Physical Review A

Comment on the Characteristic Time of Spontaneous Decay in Jaynes's Semiclassical Radiation Theory

D. Leiter

Published Jul 1, 1970• 14 citations

Abstract

We examine the implication in the fact that, in a semiclassical theory of atomic structure proposed by Jaynes and Crisp, the initial state of an atom plays a critical role in determining its average lifetime in an excited state.

DOI: 10.1103/physreva.2.259

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Physical Review A

Comment on the Characteristic Time of Spontaneous Decay in Jaynes's Semiclassical Radiation Theory

D. Leiter

Published Jul 1, 1970• 14 citations

Publisher

Abstract

DOI: 10.1103/physreva.2.259

Related papers

Radiative Decay of the 2P State of Atomic Hydrogen: A Test of the Exponential Decay Law

John M. Wessner, David K. Anderson, R. T. Robiscoe

In Jaynes's semiclassical radiation theory, the decay of an excited atomic state depends on its initial population. For the radiative decay of the $2P$ level in atomic hydrogen, produced by pulsed excitation from the $2S$ level, the theory predicts a decay rate $\ensuremath{\Gamma}=(1\ensuremath{-}f){\ensuremath{\Gamma}}_{0}+\ensuremath{\beta}P$, where $\ensuremath{\beta}=f{\ensuremath{\Gamma}}_{0}$. Here $f$ is the initial $2S$ population, ${\ensuremath{\Gamma}}_{0}$ is the natural $2P$ decay rate, and $P$ is the $2P$ population relative to $f$ . For $f=0.086\ifmmode\pm\else\textpm\fi{}0.016$, and $P$ values in the range 0.15 to 0.64, we have measured $\ensuremath{\beta}=\ensuremath{-}(2.8\ifmmode\pm\else\textpm\fi{}2.8)\ifmmode\times\else\texttimes\fi{}{10}^{6}$ ${\mathrm{sec}}^{\ensuremath{-}1}$. The corresponding value from the semiclassical theory is $\ensuremath{\beta}=+(53.8\ifmmode\pm\else\textpm\fi{}10.0)\ifmmode\times\else\texttimes\fi{}{10}^{6}$ ${\mathrm{sec}}^{\ensuremath{-}1}$.In Jaynes's semiclassical radiation theory, the decay of an excited atomic state depends on its initial population. For the radiative decay of the $2P$ level in atomic hydrogen, produced by pulsed excitation from the $2S$ level, the theory predicts a decay rate $\ensuremath{\Gamma}=(1\ensuremath{-}f){\ensuremath{\Gamma}}_{0}+\ensuremath{\beta}P$, where $\ensuremath{\beta}=f{\ensuremath{\Gamma}}_{0}$. Here $f$ is the initial $2S$ population, ${\ensuremath{\Gamma}}_{0}$ is the natural $2P$ decay rate, and $P$ is the $2P$ population relative to $f$. For $f=0.086\ifmmode\pm\else\textpm\fi{}0.016$, and $P$ values in the range 0.15 to 0.64, we have measured $\ensuremath{\beta}=\ensuremath{-}(2.8\ifmmode\pm\else\textpm\fi{}2.8)\ifmmode\times\else\texttimes\fi{}{10}^{6}$ ${\mathrm{sec}}^{\ensuremath{-}1}$. The corresponding value from the semiclassical theory is $\ensuremath{\beta}=+(53.8\ifmmode\pm\else\textpm\fi{}10.0)\ifmmode\times\else\texttimes\fi{}{10}^{6}$ ${\mathrm{sec}}^{\ensuremath{-}1}$.

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Physical ReviewMar 25, 1969211 citations

Spontaneous Emission in Semiclassical Radiation Theory

ROBERT K. NESBET

It has recently been shown by Jaynes and collaborators that semiclassical radiation theory contains a description of spontaneous emission of radiation and of radiative level shifts. The present paper gives a gauge-invariant derivation of the radiant energy production rate, using only Maxwell's equations and the usual definitions of electric current and charge density for a many-particle material system described by Schr\"odinger's equation. A complementary derivation, using the time-dependent Schr\"odinger equation, verifies the instantaneous conservation of energy and probability during the radiation process. The semiclassical rate of spontaneous emission differs from the usual formula (Einstein's $A$ coefficient) because it depends on the occupancy of both initial and final states of the material system. The implications of this with regard to thermal equilibrium and Planck's law are examined. If a new hypothesis is introduced, postulating the decomposition of the equilibrium radiation into incoherent components, each interacting with a specific pair of energy levels of the material system, then Planck's law is shown to hold for the total intensity of radiation. If this hypothesis is not introduced, the equilibrium conditions for different transitions are incompatible, and the semiclassical radiation theory is incapable of describing thermal equilibrium.It has recently been shown by Jaynes and collaborators that semiclassical radiation theory contains a description of spontaneous emission of radiation and of radiative level shifts. The present paper gives a gauge-invariant derivation of the radiant energy production rate, using only Maxwell's equations and the usual definitions of electric current and charge density for a many-particle material system described by Schr\"odinger's equation. A complementary derivation, using the time-dependent Schr\"odinger equation, verifies the instantaneous conservation of energy and probability during the radiation process. The semiclassical rate of spontaneous emission differs from the usual formula (Einstein's $A$ coefficient) because it depends on the occupancy of both initial and final states of the material system. The implications of this with regard to thermal equilibrium and Planck's law are examined. If a new hypothesis is introduced, postulating the decomposition of the equilibrium radiation into incoherent components, each interacting with a specific pair of energy levels of the material system, then Planck's law is shown to hold for the total intensity of radiation. If this hypothesis is not introduced, the equilibrium conditions for different transitions are incompatible, and the semiclassical radiation theory is incapable of describing thermal equilibrium.

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Long-Term Solution in Semiclassical Radiation Theory

Carlos R. Stroud, E. T. Jaynes

An improved form of semiclassical radiation theory is developed which includes the effect of the atom's radiation field back on the atom. This formalism is applied to the problem of a single two-level atom interacting with a monochromatic field. The resulting equations are solved without resorting to time-dependent perturbation theory, and are found to predict the behavior of the system over times long compared with the lifetime for spontaneous transitions. Not only stimulated emission and absorption, but also spontaneous emission with the proper Einstein $A$ coefficient, and a frequency shift which agrees at least semiquantitatively with the Lamb shift are described. In addition, several nonlinear effects involving the interference between spontaneous and stimulated radiation are described, and new experiments which might detect such effects are suggested.An improved form of semiclassical radiation theory is developed which includes the effect of the atom's radiation field back on the atom. This formalism is applied to the problem of a single two-level atom interacting with a monochromatic field. The resulting equations are solved without resorting to time-dependent perturbation theory, and are found to predict the behavior of the system over times long compared with the lifetime for spontaneous transitions. Not only stimulated emission and absorption, but also spontaneous emission with the proper Einstein $A$ coefficient, and a frequency shift which agrees at least semiquantitatively with the Lamb shift are described. In addition, several nonlinear effects involving the interference between spontaneous and stimulated radiation are described, and new experiments which might detect such effects are suggested.

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We consider the problem of determining the probability distribution for the time of decay for a one-dimensional Schrodinger operator.

Jul 1, 20000 citations

In Jaynes's semiclassical radiation theory, the decay of an excited atomic state depends on its initial population. For the radiative decay of the

2P

level in atomic hydrogen, produced by pulsed excitation from the

2S

level, the theory predicts a decay rate $\ensuremath{\Gamma}=(1\ensuremath{-}f){\ensuremath{\Gamma}}_{0}+\ensuremath{\beta}P$, where $\ensuremath{\beta}=f{\ensuremath{\Gamma}}_{0}$. Here

f

is the initial

2S

population, ${\ensuremath{\Gamma}}_{0}$ is the natural

2P

decay rate, and

P

is the

2P

population relative to

f

. For $f=0.086\ifmmode\pm\else\textpm\fi{}0.016$, and

P

values in the range 0.15 to 0.64, we have measured $\ensuremath{\beta}=\ensuremath{-}(2.8\ifmmode\pm\else\textpm\fi{}2.8)\ifmmode\times\else\texttimes\fi{}{10}^{6}$ ${\mathrm{sec}}^{\ensuremath{-}1}$. The corresponding value from the semiclassical theory is $\ensuremath{\beta}=+(53.8\ifmmode\pm\else\textpm\fi{}10.0)\ifmmode\times\else\texttimes\fi{}{10}^{6}$ ${\mathrm{sec}}^{\ensuremath{-}1}$.

A

coefficient for the transition. The theory also predicts that the frequency of the light radiated during a transition will have a small time dependence. The corresponding frequency shifts are compared with the Lamb shift in hydrogen. The derived equations are used to study the response of a many-level atom to an applied, monochromatic field. In the case of a three-level system, it is predicted that optical nutations are not just limited to the resonant transition, but are also present in the fluorescence involving the other level.

A

coefficient) because it depends on the occupancy of both initial and final states of the material system. The implications of this with regard to thermal equilibrium and Planck's law are examined. If a new hypothesis is introduced, postulating the decomposition of the equilibrium radiation into incoherent components, each interacting with a specific pair of energy levels of the material system, then Planck's law is shown to hold for the total intensity of radiation. If this hypothesis is not introduced, the equilibrium conditions for different transitions are incompatible, and the semiclassical radiation theory is incapable of describing thermal equilibrium.

A

coefficient, and a frequency shift which agrees at least semiquantitatively with the Lamb shift are described. In addition, several nonlinear effects involving the interference between spontaneous and stimulated radiation are described, and new experiments which might detect such effects are suggested.