Unifying Quantum and Classical Dynamics

TL;DR

The paper investigates an exact dynamical connection between quantum and classical mechanics by casting quantum observables' evolution in the Heisenberg picture into Newton-like forms. It demonstrates that for analytic potentials the operator equations reproduce Newton's laws with no explicit dependence on $\hbar$, and extends to charged particles in electromagnetic fields to recover the Lorentz force via commutator algebra. Ehrenfest’s theorem is analyzed to connect operator dynamics with expectation values, showing classical trajectories emerge only under conditions of small quantum fluctuations or linear forces. The work thus provides an exact, ħ-free unification of quantum and classical dynamics at the level of observables, clarifying the quantum–classical boundary and the role of the quantum state. This perspective offers a deeper, operator-based understanding of why classical mechanics emerges in appropriate regimes without invoking semiclassical limits.

Abstract

Classical and quantum physics represent two distinct theories; however, quantum physics is regarded as the more fundamental of the two. It is posited that classical mechanics should arise from quantum mechanics under certain limiting conditions. Nevertheless, this remains a challenging objective. In this work, we explore the potential for unifying the dynamics of classical and quantum physics. This discussion does not suggest that classical behavior emerges from quantum mechanics; rather, it demonstrates the exact equivalence between the dynamics of quantum observables and their classical counterparts. It is shown that the Heisenberg equations of motion can be cast in a form that is identical to Newton's equations of motion, with $\hbar$ being absent from the formulation. This implies that both quantum and classical dynamics are governed by the same equations, with the Heisenberg operators substituting the classical observables.