Gravitational eigenstates in weak gravity I: dipole decay rates of charged particles

The experimental demonstration that neutrons can reside in gravitational quantum stationary states formed in the gravitational field of the Earth indicates a need to examine in more detail the general theoretical properties of gravitational eigenstates. Despite the almost universal study of quantum theory applied to atomic and molecular states very little work has been done to investigate the properties of the hypothetical stationary states that should exist in similar types of gravitational central potential wells, particularly those with large quantum numbers. In this first of a series of papers, we attempt to address this shortfall by developing analytic, non-integral expressions for the electromagnetic dipole state-to-state transition rates of charged particles for any given initial and final gravitational quantum states. The expressions are non-relativistic and hence valid provided the eigenstate wavefunctions do not extend significantly into regions of strong gravity. The formulae may be used to obtain tractable approximations to the transition rates that can be used to give general trends associated with certain types of transitions. Surprisingly, we find that some of the high angular momentum eigenstates have extremely long lifetimes and a resulting stability that belies the multitude of channels available for state decay.

💡 Research Summary

The paper addresses a largely unexplored aspect of quantum mechanics: the electromagnetic dipole transition rates between stationary states that would exist for a charged particle bound in the Earth’s gravitational potential. While neutron experiments have already demonstrated that neutral particles can occupy discrete gravitational quantum levels, the authors note that the analogous problem for charged particles—where radiative decay is possible—has received little theoretical attention, especially for states with large quantum numbers.

Starting from the non‑relativistic Schrödinger equation with a central 1/r gravitational potential V(r)=−GMm/r, the authors obtain the familiar hydrogen‑like eigenfunctions ψ_{nlm}(r,θ,φ)=R_{nl}(r)Y_{lm}(θ,φ) and energies E_n=−(GMm)^2/(2ħ^2 n^2). The key novelty lies in the evaluation of the dipole matrix element ⟨n′l′m′|d̂|nlm⟩, where d̂=e r cosθ is the electric dipole operator. By separating the angular part (handled by the orthonormal spherical harmonics) the problem reduces to a radial integral

I_{n′l′,nl}=∫0^∞ R{n′l′}(r) r³ R_{nl}(r) dr.

Instead of performing this integral numerically for each pair of states, the authors derive a closed‑form, non‑integral expression using the generating function for associated Laguerre polynomials, beta‑function identities, and hypergeometric functions. The final result can be written compactly in terms of gamma functions and a ₂F₁ hypergeometric term, and it is valid for arbitrary integer quantum numbers (n,l,n′,l′). This analytic formula eliminates the need for cumbersome numerical integration and makes it possible to explore systematic trends across the entire spectrum of gravitational eigenstates.

With the matrix element in hand, the spontaneous emission rate follows the standard dipole formula

Γ_{i→f}= (4α ω³)/(3c³) |⟨f|d̂|i⟩|²,

where ω=|E_n−E_{n′}|/ħ is the transition frequency and α is the fine‑structure constant. The authors verify that the usual selection rules Δl=±1 and Δm=0,±1 remain unchanged in the gravitational context.

A thorough parametric analysis reveals several striking patterns. First, the radial matrix element scales roughly as n⁻³ l⁻¹ for large n and l, meaning that both high principal quantum numbers and high angular momentum strongly suppress the dipole coupling. Second, the energy spacing ΔE itself falls off as n⁻³, further reducing the ω³ factor in the decay rate. Consequently, states with l≈n (the “high‑l” or “circular” states) exhibit extraordinarily long lifetimes. For example, a state with n=100 and l=99 yields a calculated spontaneous‑emission rate on the order of 10⁻³⁰ s⁻¹, far exceeding the age of the universe.

The authors also compare direct large‑Δn transitions with cascades of many small‑Δn steps. Because the matrix element diminishes more slowly for Δn=±1, cascade pathways dominate the overall decay for most low‑l states, whereas for high‑l states the direct transition is already so weak that any radiative decay is essentially negligible.

Limitations of the approach are explicitly discussed. The derivation assumes that the wavefunction remains confined to regions where the gravitational field is weak, i.e., far from the Schwarzschild radius of the central mass. In strong‑gravity regimes relativistic corrections, spin‑orbit coupling, and curvature effects would have to be incorporated, invalidating the non‑relativistic formulas presented here. Moreover, experimental realization of a charged particle trapped solely by Earth’s gravity is technologically challenging; the results are therefore primarily of theoretical interest, providing benchmarks for future studies of gravitational quantum systems, possible astrophysical processes involving charged particles in weak gravitational wells, and speculative proposals for testing quantum‑gravity interplay.

In summary, this first paper in the series delivers a complete analytic framework for calculating electric‑dipole transition rates between gravitational eigenstates of a charged particle. It demonstrates that, contrary to naive expectations, many high‑angular‑momentum states are effectively metastable, with lifetimes that dwarf any realistic observational timescale. This insight enriches our understanding of quantum behavior in weak gravitational fields and opens avenues for further investigations into the stability, decoherence, and potential observational signatures of such exotic quantum states.