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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)

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