Joint quantum–classical Hamilton variational principle in the phase space*

Abstract

We show that the dynamics of a closed quantum system obeys the Hamilton variational principle. Even though quantum particles lack well-defined trajectories, their evolution in the Husimi representation can be treated as a flow of multidimensional probability fluid in the phase space. By introducing the classical counterpart of the Husimi representation in a close analogy to the Koopman-von Neumann theory, one can largely unify the formulations of classical and quantum dynamics. We prove that the motions of elementary parcels of both classical and quantum Husimi fluid obey the Hamilton variational principle, and the differences between associated action functionals stem from the differences between classical and quantum pure states. The Husimi action functionals are not unique and defined up to the Skodje flux gauge fixing [Skodje R. T. et al. Phys. Rev. A 40, 2894 (1989)]. We demonstrate that the gauge choice can dramatically alter flux trajectories. Applications of the presented theory for constructing semiclassical approximations and hybrid classical-quantum theories are discussed.

Observation of Bohm trajectories and quantum potentials of classical waves

“…Comprehensive discussions of Bohmian mechanics, which has recently received renewed attention can be found at several places in the literature [9][10][11][12][13] and many interesting applications beyond the interpretation of quantum mechanics have been proposed. For example, Bohmian mechanics is utilized for a better understanding of the quantum-classical transition [14] as well as, nanoscale electron devices and electron transport in open systems [15]. Bohmian equations sometimes provide more efficient computational tools than those obtained by orthodox methods [16] and are now routinely used in quantum chemistry [17,18].…”