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International Journal of Bifurcation and Chaos

INFORMATION ENTROPY AND NONLINEAR SEMIQUANTUM DYNAMICS

Claudia M. Sarris, Araceli N. Proto

Published Oct 1, 2009• 6 citations
Publisher

Abstract

We describe how, departing from the Shannon entropy, it is possible to deal with semiquantum time-independent nonlinear Hamiltonians. The interplay between the quantal and classical degrees of freedom can be easily seen, and the set of differential equations that govern the temporal evolution of the quantal mean values and the classical variables is obtained. We find invariants of motion and, particularly, we describe under which conditions, the uncertainty principle remains as an invariant of motion too. Through the analysis of these invariants, it is possible to follow the transition of the system from quantum to classical regime and to conclude that the uncertainty principle behaves as an indicator telling us whether the system is in regular or irregular regime. A simple example is shown.

DOI
10.1142/s0218127409024918

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We describe how, departing from the Shannon entropy, it is possible to deal with semiquantum time-independent nonlinear Hamiltonians. The interplay between the quantal and classical degrees of freedom can be easily seen, and the set of differential equations that govern the temporal evolution of the quantal mean values and the classical variables is obtained. We find invariants of motion and, particularly, we describe under which conditions, the uncertainty principle remains as an invariant of motion too. Through the analysis of these invariants, it is possible to follow the transition of the system from quantum to classical regime and to conclude that the uncertainty principle behaves as an indicator telling us whether the system is in regular or irregular regime. A simple example is shown.
We investigated the dynamics of a regular nonlinear semiclassical system by employing Shannon entropy together with two associated formulations of statistical complexity: the López-Ruiz–Mancini–Calbet (LMC) and Jensen–Shannon approaches. Additionally, Tsallis entropy was used as an alternative quantifier. In this context, quantum variables interacted with a classical environment, and both conservative and dissipative regimes were considered. To compute information-theoretic quantifiers, probability distributions were extracted from the system's temporal evolution using the Bandt–Pompe permutation method. The classical limit was characterized by a motion invariant linked to the uncertainty principle. Our analysis revealed three distinct zones that characterized the structure of transitions across the classical–quantum boundary. These findings confirmed earlier results obtained in markedly different dynamical systems. This consistency supported the idea of a possible generalization, which, if established, would have significant implications for both semiclassical and quantum theories.
We study the behavior of a nonlinear semiclassical system using Shannon entropy and two approaches to statistical complexity. These systems involve the interaction between classical variables (representing the environment) and quantum ones. Both conservative and dissipative regimes are explored. To calculate the information metrics, probability distributions are derived from the temporal evolution via the Bandt-Pompe permutation method. Additionally, we describe the classical limit in terms of a motion invariant linked to the uncertainty principle. Our analysis reveals three distinct regions, including a mesoscopic one, along with other notable findings.
We describe a general method to study the nonlinear semiquantum systems: as the dynamics of the system develops in a semiquantum space V spanned by the mean values of a Complete Set of Noncommuting observables (CSNCO) and the conjugate classical variables, X, P, it is possible to define a positive definite metric for the quantum manifold (QM). Through the components of the second rank covariant metric tensor, a close connection between this metric and the uncertainty principle can be established, meanwhile, the classical manifold (CM) of the system, is preserved as a symplectic one. We present an example of an entangled nonlinear semiquantum system where different dynamics take place.
This books presents a detailed treatment of the physics of the interaction between quantum and classical variables. Semiquantum Hamiltonians represent mixed physical systems for which one part is quantal and the other classical. These parts are coupled in nonlinear fashion. The coupling term contains both quantum and classical degrees of freedom. Physical systems that are appropriately described in this way are are to be found in molecular and solid state physics. Of course, so described is the interaction between a classical measurement's instrument and the quantum system being measured. Semiquantum systems have a technological counterpart as well, in connection with nano-technology (molecular transistors, quantum dots and SQUIDS, for instance). These special systems can be successfully modelled by quantum billiards, dynamic system in which a particle alternates between motion in a straight line and elastic reflections from a boundary. The interest focuses in vibrating quantum billiards, in which a particle of mass m is subjected to elastic interactions confined in a potential well V(q), generated by a classical mass M.
We outline a rather extraordinary series of similarities between classical and quantal behavior in the limit of adiabatic time changes. These include the power laws for the goodness of the respective invariants for isolated eigenstates and invariant tori for integrable systems, the nature of the breakdown of the invariances—level crossing in quantum systems and the role of ever present non-linear resonances is examined in the case of generically non-integrable classical dynamics—and the perhaps surprising relationship for fully chaotic systems where sufficiently slow switching in either classical or quantal systems precisely preserves the number of energy levels up to a given energy. For suitably small values of Planck's constant these similarities yield clear examples of the Bohr correspondence principle linking classical and quantum mechanics; for larger values the details in the classical picture are quenched in the quantum.