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American Journal of Physics

Atoms as sources of multipole radiation

Keh‐Ning Huang

Published Jan 1, 1994• 6 citations
Publisher

Abstract

Radiative transitions in atoms are formulated as multipole radiations from a localized charge-current distribution. The correspondence between the relativistic quantum mechanical and classical electrodynamic treatments is demonstrated explicitly. Definitions for multipole moments, line strengths, and oscillator strengths in relativistic quantum mechanics are also summarized.

DOI
10.1119/1.17715

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Radiative transitions in atoms are formulated as multipole radiations from a localized charge-current distribution. The correspondence between the relativistic quantum mechanical and classical electrodynamic treatments is demonstrated explicitly. Definitions for multipole moments, line strengths, and oscillator strengths in relativistic quantum mechanics are also summarized.
Almost everything we know about atomic energy levels comes from the spectra resulting from radiative transitions between these levels. It is possible that the electric dipole moment of a charge distribution may vanish, in which case the radiation depends on the higher moments. The source of magnetic dipole radiation is a circulating current that reverses direction harmonically, as in a loop antenna. Since a steady-current loop can be regarded as equivalent to a static magnetic dipole, the oscillating current is equivalent to an oscillating dipole. If the electric dipole radiation is impossible, but magnetic dipole radiation can occur, the radiative process would be classed as "forbidden" because of its low intrinsic intensity. An oscillating linear dipole emits no radiation along the direction of its axis of vibration. Although the physical origin of magnetic moments is circulating currents, the effect of an oscillating circular current can be calculated by considering an equivalent linear magnetic dipole.
The author reviews investigations of the photon localization and quantum phase problems based on the use of the representation of multipole photons. Section II presents a general consideration of the field quantization, including a comparison of zero-point oscillations of the plane and multipole waves of photons. Section III discusses the atom-field interaction leading to the multipole radiation. The SU(2) quantum phase representation of atomic variables is considered, as is a connection between the SU(2) quantum phase states and entanglement phenomenon. In Section IV the author describes the quantum phase of multipole radiation caused by the angular momentum conservation in the process of radiation. This approach is compared with the Pegg–Barnett formalism and with Mandel's operational approach. Section V covers the quantum polarization properties of multipole radiation. Photon localization, quantum measurements, and causality are considered in Section VI.
So far we have treated the energy levels of an atomic system as stable states with infinite lifetimes. The presence of an electromagnetic field changes this picture. Absorption of photons will excite atoms and ions to higher levels, while stimulated emission will make excited ones decay. Einstein (1917) showed, using purely statistical arguments, that an excited ion will decay through spontaneous emission in the absence of an electromagnetic field. This process can also be derived by using a rigorous quantum electrodynamic treatment, where the field is quantized. In this chapter we start by discussing transitions between atomic levels in a general sense, postulating the operators responsible for the decay. After the first few sections we will concentrate on the most important few multipoles or terms in the expansion of the electromagnetic field. We will introduce a number of different properties describing the transition, and eventually arrive at the measurable transition rate and, directly linked to it, the lifetime of an excited state. We will start, however, with the most basic property, the component strength. Our aim is to arrive at computationally useful expressions as soon as possible. For a more thorough treatment of the theory of radiative transitions we refer to the book by Sobelman (1972). For a brief review of the many factors affecting spectra of atoms, see Martin and Wiese (1996) .