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Physical Review A

Finite-time deviations from exponential decay for the Lyman-α radiation in the Dirac hydrogen atom without ignoring virtual transitions

JOSIP SEKE

Published Jan 1, 1992• 12 citations
Publisher

Abstract

By using a self-consistent projection-operator method, developed recently by the author, finite-time deviations from exponential decay for the Lyman-\ensuremath{\alpha} radiation in the Dirac hydrogen atom are calculated without neglecting retardation and virtual-transition effects.
DOI
10.1103/physreva.45.542

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Modern Physics Letters B•Jul 10, 1999•4 citations
For the first time to the author's knowledge, a rigorous mathematical treatment of the spontaneous 2P 1/2 to 1S 1/2 transition, going beyond the usual Markov approximation, is presented in the case of the Dirac hydrogen atom. The relevant transition matrix elements of the Dirac Hamiltonian are calculated without the dipole approximation. Further, a non-Markovian equation of motion for the non-decay probability of the unstable 2P 1/2 2 state is derived, by using a self-consistent projection-operator method recently developed by the present author. By a rigorous analytical treatment of the non-Markovian equation, deviations from exponential decay for asymptotic times are obtained. Moreover, by careful error estimations deviations from exponential decay for finite times with definite validity range are also calculated. Finally, for the radiative lineshape a new quasi-Lorentzian distribution is obtained.
For the first time to the authors' knowledge a mathematically rigorous method is used for treating finite-time deviations from the exponential decay in the case of spontaneous Lyman-\ensuremath{\alpha} transitions in a two-level hydrogenic atom. First, in the so-called Weisskopf-Wigner model (where the rotating-wave approximation is implied) finite-time deviations with a rigorous validity range, based on accurate error estimations, are derived. Second, in order to obtain the frequency shift (Lamb shift), counter-rotating terms are taken into account by using a projection-operator method developed a decade ago by one of the present authors [J. Seke, Phys. Rev. A 21, 2156 (1980)]. By means of this method, equations of motion for dipole-moment expectation values are derived without employing the usual Born approximation. These equations are solved by using the method of the Laplace transform and its inverse. Again finite-time deviations from the exponential decay of the dipole-moment expectation values with a definite validity range are obtained. Finally, it should be emphasized that all the results presented in this paper contain accurate error estimates which are absolutely necessary in a rigorous mathematical method.
It is shown that the application of a self-consistent projection-operator method, developed recently by the author, makes possible a complete and gauge-invariant treatment of spontaneously decaying atomic state. All the shortcomings of the Weisskopf-Wigner gauge-dependent method (Markov approximation and the neglect of virtual transition, retardation and electron-spin effects) are removed. After deriving a gauge-invariant equation of motion for the decaying state, a new physical interaction picture (in which the unobservable interaction of the free electron with the vacuum radiation field is eliminated) is introduced. In this new interaction picture, apart from a natural solution of the renormalization problem, explicit finite analytic results for the whole non-Markovian Lyman- alpha spontaneous decay are derived. Furthermore, a quasi-Lorentzian expression for the radiative line shape containing the correct (complete) finite non-relativistic frequency shift, stemming from the Lamb shifts of both levels 1S and 2P, is calculated.
It is shown that the application of a self-consistent projection-operator method, developed recently by the author, makes a complete treatment of the spontaneously decaying atomic state in relativistic QED possible. In a new physical real-transition interaction picture (in which the unobservable interaction of the free electron with the vacuum radiation field is eliminated) explicit finite analytic results for the non-Markovian time evolution and the radiative line shape in the case of relativistic Lyman-α spontaneous emission are obtained.
A recent paper by Facchi and Pascazio, in which the advantage of numerical evaluations of deviations from exponential decay was pointed out, makes a revisitation of the subject necessary. All the more, since the above-mentioned authors have not estimated the error bounds (being absolutely necessary in any calculation) for their results. On the contrary, the accuracy of all results presented in this paper is proved by error estimates. Moreover, for the first time, the power of the simple iteration method in calculating both the exponential and non-exponential time evolution of an unstable atomic state is demonstrated.