Consider a quantum system with a Hamiltonian 
. Suppose this system is subject
to an external driving force 
such that the full Hamiltonian takes the form
where B is an operator through which this coupling occurs. This is the situation, for
example, when the infrared spectrum is measured experimentally - the external force
is identified with an electric field E(t) and B is identified with
the electric dipole moment operator. If the field is inhomogeneous, then
H takes the more general form
where the sum is taken over Fourier modes. Often, B is an operator such that,
if 
, then
Suppose we take to be a monochromatic field of the form
Generally, the external field can induce transitions between eigenstates
of in the system. Consider such a transition between an initial
state 
and a final state 
, with energies 
and 
,
respectively:
(see figure below).
This transition can only occur if
