the Schrodinger equation is first order with respect to the time derivative, whereas classical mechanics is inherently second order, since forces determine accelerations and not velocities. This is not so clear a distinction as one might think, since also classical mechanics can be given the first-order form of canonical equations. I propose the opposite of such an approach, namely to give quantum dynamics a second-order form by simply differentiating the time-dependent Schrodinger equation with respect to time. To be sure, this is only a heuristic device for defining a computationally feasible approximation and not a proposal to change quantum mechanics. For the resulting equation for the second time derivative of the wave function, I then carry over and use the integration methods from the field of granular matter dynamics which I have been using in an industrial project concerning simulation of the toning process in electro-photographic copiers/printers over some years [4]. Such methods come in