Semiclassical delocalization in phase space constitutes a manifestation of the Uncertainty Principle, one indispensable part of the present understanding of Nature and the Wehrl entropy is widely regarded as the foremost localization-indicator. We readdress the matter here within the framework of the celebrated semiclassical Husimi distributions and their associatedWehrl entropies, suitably қ-deformed. We are able to show that it is possible to significantly improve on the extant phase-space classical-localization power.
We discuss thermalization of isolated quantum systems by using the Husimi–Wehrl entropy evaluated in the semiclassical treatment. The Husimi–Wehrl entropy is the Wehrl entropy obtained by using the Husimi function for the phase space distribution. The time evolution of the Husimi function is given by smearing the Wigner function, whose time evolution is obtained in the semiclassical approximation. We show the efficiency and usefulness of this semiclassical treatment in describing the entropy production of a couple of quantum-mechanical systems, whose classical counter-systems are known to be chaotic. We propose two methods to evaluate the time evolution of the Husimi–Wehrl entropy, the test-particle method and the two-step Monte Carlo method. We demonstrate the characteristics of the two methods by numerical calculations, and show that simultaneous application of the two methods ensures the reliability of the results of the Husimi–Wehrl entropy at a given time.
We discuss thermalization of isolated quantum systems by using the Husimi–Wehrl entropy evaluated in the semiclassical treatment. The Husimi–Wehrl entropy is the Wehrl entropy obtained by using the Husimi function for the phase space distribution. The time evolution of the Husimi function is given by smearing the Wigner function, whose time evolution is obtained in the semiclassical approximation. We show the efficiency and usefulness of this semiclassical treatment in describing the entropy production of a couple of quantum-mechanical systems, whose classical counter-systems are known to be chaotic. We propose two methods to evaluate the time evolution of the Husimi–Wehrl entropy, the test-particle method and the two-step Monte Carlo method. We demonstrate the characteristics of the two methods by numerical calculations, and show that simultaneous application of the two methods ensures the reliability of the results of the Husimi–Wehrl entropy at a given time.
The Heisenberg algebra is first deformed with the set of parameters
q,l,λ to generate a new family of generalized coherent states. In this framework, the matrix elements of relevant operators are exactly computed. A proof on sub-Poissonian character of the statistics of the main deformed states is provided. This property is used to determine a generalized metric. A unified method of calculating structure functions from commutation relations of deformed single-mode oscillator algebras is then presented. A natural approach to building coherent states associated to deformed algebras is deduced. Known deformed algebras are given as illustration. Futhermore, we generalize a class of two-parameter deformed Heisenberg algebras related to meromorphic functions, called
R(p,q)-deformed algebra. Relevant families of coherent states maps are probed and their corresponding hypergeometric series are computed. The latter generalizes known hypergeometric series and gives to a generalization of the binomial theorem. The involved notions of differentiation and integration generalize the usual
q- and
(p,q)-differentiation and integration. A Hopf algebra structure compatible with the
R(p,q)-algebra is deduced. We succeed in giving a new characterization of Rogers- Szegö polynomials, called
R(p,q)-deformed Rogers-Szegö polynomials, by their three-term recursion relations and the associated quantum algebra built with corresponding creation and annihilation operators. Continuous
R(p,q)-deformed Hermite polynomials and their recursion relation are also deduced. Novel algebraic relations are provided and discussed. The whole formalism is performed in a unified way, generalizing known relevant results which are straightforwardly derived as particular cases.