Non-inertial corrections to the hydrogen atom spectrum

Abstract

Aim. To study the non-inertial correction to the spectrum of the hydrogen atom. We derive the spectrum of bound states of the hydrogen atom from the stability requirement formulated in non- inertial reference frames and show how the resulting Hamiltonian reduces to the standard Coulomb problem, taking into account small corrections due to the reference frame.

Methodology. In this formulation, we introduce a stability condition for physical trajectories based on the positivity of the second variation of the action function. As an application, this approach is used to analyze bound motion in a Coulomb field. The balance between stochastic input and radiative losses selects a discrete set of stable periodic trajectories. Their characteristic frequencies exhibit scaling behavior similar to the hydrogen spectrum. In the limit of vanishing non-inertial fluctuations, the formalism reduces to the Schrödinger description.

Results of the study consist of a monotonic decrease in the absolute deviation |ΔEn| with increasing principal quantum number n, which we check at low-lying levels and summarize in a compact table, including relative errors. This structure also leads to frequency shifts for several transitions (e. g., 1S–2S and 2S–nD), expressed directly in hertz.

Research implications of the results lies in certain spectral corrections, which can be considered as consequences of dynamic stability in a fluctuating non-inertial background.