Fields and Equations of Classical Mechanics for Quantum Mechanics

James P. Finley

Published 2022-07-09Version 1

Pairs of pressures and velocities fields with kinetic energies are defined for quantum mechanical many-body systems, depending on the probability distribution and phase of a wavefunction. These functions satisfy a set of two classical energy-equations that is equivalent to the many-body time-dependent Schr\"odinger equation, and these equations also define two energy fields. An Euler equation of fluid dynamics is also derived that is satisfied by the fields. Interpretation of the Bohmian potential and the integrand of the expectation value of the kinetic energy is given, involving both pressure and classical kinetic-energy fields. The formalism is built upon Bohmian mechanics and a Bernoulli-equation description of certain quantum mechanical systems, and the three fields from these approaches are incorporation into the formalism. Definitions, using the momentum operator and the time-side of the Schr\"odinger equations are given for six fields, yielding three new fields and alternate, but equivalent, definitions for the other three. The two energy fields are shown to be conserved, over all space, for solutions of the time-dependent Schr\"odinger equation, if the Hamiltonian operator is independent of time. Both particle and fluid interpretations of classical mechanics are considered. A compressible electron-body model is considered. A notational system is introduced that removes subscripts and summation signs, yielding equation for many-body systems that, in most cases, are identical in form to the corresponding special case of one-body.