A mechanical analog of Bohr's atom based on de Broglie's double-solution approach

TL;DR

This paper presents a relativistic classical particle-wave model inspired by de Broglie's double-solution approach, capable of reproducing Bohr-Sommerfeld quantization in an atomic system, bridging hydrodynamical analogs and quantum mechanics.

Contribution

It introduces a novel relativistic classical model coupling a particle to a scalar wave-field, mimicking quantum guidance and reproducing atomic quantization rules.

Findings

Reproduces Bohr-Sommerfeld quantization formula.

Models particle guidance similar to de Broglie's phase-wave.

Connects hydrodynamical analogs with quantum atomic behavior.

Abstract

Motivated by recent developments of hydrodynamical quantum mechanical analogs [J. W. M. Bush, Annu. Rev. Fluid Mech. 47, 269-292 (2015)] we provide a relativistic model for a classical particle coupled to a scalar wave-field through a holonomic constraint. In presence of an external Coulomb field we define a regime where the particle is guided by the wave in a way similar to the old de Broglie phase-wave proposal. Moreover, this dualistic mechanical analog of the quantum theory is reminiscent of the double-solution approach suggested by de Broglie in 1927 and is able to reproduce the Bohr-Sommerfeld semiclassical quantization formula for an electron moving in a atom.