Relations between Transition Rates and Quantum Numbers in Gravitational Potentials

This paper forms part of an ongoing investigation to examine the quantum prediction that isolated baryons and electrons in the deep gravity wells of galaxy halos should exhibit reduced interaction cross-sections by virtue of the composition of the gravitational eigenspectra of their wave functions, and thereby identify a possible mechanism responsible for the origin of dark matter, without resorting to new physics or unknown particles. Relevant to this investigation are the electromagnetic state-to-state transition rates of charged particles occupying these gravitational eigenstates (EinsteinA coefficients), and, in the present work, we examine trends in these rates and net state lifetimes for particles in 1/r potential wells for values of the principal quantum number n and the angular momentum quantum number l. We find that transition rates decrease with increasing n and l, and that the rate is more steeply dependent on l when the quantum parameter Δp (≡ Δn ? Δl) is greater, in agreement with earlier work. It is also found that there is an empirical relationship between the total state lifetime τ and the eigenvalues n and l, which is given by τ ∝ n ^α l ^β , where α ≈ 3 and β ≈ 2. The results apply equally to electrical potential wells, where the phenomena of reduced cross-sections and long radiative lifetimes is well known in the case of the Rydberg states of electrons in atoms. More importantly, in the case of gravitational eigenstates discussed here, the quantum prediction of low Einstein A (and therefore B) coefficients ofmany of the stateto-state transitions will mean that a particle whose eigenspectral composition consists of many of these weakly interacting states will be less likely to undergo scattering processes such as Compton scattering. Trends in the Einstein coefficients over the range of component eigenstates are required for calculating the net visibility and interaction rates of the generalized wave functions representing charged particles in macroscopic gravitational fields.