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

• URL: http://arxiv.org/abs/2607.03897v2
• Date: Sat, 11 Jul 2026 11:45:34 GMT
• Title: Algebra of quantum mechanics via classical phonons. I: The Schrodinger equation as the Newtonian equation of motion and quantum observables as classical averages
• Abstract summary: We show that the continuous limit of the corresponding Newtonian equation of motion yields the Klein-Gordon equation for a real-valued field.<n>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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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.

Related papers

• Algebra of quantum mechanics \textit{via} classical phonons. II: Klein-Gordon dynamics, the Heisenberg formalism, the Dirac canonical commutation rule and the Poincaré algebra through the continuous Poisson bracket formalism [0.0]
  In the first part of this series we have shown how the Schrodinger equation for a single particle can be obtained from a purely classical phonon model.<n>In this work we focus instead on how the classical Hamiltonian formalism applied to the same phonon system allows to recover the feature of relativistic quantum mechanics for a single spinless particle.
  arXiv  Detail & Related papers  (2026-07-04T15:10:11Z)
• Quantum simulation of the Liouville equation in classical mechanics with discontinuous potential via Schrödingerization [42.22857110018992]
  We develop quantum simulation algorithms for the Liouville equation of classical mechanics with discontinuous potential.<n>We combine Hamiltonian-preserving schemes by Jin and Wen in Commun. Math. Sci. 3(3), 285-315 (2005) with the Schrdingerization method, which embeds the resulting nonunitary semi-discrete dynamics into a unitary Schrdinger type system.
  arXiv  Detail & Related papers  (2026-06-13T02:49:05Z)
• Euler-Korteweg vortices: A fluid-mechanical analogue to the Schrödinger and Klein-Gordon equations [0.0]
  Quantum theory and relativity exhibit several formal analogies with fluid mechanics.<n>This paper examines under which conditions a classical fluid model may reproduce the most basic mathematical formalism of both theories.
  arXiv  Detail & Related papers  (2025-12-29T07:38:30Z)
• Quantifying superluminal signalling in Schrödinger-Newton model [0.0]
  We quantify the probability of a successful superluminal bit transfer via the single-particle Schrdinger-Newton equation.<n>We prove that the Einstein-Dirac system, which yields the Schrdinger-Newton equation in the non-relativistic limit, is perfectly compatible with the relativistic causal structure.
  arXiv  Detail & Related papers  (2025-12-22T10:56:38Z)
• Revivals and quantum carpets for the relativistic Schrödinger equation [41.99844472131922]
  We investigate wavepacket dynamics for a relativistic particle in a box evolving according to the Schr"odinger equation.<n>We obtain expressions for the wavepacket revival times and explore the corresponding quantum carpets.<n>We analyze level spacing statistics as the dynamics goes from the non-relativistic regime to the ultra-relativistic limit.
  arXiv  Detail & Related papers  (2025-11-07T12:39:00Z)
• A pseudo-random and non-point Nelson-style process [49.1574468325115]
  We take up the idea of Nelson's processes, the aim of which was to deduce Schr"odinger's equation.<n>We consider deterministic processes which are pseudo-random but which have the same characteristics as Nelson's processes.
  arXiv  Detail & Related papers  (2025-04-29T16:18:51Z)
• Analogue black string in a quantum harmonic oscillator [49.1574468325115]
  We write the exact solution of the Klein-Gordon equation in the background of a chargeless, static black string.<n>The eigenvalue problem provides complex energy values for the particle, which may indicate the presence of quasinormal modes.<n>We show a simple quantum system that can imitate the particle in the black string background, whose solutions are also applications of the biconfluent Heun function.
  arXiv  Detail & Related papers  (2024-12-31T15:03:57Z)
• Klein-Gordon oscillators and Bergman spaces [55.2480439325792]
  We consider classical and quantum dynamics of relativistic oscillator in Minkowski space $mathbbR3,1$. The general solution of this model is given by functions from the weighted Bergman space of square-integrable holomorphic (for particles) and antiholomorphic functions on the K"ahler-Einstein manifold $Z_6$.
  arXiv  Detail & Related papers  (2024-05-23T09:20:56Z)
• Construction of Schrödinger, Pauli and Dirac equations from Vlasov equation in case of Lorentz gauge [0.0]
  The authors have succeeded to construct the Schr"odinger, Pauli, Dirac equation, the Hamilton-Jacobi equation and the Maxwell equations.
  arXiv  Detail & Related papers  (2024-04-21T08:38:40Z)
• Quantum mechanics without quantum potentials [0.0]
  Non-locality in quantum mechanics can be resolved by considering relativistically covariant diffusion in spacetime. We introduce the concept of momentum equilinear to replace the second-order Bohm-Newton equations of motion.
  arXiv  Detail & Related papers  (2024-01-08T18:51:38Z)
• Real-time dynamics of false vacuum decay [49.1574468325115]
  We investigate false vacuum decay of a relativistic scalar field in the metastable minimum of an asymmetric double-well potential. We employ the non-perturbative framework of the two-particle irreducible (2PI) quantum effective action at next-to-leading order in a large-N expansion.
  arXiv  Detail & Related papers  (2023-10-06T12:44:48Z)
• Free expansion of a Gaussian wavepacket using operator manipulations [77.34726150561087]
  The free expansion of a Gaussian wavepacket is a problem commonly discussed in undergraduate quantum classes. We provide an alternative way to calculate the free expansion by recognizing that the Gaussian wavepacket can be thought of as the ground state of a harmonic oscillator. As quantum instruction evolves to include more quantum information science applications, reworking this well known problem using a squeezing formalism will help students develop intuition for how squeezed states are used in quantum sensing.
  arXiv  Detail & Related papers  (2023-04-28T19:20:52Z)
• Classical and Quantum Brownian Motion [0.0]
  In quantum mechanics electrons and other point particles are no waves and the chapter of quantum mechanics originated for the force carriers. A new projector operator is proposed for the collapse of the wave function of a quantum particle moving in a classical environment. Considering the Brownian dynamics in the frames of the Bohmian mechanics, the density functional Bohm-Langevin equation is proposed.
  arXiv  Detail & Related papers  (2021-05-12T13:24:39Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.