In theoretical work that eventually led to the laser, Einstein considered a set of N two-level atoms, as shown in Figure 4.8, in a box with N_k photons, where

the energy h̄ω_k corresponds to the energy difference between the two electron levels Based on the rules (4.7.19) above, one can straightforwardly write down rate equations for the populations of electrons in the different states and the photons:

where N_lo,i and N_hi,i are the electron occupation numbers for atom i and the bars indicate their average occupation numbers.

(a) Show that the rules for fermions imply that the (1−N) factors can be dropped, under the assumption that N_i,lo + N_i,hi = 1.

(b) Show that in equilibrium these equations are satisfied if

and N_k is a Planck distribution. This was one of the original arguments for stimulated emission.

N photons hwk