The absorption of light
Our treatment above applies to a more general situation than the ammonia maser. We have treated the behavior of a molecule under the influence of an electric field, whether that field was confined in a cavity or not. So we could be simply shining a beam of “light”—at microwave frequencies—at the molecule and ask for the probability of emission or absorption. Our equations apply equally well to this case, but let’s rewrite them in terms of the intensity of the radiation rather than the electric field. If we define the intensity $\intensity$ to be the average energy flow per unit area per second, then from Chapter 27 of Volume II, we can write \begin{equation*} \intensity=\epsO c^2\abs{\Efieldvec\times\FLPB}_{\text{ave}}= \tfrac{1}{2}\epsO c^2\abs{\Efieldvec\times\FLPB}_{\text{max}}= 2\epsO c\Efield_0^2. \end{equation*}  (The maximum value of $\Efield$ is $2\Efield_0$.) The transition probability now becomes:  \begin{equation} \label{Eq:III:9:53} P(\slI\to\slII\,)=2\pi\biggl[\frac{\mu^2}{4\pi\epsO\hbar^2c}\biggr] \intensity T^2\, \frac{\sin^2[(\omega-\omega_0)T/2]} {[(\omega-\omega_0)T/2]^2}. \end{equation} 
 \begin{gather} \label{Eq:III:9:53} P(\slI\to\slII\,)=\\[1ex] 2\pi\biggl[\frac{\mu^2}{4\pi\epsO\hbar^2c}\biggr] \intensity T^2\,\frac{\sin^2[(\omega-\omega_0)T/2]} {[(\omega-\omega_0)T/2]^2}.\notag \end{gather} 
 Ordinarily the light shining on such a system is not exactly monochromatic.  It is, therefore, interesting to solve one more problem—that is, to calculate the transition probability when the light has intensity $\intensity(\omega)$ per unit frequency interval, covering a broad range which includes $\omega_0$. Then, the probability of going from $\ketsl{\slI}$ to $\ketsl{\slII}$ will become an integral:  \begin{equation} \label{Eq:III:9:54} P(\slI\to\slII\,)=2\pi\biggl[\frac{\mu^2}{4\pi\epsO\hbar^2c}\biggr]T^2 \int_0^\infty\intensity(\omega)\, \frac{\sin^2[(\omega-\omega_0)T/2]} {[(\omega-\omega_0)T/2]^2}\,d\omega. \end{equation} 
 \begin{gather} \label{Eq:III:9:54} P(\slI\to\slII\,)=\\[1ex] 2\pi\biggl[\frac{\mu^2}{4\pi\epsO\hbar^2c}\biggr]T^2\!\! \int_0^\infty\!\!\!\intensity(\omega)\, \frac{\sin^2[(\omega-\omega_0)T/2]} {[(\omega-\omega_0)T/2]^2}\,d\omega.\notag \end{gather}  In general, $\intensity(\omega)$ will vary much more slowly with $\omega$ than the sharp resonance term. The two functions might appear as shown in Fig. 9–8. In such cases, we can replace $\intensity(\omega)$ by its value $\intensity(\omega_0)$ at the center of the sharp resonance curve and take it outside of the integral. What remains is just the integral under the curve of Fig. 9–7, which is, as we have seen, just equal to $2\pi/T$. We get the result that \begin{equation} \label{Eq:III:9:55} P(\slI\to\slII\,)=4\pi^2 \biggl[\frac{\mu^2}{4\pi\epsO\hbar^2c}\biggr] \intensity(\omega_0)T. \end{equation}  This is an important result, because it is the general theory of the absorption of light by any molecular or atomic system. Although we began by considering a case in which state $\ketsl{\slI}$ had a higher energy than state $\ketsl{\slII}$, none of our arguments depended on that fact. Equation (9.55) still holds if the state $\ketsl{\slI}$ has a lower energy than the state $\ketsl{\slII}$; then $P(\slI\to\slII\,)$ represents the probability for a transition with the absorption of energy from the incident electromagnetic wave. The absorption of light by any atomic system always involves the amplitude for a transition in an oscillating electric field between two states separated by an energy $E=\hbar\omega_0$. For any particular case, it is always worked out in just the way we have done here and gives an expression like Eq. (9.55). We, therefore, emphasize the following features of this result. First, the probability is proportional to $T$. In other words, there is a constant probability per unit time that transitions will occur. Second, this probability is proportional to the intensity of the light incident on the system. Finally, the transition probability is proportional to $\mu^2$, where, you remember, $\mu\Efield$ defined the shift in energy due to the electric field $\Efield$. Because of this, $\mu\Efield$ also appeared in Eqs. (9.38) and (9.39) as the coupling term that is responsible for the transition between the otherwise stationary states $\ketsl{\slI}$ and $\ketsl{\slII}$. In other words, for the small $\Efield$ we have been considering, $\mu\Efield$ is the so-called “perturbation term” in the Hamiltonian matrix element which connects the states $\ketsl{\slI}$ and $\ketsl{\slII}$. In the general case, we would have that $\mu\Efield$ gets replaced by the matrix element $\bracketsl{\slII}{H}{\slI}$ (see Section 5–6). In Volume I (Section 42–5) we talked about the relations among light absorption, induced emission, and spontaneous emission in terms of the Einstein $A$- and $B$-coefficients. Here, we have at last the quantum mechanical procedure for computing these coefficients. What we have called $P(\slI\to\slII\,)$ for our two-state ammonia molecule corresponds precisely to the absorption coefficient $B_{nm}$ of the Einstein radiation theory. For the complicated ammonia molecule—which is too difficult for anyone to calculate—we have taken the matrix element $\bracketsl{\slII}{H}{\slI}$ as $\mu\Efield$, saying that $\mu$ is to be gotten from experiment. For simpler atomic systems, the $\mu_{mn}$ which belongs to any particular transition can be calculated from the definition \begin{equation} \label{Eq:III:9:56} \mu_{mn}\Efield=\bracket{m}{H}{n}=H_{mn}, \end{equation}  where $H_{mn}$ is the matrix element of the Hamiltonian which includes the effects of a weak electric field. The $\mu_{mn}$ calculated in this way is called the electric dipole matrix element. The quantum mechanical theory of the absorption and emission of light is, therefore, reduced to a calculation of these matrix elements for particular atomic systems. Our study of a simple two-state system has thus led us to an understanding of the general problem of the absorption and emission of light.