Time-dependent perturbation theory and transition probabilities

Time-dependent perturbation theory and transition probabilities 

In a celebrated paper, Einstein (1917) analyzed the nature of atomic transitions in a radiation field and pointed out that, in order to satisfy the conditions of thermal equilibrium, one has to have not only a spontaneous transition probability per unit time A2i from an excited state 2 to a lower state 1 and an absorption probability BUJV from 1 to 2 , but also a stimulated emission probability B2iJv from state 2 to 1 . The latter can be more usefully thought of as negative absorption, which becomes dominant in masers and lasers.1 Relations between the coefficients are found by considering detailed balancing in thermal equilibrium 

Comparison of Eq. (A3.2) with Planck s law then gives the relations 

To relate B i to the absorption coefficient av, consider a unit volume of gas bathed in a radiation field Jv. The net rate at which energy is absorbed is then 

1 Einstein defined the B coefficients in terms of radiation energy density, but following E. A. Milne it is more 

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