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

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 the 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 an atom.

Automated summary

This study presents a relativistic model describing the coupling between a classical particle and a scalar wave field, investigating the scenario where the particle is guided by the wave in the presence of an external Coulomb field. This dualistic mechanical analog is able to reproduce the Bohr-Sommerfeld semiclassical quantization formula.