Connection of microscopic with macroscopic dynamics: Towards a hydrodynamical description of quantum mechanics

We develop a novel approach to the problem of tunneling through sharp-edged potential barriers with and without dissipation. Boundary conditions can be matched without assuming continuity of the wave function. The effect of a small friction mechanism on the tunneling of a particle through a single, sharp-edged rectangular barrier diminishes the transmission coefficient.; This work provides a quantum kinetics approach to the problem of irreversibility and its connection with macroscopic equations of quantum hydrodynamics.; We also study the effect of the quantum potential and friction mechanism on nonadiabatic transport corrections to the geometric phase acquired by a wave packet undergoing an excursion in its parameter space is investigated. The macroscopic hydrodynamical equations are also applied to the study of parametric oscillations of charged particle driven by an external monochromatic wave having large phase velocity.; A new approach to the problem of scattering through arbitrarily-shaped potential wells is presented via the exact invariants of quantum hydrodynamics, where the solution preserves information about the quantum potential which accounts for quantum-wave features such as interference and diffraction effects. Within this formalism, a formula for the transmission coefficient of a steady flux of fluid-particles scattered by an arbitrarily-shaped potential well is derived and discussed for some exact and approximate solutions. This approach is extended to the problem of scattering in multilayer sharp-edged systems.; Further, the so-called Ermakov problem in the theory of invariants is derived and solved within the framework of Nelson's stochastic mechanics: a natural and straightforward particle interpretation of quantum hydrodynamics by allowing a random character to the underlying trajectories, that is, a stochastic interpretation of quantum mechanics in terms of subquantum random fluctuations resulting from the action of a stochastic invariant thermostat. Within this formalism, we investigate special types of solutions of the hydrodynamical version of a generalized Schroedinger-Langevin equation (GNLSLE) via quantum hydrodynamics. Within the same scheme of hydrodynamical differential equations, we decompose the physical process, associated with the GNLSLE, into two independent processes: a classical Langevin-type process and a pure quantum Nelson-type process. (Abstract shortened with permission of author.)...