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Physical Review

Natural Line Shape

Francis E. Low

Published Oct 1, 1952• 189 citations
Publisher

Abstract

The covariant methods of Feynman and Dyson are applied to the problem of line shape. It is shown that the line center depends almost completely on the form of the ${{S}_{\mathrm{Fe}}}^{\ensuremath{'}}$ function, that is the Feynman-type Green's function for the electron interacting with an external field ${{A}_{\ensuremath{\mu}}}^{e}$ and with the quantized radiation field. A finite equation for ${{S}_{\mathrm{Fe}}}^{\ensuremath{'}}$ is derived, from which the line center and shape may be calculated to any desired accuracy. The method is illustrated by two examples: the elastic scattering of photons by a one electron atom in its ground state and the emission of photons by the same type of atom following thermal excitation.

DOI
10.1103/physrev.88.53

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The covariant methods of Feynman and Dyson are applied to the problem of line shape. It is shown that the line center depends almost completely on the form of the SFe\ensuremath′{{S}_{\mathrm{Fe}}}^{\ensuremath{'}}SFe​\ensuremath′ function, that is the Feynman-type Green's function for the electron interacting with an external field A\ensuremathμe{{A}_{\ensuremath{\mu}}}^{e}A\ensuremathμ​e and with the quantized radiation field. A finite equation for SFe\ensuremath′{{S}_{\mathrm{Fe}}}^{\ensuremath{'}}SFe​\ensuremath′ is derived, from which the line center and shape may be calculated to any desired accuracy. The method is illustrated by two examples: the elastic scattering of photons by a one electron atom in its ground state and the emission of photons by the same type of atom following thermal excitation.
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The general method of the preceding paper is applied to the theory of spectral line shape. Illustrative calculations of the transition-energy shift and width are carried out for a twolevel atom interacting with the quantized electromagnetic field. In the thermal-equilibrium case, explicit results are given in terms of the digamma function and Bose-Einstein distribution, generalizing the standard Lamb-shift and Weisskopf-Wigner natural width results to nonzero temperature. In the nonequilibrium case expressions are obtained which exhibit the influence of nonequilibrium photon distributions on the shift and width. A general theory of the time-dependent power radiated in the decay of an excitation is developed for systems not necessarily at or near equilibrium, in terms of appropriate generalizations of the Liouvillian Green's functions of the preceding paper. This approach is illustrated by calculation of the time-dependent radiated power for the same two-level model.
A variational Hamiltonian method for quantum field theory is applied to the problem of natural line shape. Within scalar quantum electrodynamics, coupled variational equations are derived for the scattering of photons from a scalar (pionic) atom. It is shown that the coupled equations have a continuous spectrum for all energies above the atomic ground state. Excited states of the atom appear as resonances in particular partial-wave cross sections. The line center and shape are derived from the behavior of the photon-atom scattering phase shifts.
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