Energy uncertainty of the final state of a decay process

Understand

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 $\Gamma$.