Algebra of quantum mechanics via classical phonons. I: The Schrodinger equation as the Newtonian equation of motion and quantum observables as classical averages

Abstract

The Schrodinger equation for a single spinless particle is formally obtained via a classical phonon model, namely the Frenkel-Kontorova model. Starting from a one-dimensional lattice of coupled harmonic oscillators, we show that the continuous limit of the corresponding Newtonian equation of motion yields the Klein-Gordon equation for a real-valued field. By introducing a complex-valued change of variables mixing the real-valued displacement and velocity fields, and by separating fast and slow time scales, the Klein-Gordon equation is written as the Schrodinger equation within the non-relativistic limit. This complex change of variable also allows to rewrite classical global observables of the phonon field, such as the total energy or momentum, as the corresponding quantum observables. Additionally, we show that when a friction force is incorporated into the classical model, the corresponding Klein-Gordon equation can be rewritten as a Schrodinger equation with a non-Hermitian Hamiltonian. While the global approach is limited here to the non-relativistic regime and does not address the measurement problem, quantization or relativistic effects, it nonetheless illustrates how quantum algebra and complex-valued wave functions can be exactly reproduced using classical dynamics. The relativistic regime for a spinless particle and the link between commutators and Poisson brackets is addressed in the second part of this series.