Formulações semiclássicas da mecânica quântica: uma análise do oscilador singular por meio do formalismo de Weyl-Wigner e trajetórias quânticas

Abstract

The usual quantum mechanics formalism does not address relevant points of the semiclassical limit. In this work two theoretical approaches will be considered: Bohmian mechanics and the Weyl-Wigner formulation. The singular oscillator will be used as a platform to analytically obtain quantumness quantifiers for anharmonic systems. The first part will be concerned with a pure quasi-Gaussian state, for which the corresponding Bohmian trajectories will be calculated and are shown to deviate from the classical one due to the quantum potential. In the second part, a canonical ensemble will be considered by using the Weyl-Wigner formalism, with which it is possible to derive a thermalized Wigner function. Given that the potential is anharmonic, topological fluctuations on the phase space are expected and will be investigated for the Wigner flux. The influence of the thermal fluctuations will be distinguished from its quantum counterpart, which is diminished for increasing values of the temperature. Also, quantum information conservation equations will be introduced to calculate the global effect of topological fluctuations on the thermal equilibrium Wigner flux. In the last part, the Horava-Lifshitz cosmological model will be studied from both of the aforementioned theoretical perspectives. From them, the scale factor will be parameterized so that the quantumness quantifiers will depend on the spatial curvature of the universe. The results show that for increasing values of the anharmonic parameter, one observes a quasi-classical dynamics, usually associated to harmonic potentials only. Furthermore, Wigner flux topological fluctuations at thermodynamic equilibrium can be detected only locally, given that thermal fluctuations globally suppress non-Liouvillian effects.