Quantum Mechanics as a Classical Theory I: Non-relativistic Theory

TL;DR

This paper aims to derive quantum mechanics from classical principles using axiomatic methods, establishing connections between density matrices, wave functions, and classical Newtonian mechanics.

Contribution

It provides a novel axiomatic derivation of quantum mechanics from classical mechanics, including the formulation of Schrödinger's and Pauli's equations from a classical basis.

Findings

Derivation of Schrödinger's equation from classical axioms

Demonstration that density matrix operators can be reduced to wave function operators

Equivalence of Schrödinger's equation to Newton's laws for dispersion-free ensembles

Abstract

The objective of this series of three papers is to axiomatically derive quantum mechanics from classical mechanics and two other basic axioms. In this first paper, Schreodinger's equation for the density matrix is fist obtained and from it Schroedinger's equation for the wave functions is derived. The momentum and position operators acting upon the density matrix are defined and it is then demonstrated that they commute. Pauli's equation for the density matrix is also obtained. A statistical potential formally identical to the quantum potential of Bohm's hidden variable theory is introduced, and this quantum potential is reinterpreted through the formalism here proposed. It is shown that, for dispersion free {\it ensembles% }, Schroedinger's equation for the density matrix is equivalent to Newton's equations. A general non-ambiguous procedure for the construction of operators which act upon the density matrix is presented. It is also shown how these operators can be reduced to those which act upon the wave functions.