Quantum mechanics of phase and charge and quantization of the Coulomb field

P. Campos, M.G.R. Martins, J. D. M. Vianna

Gordon Baym, Tomoki Ozawa

We analyze Niels Bohr's proposed two-slit interference experiment with highly charged particles which argues that the consistency of elementary quantum mechanics requires that the electromagnetic field must be quantized. In the experiment a particle's path through the slits is determined by measuring the Coulomb field that it produces at large distances; under these conditions the interference pattern must be suppressed. The key is that, as the particle's trajectory is bent in diffraction by the slits, it must radiate and the radiation must carry away phase information. Thus, the radiation field must be a quantized dynamical degree of freedom. However, if one similarly tries to determine the path of a massive particle through an inferometer by measuring the Newtonian gravitational potential the particle produces, the interference pattern would have to be finer than the Planck length and thus indiscernible. Unlike for the electromagnetic field, Bohr's argument does not imply that the gravitational field must be quantized.

Waldemar Wolney Filho

In this work we are presenting a modified Coulomb potential function to describe the interaction between two micro-scopic electric charges. In particular, concerning the interaction between the proton and the electron in the hydrogen atom. The modified potential function is the product of the classical Coulomb potential and an oscillatory function dependent on a quantized phase factor. The oscillatory function picks up only selected points along the Coulomb potential, creating potential wells and barriers around the nucleus of the atom. The new potential reveals us new features of the hydrogen atom. Searching for a manner to determine the phase factor, we are using the concept of the de Broglie particle wavelike behavior and the quantum analogue of the virial theorem for describing the bound motion of a particle in a central force field. This procedure is a kind of feedback action, where we are making use of well established concepts of the quantum mechanics aiming to determine the phase factor of the new interaction potential.

H. van Haeringen

Jochen Vogt

Στυλιανός Σασλόγλου

1. Abstract We investigate the dynamical consequences of a single scalar charge-phase field, θcharge and show that both electromagnetism and non-relativistic quantum mechanics emerge from its phase geometry and evolution. Starting from a local phase-field Lagrangian, the Euler-Lagrange structure generates a continuity equation together with the associated charge density and current density. Spatial gradients of the charge phase define a propagating spin-1 bond mode which, in the weak-field and long-range limit, gives rise to electromagnetic radiation and reduces to Maxwell electrodynamics. The same charge-phase structure also admits a mechanical representation. The corresponding phase dynamics naturally reproduce the Hamilton-Jacobi equation, establishing a direct connection between charge-phase evolution and dynamical motion. Combining the Hamilton-Jacobi structure with the continuity equation yields a quantum wave description in which the wavefunction emerges from the conserved density and charge phase. Granular corrections associated with the underlying phase structure generate the quantum potential and produce the Schrödinger equation as the effective quantum evolution law. Angular-momentum quantization follows from phase single-valuedness and periodicity, without the introduction of independent quantization postulates. Within this framework, Maxwell electrodynamics, Hamilton-Jacobi mechanics, Schrödinger dynamics, and quantum quantization appear as different dynamical limits of the same charge-phase field. Electromagnetism corresponds to the long-range linear propagation of phase bonds, whereas quantum mechanics reflects the oscillatory and granular dynamics of the underlying phase geometry.