# 12.5: Harmonic Perturbation

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Consider a (Hermitian) perturbation that oscillates sinusoidally in time. This is usually termed a *harmonic perturbation*. Such a perturbation takes the form

\[\label{e13.51} H_1(t) = V\,\exp(\,{\rm i}\,\omega\,t) + V^\dagger\,\exp(-{\rm i}\,\omega\,t),\] where \(V\) is, in general, a function of position, momentum, and spin operators.

It follows from Equations ([e13.48]) and ([e13.51]) that, to first-order, \[c_f(t) = - \frac{\rm i}{\hbar}\int_0^t\left[V_{fi}\,\exp(\,{\rm i}\,\omega\,t') + V_{fi}^\dagger\,\exp(-{\rm i}\,\omega\,t')\right] \exp(\,{\rm i}\,\omega_{fi}\,t')\,dt',\] where

\[\begin{aligned} \label{e13.53} V_{fi}&= \langle f|V|i\rangle,\\[0.5ex] V_{fi}^\dagger &=\langle f|V^\dagger|i\rangle = \langle i|V|f\rangle^\ast.\end{aligned}\] Integration with respect to \(t'\) yields \[\begin{aligned} c_f(t)&= - \frac{{\rm i}\,t}{\hbar}\left(V_{fi}\,\exp\left[\,{\rm i}\,(\omega+\omega_{fi})\,t/2\right]{\rm sinc}\left[(\omega+\omega_{fi})\,t/2\right]\right.\nonumber\\[0.5ex]& \left.\phantom{=}+V_{fi}^\dagger\,\exp\left[-{\rm i}\,(\omega-\omega_{fi})\,t/2\right]{\rm sinc}\left[(\omega-\omega_{fi})\,t/2\right]\right),\label{e13.55}\end{aligned}\] where \[{\rm sinc}\, x\equiv \frac{\sin\,x}{x}.\]

**Figure 25:** *The functions *\(\begin{equation}\operatorname{sinc}(x)\end{equation}\) *(dashed curve) and *\(\begin{equation}\operatorname{sinc}^{2}(x)\end{equation}\) *(solid curve). The vertical dotted lines denote the region *\(\begin{equation}|x| \leq \pi\end{equation}\)

Now, the function \({\rm sinc}(x)\) takes its largest values when \(\begin{equation}|x| \lesssim \pi\end{equation}\), and is fairly negligible when \(|x|\gg \pi\). (See Figure [fsinc].) Thus, the first and second terms on the right-hand side of Equation ([e13.55]) are only non-negligible when

\begin{equation}\left|\omega+\omega_{f i}\right| \lesssim \frac{2 \pi}{t}\end{equation} and \begin{equation}\left|\omega-\omega_{f i}\right| \lesssim \frac{2 \pi}{t}\end{equation} respectively.

Clearly, as \(t\) increases, the ranges in \(\omega\) over which these two terms are non-negligible gradually shrink in size. Eventually, when \(t\gg 2\pi/|\omega_{fi}|\), these two ranges become strongly non-overlapping. Hence, in this limit, \(P_{i\rightarrow f}=|c_f|^{\,2}\) yields

\[\label{e13.49} P_{i\rightarrow f}(t) = \frac{t^{\,2}}{\hbar^{\,2}}\left\{ |V_{fi}|^{\,2}\,{\rm sinc}^2\left[(\omega+\omega_{fi})\,t/2\right] + |V_{fi}^\dagger|^{\,2}\,{\rm sinc}^2\left[(\omega-\omega_{fi})\,t/2\right]\right\}.%\label{e13.59}\]

Now, the function \({\rm sinc}^2(x)\) is very strongly peaked at \(x=0\), and is completely negligible for \(\begin{equation}|x| \gg \pi\end{equation}\). (See Figure [fsinc].) It follows that the previous expression exhibits a resonant response to the applied perturbation at the frequencies \(\omega=\pm\omega_{fi}\). Moreover, the widths of these resonances decease linearly as time increases. At each of the resonances (i.e., at \(\omega=\pm\omega_{fi}\)), the transition probability \(P_{i\rightarrow f}(t)\) varies as \(t^{\,2}\) [because \({\rm sinh} (0)=1\)]. This behavior is entirely consistent with our earlier result ([e13.28]), for the two-state system, in the limit \(\gamma\,t\ll 1\) (recall that our perturbative solution is only valid as long as \(P_{i\rightarrow f}\ll 1\)).

The resonance at \(\omega=-\omega_{fi}\) corresponds to \[E_f - E_i = -\hbar\,\omega.\] This implies that the system loses energy \(\hbar\,\omega\) to the perturbing field, while making a transition to a final state whose energy is less than the initial state by \(\hbar\,\omega\). This process is known as *stimulated emission*. The resonance at \(\omega=\omega_{fi}\) corresponds to \[E_f - E_i = \hbar\,\omega.\] This implies that the system gains energy \(\hbar\,\omega\) from the perturbing field, while making a transition to a final state whose energy is greater than that of the initial state by \(\hbar\,\omega\). This process is known as *absorption*.

Stimulated emission and absorption are mutually exclusive processes, because the first requires \(\omega_{fi}<0\), whereas the second requires \(\omega_{fi}>0\). Hence, we can write the transition probabilities for both processes separately. Thus, from Equation ([e13.49]), the transition probability for stimulated emission is \[P_{i\rightarrow f}^{stm}(t) = \frac{t^{\,2}}{\hbar^{\,2}}\, |V_{if}^\dagger|^{\,2}\,{\rm sinc}^2\left[(\omega-\omega_{if})\,t/2\right],\] where we have made use of the facts that \(\omega_{if}=-\omega_{fi}>0\), and \(|V_{fi}|^{\,2}=|V_{if}^\dagger|^{\,2}\). Likewise, the transition probability for absorption is

\[\label{e13.63} P_{i\rightarrow f}^{abs}(t) = \frac{t^{\,2}}{\hbar^{\,2}}\, |V_{fi}^\dagger|^{\,2}\,{\rm sinc}^2\left[(\omega-\omega_{fi})\,t/2\right].\]

## Contributors and Attributions

Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

\( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)