where H'mn(t) = J* ftJ„u(r, t)ftn dt is the integral over volume (with dt representing an element of volume), and wmn = (Em — En)/h. For an interaction between a particle and an electromagnetic field of angular frequency w, the time-dependent perturbation varies, to a first approximation, sinusoidally with time as u(r, t) = u'(r) cos wt. (6.5)

Substituting this into the previous equations and defining H'n = Jft'^'(r)ftn dt we find that the probability that the particle will be in state m some time t after being in state n is given by

This probability is negligible except when u is close to umn, provided that the time t is long compared with the period of the perturbation, and thus it can be seen that the absorption or emission of a photon of appropriate frequency may change the state of a particle. In reality, various broadening effects such as Doppler broadening mean that there are a number of pairs of energy levels which have an energy difference «umn and thus we need an additional term, known as the density of states g(umn), where g(umn) dumn is defined as the number of pairs of energy levels with energy difference between «umn and «(umn + dumn). The total probability for the transition to take place is then

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