where is, in general, a function of position, momentum, and spin operators.

It follows from Eqs. (1064) and (1067) that, to first-order,

(1068) |

Integration with respect to yields

where

(1072) |

Now, the function takes its largest values when
, and is fairly negligible when (see Fig. 25). Thus, the first and second terms
on the right-hand side of Eq. (1071) are only
non-negligible when

(1073) |

(1074) |

Now, the function
is very strongly peaked at ,
and is completely negligible for
(see Fig. 25). It follows that the above expression exhibits a
*resonant response* to the applied perturbation at the frequencies
. Moreover,
the widths of these resonances decease linearly as time increases. At each
of the resonances (*i.e.*, at
), the transition
probability
varies as [since
]. This behaviour
is entirely consistent with our earlier result (1044), for the two-state
system, in the limit
(recall that our perturbative
solution is only valid as long as
).

The resonance at
corresponds to

(1076) |

(1077) |

Stimulated emission and absorption are mutually exclusive processes, since the
first requires , whereas the second requires . Hence, we can write the transition probabilities for
both processes separately. Thus, from (1075), the
transition probability for stimulated emission is

(1078) |