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arXiv:1010.1494

Geometry and Statistics of Ehrenfest dynamics

José L. Alonso, Alberto Castro, Jesús Clemente-Gallardo, Juan C. Cuchí, P. M. Echenique, Fernando Falceto

Published Oct 7, 2010• 2 citations
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Abstract

Quantum dynamics (e.g., the Schrödinger equation) and classical dynamics (e.g., Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. The difference between both worlds is due to the presence of extra structure in the quantum case, that leads to the appearance of the probabilistic nature of the measurements and the indetermination and superposition principles. In this paper we first show that the quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well defined Poisson bracket allows to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann's distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. This work provides the basis for extending stochastic methods to Ehrenfest dynamics.

DOI
10.48550/arxiv.1010.1494

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Quantum dynamics (e.g., the Schrödinger equation) and classical dynamics (e.g., Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. The difference between both worlds is due to the presence of extra structure in the quantum case, that leads to the appearance of the probabilistic nature of the measurements and the indetermination and superposition principles. In this paper we first show that the quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well defined Poisson bracket allows to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann's distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. This work provides the basis for extending stochastic methods to Ehrenfest dynamics.
Quantum dynamics (i.e. the Schrödinger equation) and classical dynamics (i.e. Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. In this paper, we first show that the hybrid quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well-defined Poisson bracket allows us to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. Once we know the canonical distribution of an Ehrenfest system, it is straightforward to extend the formalism of Nosé (invented to do constant temperature molecular dynamics by a non-stochastic method) to our Ehrenfest formalism. This work also provides the basis for extending stochastic methods to Ehrenfest dynamics.
Quantum dynamics (i.e. the Schrödinger equation) and classical dynamics (i.e. Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. In this paper, we first show that the hybrid quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well-defined Poisson bracket allows us to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. Once we know the canonical distribution of an Ehrenfest system, it is straightforward to extend the formalism of Nosé (invented to do constant temperature molecular dynamics by a non-stochastic method) to our Ehrenfest formalism. This work also provides the basis for extending stochastic methods to Ehrenfest dynamics.
We introduce the Poisson bracket operator, which is an alternative quantum counterpart of the Poisson bracket. This operator is defined using the operator derivative formulated in quantum analysis and is equivalent to the Poisson bracket in the classical limit. Using this, we derive the quantum canonical equation, which describes the time evolution of operators. In the standard applications of quantum mechanics, the quantum canonical equation is equivalent to the Heisenberg equation. At the same time, this equation is applicable to ccc-number canonical variables and then coincides with the canonical equation in classical mechanics. Therefore, the Poisson bracket operator enables us to describe classical and quantum behaviors in a unified way. Moreover, the quantum canonical equation is applicable to nonstandard system where the Heisenberg equation is not defined. As an example, we consider the application to the system where ccc-number and qqq-number particles coexist. The derived dynamics satisfies the Ehrenfest theorem and the energy and momentum conservations.
The symmetry properties of quantum variational principles are considered. Euler-Poincaré reduction theory is applied to the Dirac-Frenkel variational principle for Schrödinger's dynamics producing new variational principles for the different pictures of quantum mechanics (Schrödinger's, Heisenberg's and Dirac's) as well as for the Wigner-Moyal formulation. In addition, new variational principles for mixed states dynamics have been formulated. The already known geometric characterization of quantum mechanics on the complex projective space is shown to emerge naturally from the Euler-Poincaré variational principle. Semidirect-product structures are seen to produce new variational principles for Dirac's picture. \n \nOn the other hand, the variational and Hamiltonian approach to Ehrenfest expectation values dynamics is proposed. In the Lagrangian framework, the quantum variational principle for Schrödinger's dynamics is extended to account for both classical and quantum degrees of freedom. First, it is shown that the mean field model of any quantum mechanical system can be derived from a classical-quantum Euler-Poincaré Lagrangian on the direct sum Lie algebra of the Heisenberg and unitary groups. Then, the semidirect-product structure (named Ehrenfest group), is constructed using the displacement operator from the theory of coherent quantum states ( the unitary action of the Heisenberg group on the space of wavefunctions). New classical-quantum equations for Ehrenfest's expectation values dynamics are derived redefining the mean-field model Euler-Poincaré Lagrangian on the Lie algebra of the Ehrenfest group. In the Hamiltonian framework, first expectation values of the canonical observables are shown to be equivariant momentum maps for the unitary action of the Heisenberg group on quantum states. Then, the Hamiltonian structure for Ehrenfest's dynamics is shown to be Lie-Poisson for the Ehrenfest group. The variational formulation is then given a corresponding Hamiltonian structure. The classical-quantum Ehrenfest dynamics equations produce classical and quantum dynamics as special limit cases. \n \nIn the particular case of Gaussian states, expectation values couple to second order moments, so that GS are completely characterized by first and second moments. When the total energy is computed with respect to a Gaussian state, higher moments can be expressed in terms of the first two, so that the moment hierarchy closes for Gaussian states. Second moments are shown to be equivariant momentum maps for the action of the symplectic group on the space of Wigner functions. Eventually, Gaussian states are shown to possess a Lie-Poisson structure on the Jacobi group. This structure produces an energy-conserving variant of a class of Gaussian moment models that have appeared in the chemical physics literature.
In the first part of this series we have shown how the Schrodinger equation for a single particle and the corresponding non relativistic quantum observables can be obtained from a purely classical phonon model through the Newtonian equations of motion. In this work we focus instead on how the classical Hamiltonian formalism applied to the same phonon system allows to recover the feature of relativistic quantum mechanics for a single spinless particle. Using the classical nature of the phonon model, we naturally define continuous Poisson brackets between classical observables, which allows to recover the dynamics of such observables, i.e. the Ehrenfest relations associated to real-valued Klein-Gordon fields. The Poisson brackets also permits to obtain the generic form of constants of motions, thus generalizing the concept of inner products and momentum on Klein-Gordon fields. We then connect the formalism of real-valued classical functionals with that of hermitian operators and complex-valued wave functions. This is done through the introduction of a non-local complex-valued change of variables which allows to rewrite the real-valued Klein-Gordon equation in a form akin to the Schrodinger equation, and the classical observables as quantum expectation values. Then, we show how this change of variables allows to rewrite the classical Poisson brackets as commutators of hermitian operators. This points out the strict equivalence between the Heisenberg formalism and the formalism of classical Poisson bracket. Eventually, we illustrate how the Poisson brackets allows to recover the transformations of Poincare group in 1+1 dimension together with its algebra. The latter makes the link between the Lorentz invariant inner product of Mostafazadeh and the Casimir invariant associated to the mass of particle.
The dynamics of quantum expectation values is considered in a geometric setting. First, expectation values of the canonical operators are shown to be equivariant momentum maps for the action of the Heisenberg group on quantum states. Then, the Hamiltonian structure of Ehrenfest's theorem is shown to be Lie-Poisson for a semidirect-product Lie group, named the `Ehrenfest group'. In addition, quantum dynamics is expressed in the frame of the expectation values, in which the latter undergo canonical Hamiltonian motion. In the case of Gaussian states, expectation values dynamics couples to second-order moments, which also enjoy a momentum map structure. Eventually, Gaussian states are shown to possess a Lie-Poisson structure associated to a semidirect-product subgroup of the Ehrenfest group, which is called the Jacobi group. This structure produces new energy-conserving terms in a class of Gaussian moment models (previously appeared in the chemical physics literature) that suffer from lack of energy conservation in the general case.