On emission from a hydrogen-like atom
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, PARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS
On Emission from a Hydrogen-Like Atom V. V. Skobelev Moscow State University of Mechanical Engineering (MAMI), Moscow, 115280 Russia e-mail: [email protected] Received September 4, 2014; in final form, September 21, 2015
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)* → (Ze) + νν , 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 (G m 2p )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. DOI: 10.1134/S1063776116020126
1. INTRODUCTION The problem of a hydrogen-like atom has been of the highest priority from the very beginning of quantum physics and mechanics. This is due both to its fundamental importance as a test of any theory in application to the simplest atom and to possible astrophysical aspects because hydrogen is the most abundant element in the Universe. In addition to the solved problem of the determination of the spectrum of electromagnetic radiation from the hydrogen-like atom, it is important to calculate the probabilities W(ν) of a transition of the electron from excited states with the emission of a photon per unit time and the corresponding lifetimes τ(γ) = [W(γ)]–1 in these excited states. As far as I know, this problem has not yet been solved completely in an analytical form using exact QED methods with the interaction Lagrangian
L = e[Ψ e γ μ Ψ e ]Aμ.
(1)
This is partially because of the absence of a solution Ψ (Ψe = exp(–iEt)Ψ]) of the Dirac equation for the electron in the field of the point nucleus Ze that could be used for real calculations. Estimates are usually calculated with the following expression obtained for semiclassical reasons (see, e.g., [1]):
3 W (γ) = 4 αΔ3 E2 |〈n '| r| n〉| 2, 3 c
(2)
where 2
α = e ≈ 1 , Δ E = E n − E n ', c 137 (2a) m(Ze 2 ) 2 (Z α) 2 2 =− E ≡ En = − mc . 2 2 2 2 n 2n Spin effects are naturally disregarded in this expression. However, the matrix element of the emission of a transversely polarized photon from the hydrogen-like atom with the wavefunction Ψ obtaine
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