Spontaneous Emission in Semiclassical Radiation Theory

Abstract

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.