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arXiv:quant-ph/0511229 · The Journal of Chemical Physics

Quantum-classical dynamics of wave fields

Alessandro Sergi

Published Feb 21, 2007• 7 citations
arXivHTMLPDF

Abstract

An approach to the quantum-classical mechanics of phase space dependent operators, which has been proposed recently, is remodeled as a formalism for wave fields. Such wave fields obey a system of coupled nonlinear equations that can be written by means of a suitable non-Hamiltonian bracket. As an example, the theory is applied to the relaxation dynamics of the spin-boson model. In the adiabatic limit, a good agreement with calculations performed by the operator approach is obtained. Moreover, the theory proposed in this paper can take nonadiabatic effects into account without resorting to surface-hopping approximations. Hence, the results obtained follow qualitatively those of previous surface-hopping calculations and increase by a factor of (at least) 2, the time length over which nonadiabatic dynamics can be propagated with small statistical errors. Moreover, it is worth to note that the dynamics of quantum-classical wave fields proposed here is a straightforward non-Hamiltonian generalization of the formalism for nonlinear quantum mechanics that Weinberg introduced recently.

DOI
10.1063/1.2472363

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An approach to the quantum-classical mechanics of phase space dependent operators, which has been proposed recently, is remodeled as a formalism for wave fields. Such wave fields obey a system of coupled nonlinear equations that can be written by means of a suitable non-Hamiltonian bracket. As an example, the theory is applied to the relaxation dynamics of the spin-boson model. In the adiabatic limit, a good agreement with calculations performed by the operator approach is obtained. Moreover, the theory proposed in this paper can take nonadiabatic effects into account without resorting to surface-hopping approximations. Hence, the results obtained follow qualitatively those of previous surface-hopping calculations and increase by a factor of (at least) 2, the time length over which nonadiabatic dynamics can be propagated with small statistical errors. Moreover, it is worth to note that the dynamics of quantum-classical wave fields proposed here is a straightforward non-Hamiltonian generalization of the formalism for nonlinear quantum mechanics that Weinberg introduced recently.
The symplectic structure of Weinberg's formalism for nonlinear quantum mechanics is first unveiled and then generalized to introduce non-Hamiltonian quantum mechanics. By exploiting the correspondence between wave and matrix mechanics, a link between this generalization and a non-Hamiltonian commutator, proposed recently by this author, is found. The general correspondence between operator and wave formalisms in non-Hamiltonian quantum mechanics is exploited to introduce a quantum-classical theory of wave fields. This can be considered as a first step toward a deeper understanding of the relation between operator quantum-classical mechanics, introduced some time ago, and the original quantum-classical scheme of motion where wave functions are evolved in time and the classical degrees of freedom follows surface-hopping trajectories on single quantum states.
The coupled Schrödinger equations that describe nonadiabatic dynamics are recast using the Bohm formulation of quantum mechanics. The resulting coupled Bohm equations are solved numerically for two scattering models, giving results that are essentially identical to wave-packet solution of the original coupled Schrödinger equations. The classical limit of the set of coupled Bohm equations is then described, producing a mixed quantum-classical theory incorporating classical-like motion on each potential energy surface accompanied by quantum transitions between the quantum states. Numerical tests of the mixed quantum-classical method are in excellent agreement with the accurate full-quantum results for the model problems. The method is contrasted with the related surface-hopping approach. It is shown that computing the dynamics of a distribution of classical particles is more consistent and more accurate than computing the motion of independent point particles as with surface hopping.
The spin-boson model is solved within the framework of quantum-classical dynamics using our recently-developed surface-hopping scheme. The quantum-classical equation of motion is expressed in an adiabatic basis and its solution is constructed from an ensemble of trajectories which undergo nonadiabatic transitions and evolve coherently on the adiabatic surfaces. Details of the algorithm for the simulation of the dynamics are presented and the method of simple Monte Carlo sampling used to evaluate the expectation values of observables is discussed. The simulation method is applied to a spin-boson system with a harmonic bath composed of ten oscillators with an Ohmic spectral density. For the spin-boson model the present implementation of quantum-classical dynamics is exact and the results of our surface-hopping simulations are in accord with previous numerically exact results for this model.
The quantum-classical Liouville equation describes the dynamics of a quantum subsystem coupled to a classical environment. It has been simulated using various methods, notably, surface-hopping schemes. A representation of this equation in the mapping Hamiltonian basis for the quantum subsystem is derived. The resulting equation of motion, in conjunction with expressions for quantum expectation values in the mapping basis, provides another route to the computation of the nonadiabatic dynamics of observables that does not involve surface-hopping dynamics. The quantum-classical Liouville equation is exact for the spin-boson system. This well-known model is simulated using an approximation to the evolution equation in the mapping basis, and close agreement with exact quantum results is found.
Quantum mechanics is not logically closed with respect to the classical world.Its formalism unfolds as the quantization of a sub-set of classical Hamiltonians.The interpretation of quantum theory in terms of the measurement process inevitably requires to deal with systems composed by a mixture of both classical and quantum degrees of freedom.Moreover, when energy can flow between the quantum and classical degrees of freedom (i.e., in the case of nonadiabatic dynamics), there are more theoretical difficulties in order to obtain a fully consistent quantum-classical formalism.In order to perform calculations, one can renounce to the usual Lie algebraic structure of well-established physical theories, adopt non-Hamiltonian brackets, and obtain a formalism for the dynamics and statistics of quantum-classical systems that has an affordable computational complexity.Recent progress in the algorithms for the sampling of nonadiabatic dynamics of quantum-classical systems at long time is reviewed here.
Abstract This chapter the reviews essential concepts and methods of time-dependent quantum mechanics. (1) Formal solutions of the time-dependent Schrödinger equation. (2) The two-level system. (3) Interacting systems: The time-dependent self-consistent field approximation and the Ehrenfest approximation. (4) The Born–Oppenheimer approximation and nuclear potential surfaces. (5) Time-dependent Hamiltonians: A two-level system in a time-dependent field. (6) Non-adiabatic dynamics: The surface-hopping method. (7) Expressing the time evolution in terms of the Green operator. (8) Representations (the Schrödinger and Heisenberg representations, the interaction representation). (9) Time-dependent perturbation theory. (10) Quantum dynamics of the free particles. (11) Quantum dynamics of the harmonic oscillator. (12) The raising/lowering operators formalism. (13) Tunneling.