Wave Function of the Dirac Equation for an Electron in the Field of a Nucleus Expressed in Terms of an Eigenfunction of the Spin Projection Operator and a Wave Function of the Schrödinger Equation. Radiative Processes of a Hydrogen-Like Atom and Selection Rules

A solution of the Dirac equation for an electron in the field of a point nucleus (Ze), expressed in terms of an eigenfunction of the operator of the spin projection onto the third axis and the corresponding solution of the Schrödinger equation is derived. This solution is suitable for practical calculations. On the basis of this solution, using ordinary methods of QED and field theory, general principles for the emission of photons, axions, and neutrinos ${(Ze)}^{*}\to (Ze)+\gamma, a,\kern0.5em v\overline{v}$ by a hydrogen-like atom are formulated which take into account the spin state of the electron and, in the case of photons, their polarization. This range of questions pertaining to a comparative characteristic of processes of emission of massless or almost massless particles has, to this day, not been discussed from this point of view in the literature. Selection rules for $\gamma, a,v\overline{v}$ emission processes are also obtained, where for axions and neutrinos they coincide with the existing selection rules in the literature ∆m = 0,±1; with ∆l = ±1 pertaining to photons, but for photon emission a few of them do in fact differ from them with the hypothesis of odd values of ∆l, not established by us and additional to the usual values ∆l = ±1 of variation of the azimuthal quantum number l due to the appearance of “new” integrals over the spherical angle $\theta$ for ∆m = ±1, where for ∆m = 0, as before, ∆l = ±1. Moreover, the dependence of the amplitude of the photon emission process on the quantum numbers is in principle different than in the previously adopted approach to the problem although the lifetime in the excited state for small values of the quantum numbers coincides in order of magnitude with the accepted value ~10^–9 s.