The Hydrogen Atom as Pure Vibration:

A Classical Field Model with a Q-Ball Nucleus and Quantised Electron Waves

Abstract

We construct a mathematically complete model of the hydrogen atom in which no point particles appear. The nucleus is a stable, gauged Q‑ball soliton of a complex scalar field; the electron is a charged scalar field vibration; and the electromagnetic interaction is carried by a massless vector field. Imposing finite total energy for stationary electron waves leads directly to a quantised frequency spectrum that exactly matches the Bohr levels. The sign of the Coulomb force is corrected, giving attraction. The electron self‑interaction is treated rigorously: a classical mass renormalisation is performed, and the finite residual corresponds to a classical Lamb‑shift‑like correction. Spontaneous emission is derived as the deterministic radiation coming from the beat frequency of two stationary vibrations; the Larmor‑formula decay rate for the hydrogen 2P to 1S transition is computed and shown to coincide with the quantum electrodynamics result. Throughout, the model reproduces many quantum phenomena within a purely vibrational, classical field theory, while the Born rule is identified as the remaining conceptual frontier. A full bibliography of 28 references supports the work.