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This research demonstrates a structural equivalence between Heisenberg's quantum equations of motion and classical Newtonian or Hamiltonian dynamics by showing that the latter emerges when Planck's constant vanishes. The study provides theoretical insight into how fundamental physics unifies these distinct regimes without claiming that macroscopic behavior directly arises from microscopic quantum mechanics in this specific formulation.
Unifying Quantum and Classical Dynamics
arXiv:2601.10423v3 Announce Type: replace 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 structurally equivalent to Newton's equations of motion, with $\hbar$ dropping out from the equations. 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 form of equations, with the Heisenberg operators substituting the classical observables.
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 structurally equivalent to Newton's equations of motion, with $\hbar$ dropping out from the equations. 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 form of equations, with the Heisenberg operators substituting the classical observables.
Accepted in Am. J. Phys
arXiv:2601.10423 [quant-ph]
(or arXiv:2601.10423v3 [quant-ph] for this version)
Full text under the publisher's open licence · 175 words
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