(1031) | |||

(1032) |

Suppose, for the sake of simplicity, that the diagonal elements of the interaction Hamiltonian, , are zero:

(1033) |

(1034) |

For a two-state system, Eq. (1028) reduces to

where . The above two equations can be combined to give a second-order differential equation for the time-variation of the amplitude :

Once we have solved for , we can use Eq. (1036) to obtain the amplitude . Let us search for a solution in which the system is certain to be in state 1 (and, thus, has no chance of being in state 2) at time . Thus, our initial conditions are and . It is easily demonstrated that the appropriate solutions to (1037) and (1036) are

(1038) | |||

(1039) |

where

(1040) |

Now, the probability of finding the system in state 1 at time is
simply
. Likewise, the probability of finding the system
in state 2 at time is
. It follows that

This result is known as

Equation (1042) exhibits all the features of a classic *resonance*.
At resonance, when the oscillation frequency of the perturbation, ,
matches the frequency , we find that

According to the above result, the system starts off in state 1 at . After a time interval it is certain to be in state 2. After a further time interval it is certain to be in state 1 again, and so on. Thus, the system periodically flip-flops between states 1 and 2 under the influence of the time-dependent perturbation. This implies that the system alternatively absorbs and emits energy from the source of the perturbation.

The absorption-emission cycle also takes place away from the resonance,
when
. However, the amplitude of the
oscillation in the coefficient is reduced. This means that the maximum
value of is no longer unity, nor is the minimum of
zero. In fact, if we plot the maximum value of as a function
of the applied frequency, , we obtain a resonance curve whose
maximum (unity) lies at the resonance, and whose full-width half-maximum
(in frequency) is . Thus, if the applied frequency differs
from the resonant frequency by substantially more than
then the probability of the system jumping from state 1 to state 2 is
always very small. In other words, the time-dependent
perturbation is only effective at causing transitions between states 1 and 2
if its frequency of oscillation lies in the approximate
range
. Clearly, the weaker
the perturbation (*i.e.*, the smaller becomes), the narrower
the resonance.