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

Abstract

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.