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Consider a system in which the time-independent Hamiltonian possesses
two eigenstates, denoted
Suppose, for the sake of simplicity, that the diagonal elements of
the interaction Hamiltonian, , are zero: i.e.,
|
(1033) |
The off-diagonal elements are assumed to oscillate sinusoidally
at some frequency : i.e.,
|
(1034) |
where and are real. Note that it is only the off-diagonal
matrix elements which give rise to the effect which we are interested
in--namely, transitions between states 1 and 2.
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 : i.e.,
|
(1037) |
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
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 Rabi's formula.
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.
Next: Spin Magnetic Resonance
Up: Time-Dependent Perturbation Theory
Previous: Preliminary Analysis
Richard Fitzpatrick
2010-07-20