Stimulated Emission and Bose-Einstein Statistics

Abstract

Einstein's derivation of Planck's blackbody formula using the A and B coefficients for spontaneous and stimulated transition probabilities is reviewed. Application of the Einstein-Fowler equation for statistical fluctuations, 〈(ΔE)2〉 = kT2(δE/δT), shows that two terms contribute to the fluctuations, one from the particle nature of the system and one from its wave nature. When expressed in terms of the average number of quanta n̄, occupying g phase cells, the fluctuations become 〈(Δn)2〉 = n̄ + (n̄2/g). This is exactly the first order variance 〈(Δn)2〉 of a system of n̄ particles distributed among g phase cells in Bose-Einstein statistics. If stimulated emission is omitted in the Einstein derivation, the Wien formula results, and the fluctuations become equal to the single term n̄, which is characteristic of classical statistics. Since the Planck law, which depends on both stimulated emission and stimulated absorption, gives fluctuation in accord with the Bose-Einstein statistics, whereas Wien's law, which depends on stimulated absorption alone, gives fluctuations in accord with classical statistics, it is concluded that the stimulated emission process is the mechanism responsible for the appearance of the Bose-Einstein statistics in blackbody radiation.