A Formulation of Quantum Fluid Mechanics and Trajectories

TL;DR

This paper develops a classical mechanics formalism for quantum many-body states, linking fluid flow and trajectories with the Schrödinger equation, and deriving classical-like equations for quantum systems.

Contribution

It introduces a novel formalism connecting quantum mechanics with fluid dynamics and trajectories, including generalized energy and Euler equations for quantum states.

Findings

Derives classical fluid equations from quantum mechanics.

Obtains trajectories for one-electron atoms with closed orbits.

Generalizes Hartree-Fock equations with quantum Coulomb law.

Abstract

A formalism of classical mechanics is given for time-dependent many-body states of quantum mechanics, describing both fluid flow and point mass trajectories. The familiar equations of energy, motion, and those of Lagrangian mechanics are obtained. An energy and continuity equation is demonstrated to be equivalent to the real and imaginary parts of the time dependent Schroedinger equation, respectively, where the Schroedinger equation is in density matrix form. For certain stationary states, using Lagrangian mechanics and a Hamiltonian function for quantum mechanics, equations for point-mass trajectories are obtained. For 1-body states and fluid flows, the energy equation and equations of motion are the Bernoulli and Euler equations of fluid mechanics, respectively. Generalizations of the energy and Euler equations are derived to obtain equations that are in the same form as they are in classical mechanics. The fluid flow type is compressible, inviscid, irrotational, with the nonclassical element of local variable mass. Over all space mass is conserved. The variable mass is a necessary condition for the fluid flow to agree with the zero orbital angular momentum for s states of hydrogen. Cross flows are examined, where velocity directions are changed without changing the kinetic energy. For one-electron atoms, the velocity modification gives closed orbits for trajectories, and mass conservation, vortexes, and density stratification for fluid flows. For many body states, Under certain conditions, and by hypotheses, Euler equations of orbital-flows are obtained. One-body Schroedinger equations that are a generalization of the Hartree-Fock equations are also obtained. These equations contain a quantum Coulomb's law, involving the 2-body pair function of reduced density matrix theory that replace the charge densities.