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Consider a (Hermitian) perturbation which oscillates sinusoidally
in time. This is usually termed a harmonic perturbation. Such
a perturbation takes the form
![\begin{displaymath}
H_1(t) = V \exp( {\rm i} \omega t) + V^\dag \exp(-{\rm i} \omega t),
\end{displaymath}](img2481.png) |
(1067) |
where
is, in general, a function of position, momentum, and spin operators.
It follows from Eqs. (1064) and (1067) that, to first-order,
![\begin{displaymath}
c_f(t) = - \frac{\rm i}{\hbar}\int_0^t\left[V_{fi} \exp( {...
... i} \omega t')\right]
\exp( {\rm i} \omega_{fi} t') dt',
\end{displaymath}](img2482.png) |
(1068) |
where
Integration with respect to
yields
where
![\begin{displaymath}
{\rm sinc} x\equiv \frac{\sin x}{x}.
\end{displaymath}](img2491.png) |
(1072) |
Figure 25:
The functions
(dashed curve) and
(solid curve). The vertical dotted lines denote
the region
.
![\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter13/fig01.eps}}
\end{figure}](img2494.png) |
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
![\begin{displaymath}
\vert\omega+\omega_{fi}\vert\stackrel {_{\normalsize <}}{_{\normalsize\sim}}\frac{2\pi}{t},
\end{displaymath}](img2497.png) |
(1073) |
and
![\begin{displaymath}
\vert\omega-\omega_{fi}\vert\stackrel {_{\normalsize <}}{_{\normalsize\sim}}\frac{2\pi}{t},
\end{displaymath}](img2498.png) |
(1074) |
respectively.
Clearly, as
increases, the ranges in
over which these two
terms are non-negligible gradually shrink in size.
Eventually, when
, these two ranges become strongly non-overlapping.
Hence, in this limit,
yields
![\begin{displaymath}
P_{i\rightarrow f}(t) = \frac{t^2}{\hbar^2}\left\{
\vert V_{...
...} {\rm sinc}^2\left[(\omega-\omega_{fi}) t/2\right]\right\}.
\end{displaymath}](img2501.png) |
(1075) |
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
![\begin{displaymath}
E_f - E_i = -\hbar \omega.
\end{displaymath}](img2510.png) |
(1076) |
This implies that the system loses energy
to the
perturbing field, whilst making a transition to a final state whose
energy is less than the initial state by
. This process is
known as stimulated emission. The resonance at
corresponds to
![\begin{displaymath}
E_f - E_i = \hbar \omega.
\end{displaymath}](img2512.png) |
(1077) |
This implies that the system gains energy
from the
perturbing field, whilst making a transition to a final state whose
energy is greater than that of the initial state by
. This
process is known as absorption.
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
![\begin{displaymath}
P_{i\rightarrow f}^{stm}(t) = \frac{t^2}{\hbar^2}
\vert V_...
...ert^{ 2} {\rm sinc}^2\left[(\omega-\omega_{if}) t/2\right],
\end{displaymath}](img2515.png) |
(1078) |
where we have made use of the facts that
,
and
. Likewise, the transition probability
for absorption is
![\begin{displaymath}
P_{i\rightarrow f}^{abs}(t) = \frac{t^2}{\hbar^2}
\vert V_...
...ert^{ 2} {\rm sinc}^2\left[(\omega-\omega_{fi}) t/2\right].
\end{displaymath}](img2518.png) |
(1079) |
Next: Electromagnetic Radiation
Up: Time-Dependent Perturbation Theory
Previous: Perturbation Expansion
Richard Fitzpatrick
2010-07-20