Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Spontaneous decay of an unstable atomic state in non-relativistic QED: a complete treatment including gauge invariance, renormalization and non-Markovian time behaviour

JOSIP SEKE

Published Jan 21, 1994• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/27/2/012

Related papers

NEW NON-MARKOVIAN TREATMENT OF THE SPONTANEOUS DECAY OF AN UNSTABLE ATOMIC STATE IN RELATIVISTIC QED

JOSIP SEKE

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.

Modern Physics Letters B•Aug 30, 1996•0 citations

Non-Markovian treatment of the spontaneous 2P1/2to 1S1/2transition in the Dirac hydrogen atom

JOSIP SEKE

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.

Journal of Physics A Mathematical and General•Feb 7, 1992•14 citations

Improved Heisenberg equations-of-motion approach for nonequilibrium decay phenomena

C. F. Hart

It is pointed out that the usual Heisenberg equations-of-motion or Liouvillian superoperator perturbation-theory approaches to systems with arbitrary statistical ensembles can give spurious, nonconvergent results when attempts are made to calculate the expectation value for certain operators which commute with the zeroth-order Hamiltonian. The non-Markovian, long-time tail of the Weisskopf-Wigner spontaneous-emission decay of a two-level atom is presented as an example of this failure. The general reason for such failures is elucidated and a new method applicable to arbitrary equilibrium or nonequilibrium statistical ensembles is formulated. The new method is applied to the same example of spontaneous emission to derive the correct short- and long-time behavior. The physical model of a two-level atom is discussed and some mathematical techniques of general usefulness are presented. The present results are also briefly discussed in the context of previous work on nonexponential decay. A fully quantum-mechanical definition of time-dependent spectral line shape is given in terms of the response of a two-level---atom detector with variable frequency interacting with electromagnetic radiation. The non-Lorentzian line shape recorded by such a detector for the spontaneous emission of a source two-level atom is calculated as another example of a problem for which the usual methods give incorrect results. The extension of the present method to Liouvillian or master-equation approaches is discussed.

Physical Review A•Jul 1, 1983•2 citations

Finite-time deviations from exponential decay in the case of spontaneous emission from a two-level hydrogenic atom

JOSIP SEKE, Wolfgang N. Herfort

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.

Physical Review A•Aug 1, 1989•35 citations

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

JOSIP SEKE

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.

Physical Review A•Jan 1, 1992•12 citations

Quantum Field Theory Description of the Hydrogen Lyman α Transition: From QED to the Bohr–Maxwell Correspondence

Daniel Scretching

This paper presents a quantum-field-theoretic formulation of the hydrogen 2p → 1s Lyman-α transition, replacing the semiclassical Bohr–Maxwell description with the modern framework of quantum electrodynamics. The initial and final bound-electron states are described by the Dirac equation in the proton’s Coulomb field, with the Schrödinger equation recovered as the leading nonrelativistic limit. The electromagnetic radiation field is quantized, and photon emission is generated by the QED interaction Hamiltonian. The transition amplitude is evaluated from the atomic current and quantized electromagnetic field, leading through Fermi’s golden rule to the spontaneous-emission rate and Einstein coefficient A₂₁. Energy and momentum conservation require the emitted photon to carry the atomic transition energy, including the small correction associated with proton recoil. At leading order, the Lyman-α photon has ΔE ≈ 10.2 eV, ν ≈ 2.47 × 10¹⁵ Hz, and λ ≈ 121.6 nm. In the electric-dipole approximation, the field-theoretic calculation reduces to the standard matrix element of the interaction Hamiltonian H_int = −μ·E, thereby connecting QED directly to the transition dipole moment, oscillator strength, Einstein A coefficient, and radiative lifetime. The calculation reproduces the accepted numerical results for hydrogen while identifying their origin in the coupling between the quantized electron field and the quantized electromagnetic field. The paper also explains how classical electrodynamics is recovered from QED. Maxwell’s equations govern the expectation values of the electromagnetic-field operators in the appropriate classical-field limit, particularly for coherent states containing many photons. A single spontaneously emitted Lyman-α photon, however, remains a genuinely quantum excitation rather than a classical electromagnetic wave. Leading corrections beyond the nonrelativistic Coulomb approximation are then considered, including relativistic fine structure, the Lamb shift, and atomic recoil. The resulting treatment provides a rigorous bridge from the historical Bohr model and Schrödinger wave mechanics to Dirac theory and full QED, verifying the Lyman-α energy, frequency, wavelength, transition probability, and radiative decay structure within a unified quantum-field framework.

•Jul 12, 2026•0 citations
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.
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.
It is pointed out that the usual Heisenberg equations-of-motion or Liouvillian superoperator perturbation-theory approaches to systems with arbitrary statistical ensembles can give spurious, nonconvergent results when attempts are made to calculate the expectation value for certain operators which commute with the zeroth-order Hamiltonian. The non-Markovian, long-time tail of the Weisskopf-Wigner spontaneous-emission decay of a two-level atom is presented as an example of this failure. The general reason for such failures is elucidated and a new method applicable to arbitrary equilibrium or nonequilibrium statistical ensembles is formulated. The new method is applied to the same example of spontaneous emission to derive the correct short- and long-time behavior. The physical model of a two-level atom is discussed and some mathematical techniques of general usefulness are presented. The present results are also briefly discussed in the context of previous work on nonexponential decay. A fully quantum-mechanical definition of time-dependent spectral line shape is given in terms of the response of a two-level---atom detector with variable frequency interacting with electromagnetic radiation. The non-Lorentzian line shape recorded by such a detector for the spontaneous emission of a source two-level atom is calculated as another example of a problem for which the usual methods give incorrect results. The extension of the present method to Liouvillian or master-equation approaches is discussed.
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.
This paper presents a quantum-field-theoretic formulation of the hydrogen 2p → 1s Lyman-α transition, replacing the semiclassical Bohr–Maxwell description with the modern framework of quantum electrodynamics. The initial and final bound-electron states are described by the Dirac equation in the proton’s Coulomb field, with the Schrödinger equation recovered as the leading nonrelativistic limit. The electromagnetic radiation field is quantized, and photon emission is generated by the QED interaction Hamiltonian. The transition amplitude is evaluated from the atomic current and quantized electromagnetic field, leading through Fermi’s golden rule to the spontaneous-emission rate and Einstein coefficient A₂₁. Energy and momentum conservation require the emitted photon to carry the atomic transition energy, including the small correction associated with proton recoil. At leading order, the Lyman-α photon has ΔE ≈ 10.2 eV, ν ≈ 2.47 × 10¹⁵ Hz, and λ ≈ 121.6 nm. In the electric-dipole approximation, the field-theoretic calculation reduces to the standard matrix element of the interaction Hamiltonian H_int = −μ·E, thereby connecting QED directly to the transition dipole moment, oscillator strength, Einstein A coefficient, and radiative lifetime. The calculation reproduces the accepted numerical results for hydrogen while identifying their origin in the coupling between the quantized electron field and the quantized electromagnetic field. The paper also explains how classical electrodynamics is recovered from QED. Maxwell’s equations govern the expectation values of the electromagnetic-field operators in the appropriate classical-field limit, particularly for coherent states containing many photons. A single spontaneously emitted Lyman-α photon, however, remains a genuinely quantum excitation rather than a classical electromagnetic wave. Leading corrections beyond the nonrelativistic Coulomb approximation are then considered, including relativistic fine structure, the Lamb shift, and atomic recoil. The resulting treatment provides a rigorous bridge from the historical Bohr model and Schrödinger wave mechanics to Dirac theory and full QED, verifying the Lyman-α energy, frequency, wavelength, transition probability, and radiative decay structure within a unified quantum-field framework.