An Internal-Clock Route to the Hydrogenic Energy Ladder

Published version of: Carlos A. Gómez-Uribe, “An Internal-Clock Route to the Hydrogenic Energy Ladder,” International Journal of Quantum Foundations 12(3), 198–250 (2026). This paper gives a conditional route to the hydrogenic action ladder from a localized internal clock carried with a field-dressed charge. The starting point is de Broglie’s old phase-harmony idea: a particle may carry an internal periodicity, while standard quantum and semiclassical treatments impose global consistency on an associated matter-wave phase. Here the matter wave is not assumed. Instead, the paper asks how far one can get by following the phase of a localized clock whose ticking advances with proper time. For a nonrelativistic Coulomb orbit, special-relativistic time dilation makes the internal clock lag slightly behind laboratory time. After removing the trivial rapid ticking at the clock frequency ω₀, the remaining slow phase drift is proportional to the Kepler action accumulated around the independent radial and angular cycles of the bound orbit. Requiring this slow clock phase to close consistently on those cycles, including the standard semiclassical half-shifts at the radial turning point and Coulomb origin, gives the usual Coulomb action lattice. Substitution into the classical Coulomb Hamiltonian then reproduces the hydrogenic inverse-square energy law, Eₙ ∝ −1/n², with the expected reduced-mass scaling. The result is explicitly conditional. The existence of a long-lived internal clock, the local half-shift corrections, and the action scale are inputs; matching the empirical hydrogen spectrum calibrates that action scale to Planck’s constant ħ. The paper does not claim to derive ħ, full wave functions, radiative stability, fine structure, Lamb shifts, or measurement statistics. Its narrower purpose is to show that, once such an internal clock is admitted, closed-loop phase consistency is enough to reproduce the Coulomb EBK action ladder. The paper also discusses a possible selection mechanism. In a frequency-local protected-clock regime, radiation is suppressed at the clock frequency but nearby detuned sidebands can leak. A mismatch in the closed-loop phase shifts spectral weight into these sidebands and produces a quadratic near-closure leakage cost. This provides a physical bias toward the closed templates, while leaving a full kinetic relaxation theory as an open problem.