 Solution

An unstable isolated level has survival amplitude $a(t)=\theta (t)e^{-i{E}_{0}t/\hslash }{e}^{-\Gamma t/(2\hslash )}$, because its survival probability falls as $e^{-\Gamma t/\hslash }$. Its energy-domain amplitude is proportional toTaking the modulus squared gives the denominator of the Breit-Wigner resonance cross sectionAt $E-E_0=\pm \Gamma /2$ 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.