Energy uncertainty of the final state of a decay process

We derive the expression for the energy uncertainty of the final state of a decay of an unstable quantum state prepared at the initial time $t=0$. This expression is function of the time $t$ at which a measurement is performed to determine if the state has decayed and, if yes, in which one of the infinitely many possible final states. For large times the energy spread is, as expected, given by the decay width $\Gamma$ of the initial unstable state. However, if the measurement of the final state is performed at a time $t$ comparable to (or smaller than) the mean lifetime of the state $1/\Gamma$, then the uncertainty on the energy of the final state is much larger than the decay width $\Gamma$. Namely, for short times an uncertainty of the type $1/t$ dominates, while at large times the usual spread $\Gamma$ is recovered. Then, we turn to a generic two-body decay process and describe the energy uncertainty of each one of the two outgoing particles. We apply these formulas to the two-body decays of the neutral and charged pions and to the spontaneous emission process of an excited atom. As a last step, we study a case in which the non-exponential decay is realized ad show that for short times eventual asymmetric terms are enhanced in the spectrum.