Fields and Equations of Classical Mechanics for Quantum Mechanics

TL;DR

This paper derives a generalized fluid dynamic framework for quantum mechanics using a two-substate model, connecting classical fluid equations with the Schrödinger and Bohmian equations, revealing new energy and pressure fields.

Contribution

It introduces a novel two-substate Eulerian equation for quantum states, linking classical fluid dynamics with quantum mechanics and Bohmian interpretations, including new energy and pressure field concepts.

Findings

The generalized Euler equation models quantum states as interacting substates.

The total-energy equation generalizes Bernoulli's equation for quantum systems.

Energy conservation is demonstrated for nondegenerate eigenstate superpositions.

Abstract

A generalized Euler equation of fluid dynamics is derived for describing many-body states of quantum mechanics. The Eulerian Eq. can be viewed as representing the interaction of two substates, where each substate has its own velocity and pressure fields. These field quantities are given by maps of the wavefunction. For one-body systems, the Eulerian Eq. can model either a fluid or particle description of quantum states. The generalized Euler Eq. is shown to be the gradient of an equation representing the total-energy of the two substates, having two energy fields. This total-energy Eq. is a generalization of the Bernoulli Eq. of fluid dynamics. The total-energy Eq., along with a continuity-equation, is equivalent to the time-dependent Schroedinger Eq. An equation is also derived that is equivalent to the main equation of Bohmian mechanics with additional identifications: The quantum potential of Bohmian mechanics is given as a sum of a kinetic energy and pressure fields. Also, the time derivative of the wavefunction phase is replaced by an energy field. In the formalism, field quantities are identified from their placement in equations of classical mechanics and separately, by definitions that involve the wavefunction and operators of quantum mechanics. This approach yields, unintended, and unknown energy and pressure fields. These fields, however, are shown to satisfy a continuity Eq., an equation that is equivalent to the other equation of Bohmian mechanics. It is also demonstrated that energy conservation holds for both of these energy fields, if the wavefunction is a linear-combination of eigenvectors, where the eigenvectors can be nondegenerate. A detailed investigation is given on the possible behavior, or source, of an electron that has one of the velocity fields. Alternate formulae for this velocity fields are also considered.