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-tostate 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.
This paper develops further approximate methods for obtaining the dipole matrix elements and corresponding transition and decay rates of the highn, high-l gravitational eigenstates. These methods include (1) investigation of the polar spreads of the angular components of the high-n, high-l eigenstates and the effects these have on the limiting values of the angular components of the dipole matrix elements in the case of large l and m and (2) investigation of the rapid cut off and limited width of the low-p, high-n radial eigenfunctions, and the development of an equation to determine the width, position and oscillatory behaviour of those eigenfunctions in cases of arbitrarily large values of n, l and p. The methods have wider applicability than dipole transition rate estimates and may be also used to determine limits on the rates for more general interactions. Combining the methods enables the establishment of upper limits to the total dipole decay rates of many high-n, low-p states on the state diagram to be determined, even those that have many channels available for decay. The results continue to support the hypothetical existence of a specialized set of highn, low-p gravitational eigenfunctions that are invisible and stable, both with respect to electromagnetic decay and gravitational collapse, making them excellent dark matter candidates.
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