= Solution
{c}
Let the upper-state population lifetime be $T_1=1/\Gamma$, and take the lower <energy level> to be stable. <Spontaneous emission> gives survival <probability> $e^{-\Gamma t}$, hence a field or excited-state <probability amplitude> proportional to $e^{-\Gamma t/2}e^{-i\omega_0t}$ for $t\geq0$. Its <Fourier transform> at <angular frequency> $\omega$ is proportional to
$$
\int_0^\infty e^{-\Gamma t/2}e^{i(\omega-\omega_0)t}\,dt
=\frac1{\Gamma/2-i(\omega-\omega_0)}.
$$
The squared modulus therefore has the <Lorentzian line shape>
$$
\boxed{L(\omega)=\frac1\pi\frac{\Gamma/2}{(\omega-\omega_0)^2+(\Gamma/2)^2},
\qquad\Delta\omega_{\rm FWHM}=\Gamma=\frac1{T_1}.}
$$
Here $L$ is normalized over the full mathematical <frequency> axis. The <full width at half maximum> in ordinary <frequency> is $1/(2\pi T_1)$. If both <energy levels> decay, their population decay rates add in the <natural linewidth>; pure <dephasing> adds to the amplitude-decay rate instead.
Impact-regime <pressure broadening> provides another example. Independent phase-randomizing collisions with rate $\gamma_c$ give optical correlation proportional to $e^{-\gamma_c|t|}$, whose <Fourier transform> is a <Lorentzian line shape> of <full width at half maximum> $2\gamma_c$. This assumes short, uncorrelated impacts: a general static distribution of perturbing fields need not give a <Lorentzian line shape>.
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