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arXiv:1305.4467 · Physical Review A

Energy uncertainty of the final state of a decay process

Francesco Giacosa

Published Nov 27, 2013• 17 citations
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Abstract

We derive the expression for the energy uncertainty of the final state of a decay of an unstable quantum state prepared at initial time $t=0$. This expression is a 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 $\ensuremath{\Gamma}$ 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 $1/\ensuremath{\Gamma}$, then the uncertainty in the energy of the final state is much larger than the decay width $\ensuremath{\Gamma}$. Namely, for short times an uncertainty of type $1/t$ dominates, while at long times the usual spread $\ensuremath{\Gamma}$ is recovered. Then we turn to a generic two-body decay process and describe the energy uncertainty of each of the two outgoing particles. We apply these formulas to the two-body decays of neutral and charged pions and to the spontaneous emission process of an excited atom. As the last step, we study a case in which the nonexponential decay is realized and show that for short times eventual asymmetric terms are enhanced in the spectrum.

DOI
10.1103/physreva.88.052131

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We derive the expression for the energy uncertainty of the final state of a decay of an unstable quantum state prepared at initial time t=0t=0t=0. This expression is a function of the time ttt 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 \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ of the initial unstable state. However, if the measurement of the final state is performed at a time ttt comparable to (or smaller than) the mean lifetime of the state 1/\ensuremathΓ1/\ensuremath{\Gamma}1/\ensuremathΓ, then the uncertainty in the energy of the final state is much larger than the decay width \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ. Namely, for short times an uncertainty of type 1/t1/t1/t dominates, while at long times the usual spread \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ is recovered. Then we turn to a generic two-body decay process and describe the energy uncertainty of each of the two outgoing particles. We apply these formulas to the two-body decays of neutral and charged pions and to the spontaneous emission process of an excited atom. As the last step, we study a case in which the nonexponential decay is realized and show that for short times eventual asymmetric terms are enhanced in the spectrum.
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On the basis of the principles proposed by the first author in 1970 and 1979 the time-energy uncertainty relationship in quantum mechanics is constructed explicitly for transitions between adequately prescribed states of a simple system of a nonrelativistic particle moving in a constant uniform field of force. The implications of the uncertainties in time and energy contemplated for transitions are clarified by contrasting them with ordinary uncertainties of various dynamical variables evaluated in one single state of the system. Decay process and the mean lifetime related to it are discussed too in a particular case when a pair of states, initial and final, for a transition happen to be identical.