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Phase-space Weyl-Wigner's formalism for the calculation of quantum distortions in non-linear cyclic Hamiltonian systems and extensions to the Earth System

Grant number: 20/01976-5
Support type:Scholarships abroad - Research
Effective date (Start): February 01, 2022
Effective date (End): July 31, 2022
Field of knowledge:Physical Sciences and Mathematics - Physics - General Physics
Principal researcher:Alex Eduardo de Bernardini
Grantee:Alex Eduardo de Bernardini
Host: Orfeu Bertolami Neto
Home Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
Research place: Universidade do Porto (UP), Portugal  

Abstract

This project proposes to implement, in an analytical way, the Weyl-Wigner (WW) formalism of quantum mechanics (MQ) in phase space, x-p, in non-linear systems of Hamiltonian character, H, generically described by H(x,p) = g(p) + f(x). In this type of research, the knowledge of the classical background is fundamental, both for testing the classical-quantum limits of exact quantum solutions, as well as for establishing the reference for the (quantum) information quantifiers, characteristic of formalism. The starting point are Hamiltonian systems that exhibit anharmonic periodic behaviours deriving from non-linearity coming exclusively from the position coordinate, for which we already have the results and the potentiality of developed applications (cf. singular oscillator, Poschl-Teller, etc.). The biggest challenge, however, is the implementation of WW formalism, the description of probability currents, and the construction of information (quantum) flow for more general Hamiltonian systems as in H(x,p) = g(p) + f(x), which can be, in particular, characteristic of Harper-type and Lotka-Volterra-type periodic systems, which do not have definitive quantum analytical frameworks, mostly due to contributions of nonlinearity arising from the momentum coordinate, p (which, distinctly from Schroedinger-like Hamiltonian, is not displayed in quadratic form).In addition to the classical-quantum limit, our proposal is to evaluate the thermodynamic regimes in the phase space for the systems mentioned above.As a second hypothesis, we intend to evaluate the implementation of cyclic (anharmonic) systems in the description of the Earth System, also with the purpose of describing thermodynamic statistical limits.On the one hand, this is a project of Fundamental Physics, as we seek answers to quantum distortions about classic anharmonic (and not usual in the moment coordinate) Hamiltonian and its corresponding statistical ensembles, as well as a seminal generalization for the description of Wigner's currents. On the other hand, the proposal advances on appealing issues such as quantum chaos, and electrons in 2D crystal networks, as well as on topics of a multidisciplinary nature, as we seek to implement the classical results and the systematic of WW formalism to the study of Lotka-Volterra systems, among others, and the Earth System itself.As a foundation for the execution of this project, we consider fundamental the scientific interface of more than a decade involving the Dept. of Physics and Astronomy of the University of Porto, through which, with the support of several theoretical platforms, we intend to extend the scope of the aforementioned framework. (AU)