A Formulation of Quantum Fluid Mechanics and Trajectories
James P. Finley
TL;DR
This work builds a comprehensive Newtonian-like framework for quantum states by recasting time-dependent many-body quantum dynamics as fluid- and trajectory-based mechanics. It establishes a precise correspondence between the Schrödinger equation and generalized Bernoulli/Euler-type equations through dual velocity fields and variable mass, yielding a fluid-dynamical view of quantum motion that encompasses both one-body and many-body states. The formalism introduces cross flows, Lagrangian/Hamiltonian structures, and reduced-density-matrix concepts to derive 1-body Schrödinger equations that generalize Hartree–Fock via a quantum Coulomb law. By analyzing quantum flow mixtures and orbital flows, the paper connects trajectory-like descriptions with orbital-based quantum states, offering a versatile toolkit for interpreting quantum dynamics while highlighting the need for careful interpretation and potential limitations of trajectory pictures.
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.