On emission from a hydrogen-like atom

Abstract

A solution of the Dirac equation for an electron in the field of a point nucleus (Ze) has been obtained as an eigenfunction of the Schrödinger Hamiltonian and the spin projection operator Σ₃. With the use of this solution, the probability W^(ν) of the emission of a neutrino per unit time from a hydrogen-like atom, \((Ze)* \to (Ze) + \nu \bar \nu\), has been calculated for the first time in the first order of the parameter Ze ≪ 1. The probability W^(ν) appears to be rather small, and the corresponding lifetime τ^(ν) = [W^(ν)]^–1 is much larger than the age of the Universe; correspondingly, this process cannot affect the balance of low-energy neutrinos. The smallness of W^(ν) is due not only to the presence of the obvious “weak” factor (Gm_p²)²(m/mp)⁴ in the expression for W^(ν), but also primarily to the “electromagnetic” factor (Zα)¹², which can be revealed only in a particular calculation. It has been argued within quantum electrodynamics with the mentioned wavefunctions that photon emission, (Ze)* → (Ze) + γ, can be absent (analysis of photon emission requires the further development of the method), whereas axion emission, (Ze)* → (Ze) + a, can occur, although the last two effects have not been considered in detail.