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Consider a perturbation which oscillates sinusoidally in time.
This is usually called a harmonic perturbation. Thus,

(812) 
where is, in general, a function of position, momentum, and
spin operators.
Let us initiate the system in the eigenstate of the unperturbed
Hamiltonian, , and switch on the harmonic perturbation at .
It follows from Eq. (787) that
where
This formula is analogous to Eq. (794), provided that

(816) 
Thus, it follows from the previous analysis that
the transition probability
is only appreciable in the limit
if
Clearly, (817) corresponds to the first term on the righthand side
of Eq. (813), and (818) corresponds to the second term. The former
term describes a process by which the system gives up energy
to the perturbing field, whilst making a transition
to a final state whose energy level is less than that of the initial
state by . This process is known as stimulated emission.
The latter term describes a process by which the system gains
energy from the perturbing field, whilst making a transition
to a final state whose energy level exceeds that of the initial
state by . This process is known as absorption. In
both cases, the total energy (i.e., that of the system plus
the perturbing field) is conserved.
By analogy with Eq. (807),
Equation (819) specifies the transition rate for stimulated emission, whereas
Eq. (820) gives the transition rate for absorption.
These equations are more usually written
It is clear from Eqs. (814)(815) that
.
It follows from Eqs. (819)(820) that

(823) 
In other words, the rate of stimulated emission, divided by the density
of final states for stimulated emission, equals the rate of absorption,
divided
by the density of final states for absorption. This result, which
expresses a fundamental symmetry between absorption and stimulated
emission, is known as detailed balancing, and is very important in
statistical mechanics.
Next: Absorption and stimulated emission
Up: Approximation methods
Previous: Constant perturbations
Richard Fitzpatrick
20060216