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arXiv:2601.10423

Unifying Quantum and Classical Dynamics

Abdul Rahaman Shaikh, Tabish Qureshi

quant-phSubmitted Jan 15, 2026• Announced Jan 16, 2026• 0 citations• Revised Jan 20, 2026 (v2)

v1 · Jan 15, 2026 v2 · Jan 20, 2026 v3 · Jul 30, 2026

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. In a generalized analysis, the Heisenberg equations are cast in a form that is identical to the classical Hamilton's equations of motion. This implies that both quantum and classical dynamics are governed by the same equations, with the Heisenberg operators substituting the classical observables.

Comments: 5 pages, thoroughly revised, 2 new sections added

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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. In a generalized analysis, the Heisenberg equations are cast in a form that is identical to the classical Hamilton's equations of motion. This implies that both quantum and classical dynamics are governed by the same equations, with the Heisenberg operators substituting the classical observables.

Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations

F=\frac{dP}{dt}

,

P=m\frac{dV}{dt}

, and

\left[X,P\right]=i

. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.

We show that the classical Hamilton equations of motion can be derived from the energy conservation condition. A similar argument is shown to carry to the quantum formulation of Hamiltonian dynamics. Hence, showing a striking similarity between the quantum formulation and the classical formulation. Furthermore, it is shown that the fundamental commutator can be derived from the Heisenberg equations of motion and the quantum Hamilton equations of motion. Also, that the Heisenberg equations of motion can be derived from the Schrödinger equation for the quantum state, which is the fundamental postulate. These results are shown to have important bearing for deriving the quantum Maxwell’s equations.

An analysis is made of the relation between quantum theory and classical mechanics, in the context of the limit ℏ→0. Several ways in which this limit may be performed are considered. It is shown that Schrödinger's equation for a single particle moving in an external potential V does not, except in special cases, lead, in this limit, to Newton's equation of motion for the particle. This shows that classical mechanics cannot be regarded as emerging from quantum mechanics—at least in this sense—upon straightforward application of the limit ℏ→0.

Classical mechanics admits multiple equivalent formulations, from Newton's equations to the variational Lagrange-Hamilton framework and the scalar Hamilton-Jacobi (HJ) theory. In the HJ formulation, classical ensembles evolve through the continuity equation for a real density

ρ= R^{2}

coupled to Hamilton's principal function

S

. Here we develop a complementary formulation, the Hamilton-Jacobi-Schrödinger (HJS) theory, by embedding the pair

(R,S)

into a single complex field. Starting from a completely general complex ansatz

ψ= f(R,S)\, e^{i g(R,S)},

and imposing two minimal structural requirements, we obtain a unique map $ψ= R\, e^{iS/κ}\, $ together with a linear HJS equation whose

|κ| \to 0

limit reproduces the HJ formulation exactly. Remarkably, when

\mathrm{Re}(κ)\neq 0

, essential features of quantum mechanics, superposition, operator algebra, commutators, the Heisenberg uncertainty principle, Born's rule and unitary evolution, follow naturally as structural consistency conditions. HJS thus provides a unified mathematical viewpoint in which classical and quantum dynamics appear as different limits of a single underlying structure.

Quantum-classical equations of motion satisfying Jacobi identity were recently proposed by V. V. Kisil ( Europhys. Lett. , 72 (2005) 873), generalizing the Heisenberg group formulation of quantum mechanics. In the attempt to provide a physical interpretation for the resulting quantum-classical dynamics, we have reconstructed and analyzed the derivation, finding that there is a suitable representation from which the purely classical nature of the supposedly quantum-classical equations of motion clearly emerges.

We develop a dynamical theory, based on a system of ordinary differential equations describing the motion of particles which reproduces the results of quantum mechanics. The system generalizes the Hamilton equations of classical mechanics to the quantum domain, and turns into them in the classical limit

\hbar\rightarrow 0

. The particles' motions are completely determined by the initial conditions. In this theory, the wave function

ψ

of quantum mechanics is equal to the exponent of an action function, obtained by integrating some Lagrangian function along particle trajectories, described by equations of motion. Consequently, the equation for the logarithm of a wave function is related to the equations of motion in the same way as the Hamilton-Jacobi equation is related to the Hamilton equations in classical mechanics. We demonstrate that the probability density of particles, moving according to these equations, should be given by a standard quantum-mechanical relation,

ρ=|ψ|^2

. The theory of quantum measurements is presented, and the mechanism of nonlocal correlations between results of distant measurements with entangled particles is revealed. In the last section, we extend the theory to particles with nonzero spin.