[Paper Review] Quantum Mechanics as a Classical Theory I: Non-relativistic Theory

L. S. F. Olavo|ArXiv.org|Mar 28, 1995

Quantum Mechanics and Applications14 references3 citations

TL;DR

This paper proposes a derivation of non-relativistic quantum mechanics from classical mechanics using two additional axioms, showing that Schrödinger's equation for the density matrix emerges from classical dynamics. It demonstrates that for dispersion-free ensembles, the density matrix equation reduces to Newton's equations, and reinterprets the quantum potential as a statistical construct within a classical framework.

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.

Motivation & Objective

• To axiomatically derive quantum mechanics from classical mechanics and two fundamental postulates.
• To show that Schrödinger's equation for the density matrix arises naturally from classical dynamics.
• To reinterpret Bohm's quantum potential as a statistical potential within a classical formalism.
• To establish a general procedure for constructing operators acting on the density matrix and reducing them to wave function operators.
• To demonstrate equivalence between the density matrix equation and Newton's equations in dispersion-free ensembles.

Proposed method

• Derives the Schrödinger equation for the density matrix from classical mechanics using two additional axioms.
• Defines momentum and position operators acting on the density matrix and proves their commutativity.
• Derives Pauli's equation for the density matrix within the proposed formalism.
• Introduces a statistical potential formally identical to Bohm's quantum potential, reinterpreted as a classical statistical effect.
• Develops a general, unambiguous procedure for constructing operators on the density matrix.
• Reduces operators on the density matrix to standard wave function operators through a consistent mapping.

Experimental results

Research questions

• RQ1 Can non-relativistic quantum mechanics be derived from classical mechanics using only two additional axioms?
• RQ2 How do the momentum and position operators acting on the density matrix relate to classical dynamics?
• RQ3 What is the role of the quantum potential in this classical framework, and can it be reinterpreted as a statistical effect?
• RQ4 Under what conditions does the density matrix equation reduce to Newton's equations?
• RQ5 What is the general procedure for constructing and reducing operators acting on the density matrix to those acting on wave functions?

Key findings

• Schrödinger's equation for the density matrix is derived as a consequence of classical mechanics and two additional axioms.
• The momentum and position operators acting on the density matrix are shown to commute, preserving consistency with standard quantum formalism.
• Pauli's equation for the density matrix is successfully derived within the proposed classical framework.
• A statistical potential is introduced that is formally identical to Bohm's quantum potential, but reinterpreted as a classical statistical effect.
• For dispersion-free ensembles, the density matrix equation reduces exactly to Newton's equations, establishing a classical-quantum correspondence.
• A general, unambiguous procedure is established for constructing operators on the density matrix and reducing them to wave function operators.