= Solution
An unstable isolated level has survival amplitude $a(t)=\theta(t)e^{-iE_0t/\hbar}e^{-\Gamma t/(2\hbar)}$, because its <survival probability> falls as $e^{-\Gamma t/\hbar}$. Its energy-domain amplitude is proportional to
$$
\int_0^\infty e^{i(E-E_0)t/\hbar}e^{-\Gamma t/(2\hbar)}dt
=\frac{\hbar}{\Gamma/2-i(E-E_0)}.
$$
Taking the modulus squared gives the denominator of the <Breit-Wigner resonance cross section>
$$
\boxed{(E-E_0)^2+(\Gamma/2)^2}.
$$
At $E-E_0=\pm\Gamma/2$ the resonant factor is half its peak, so $\Gamma$ is the <full width at half maximum>. <exponential decay> is the isolated-level approximation; energy-dependent thresholds or overlapping levels need a more general lineshape.
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