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Journal of Mathematical Physics

A theorem on the rate of regeneration in decay processes

Michio Nishioka

Published Aug 1, 1988• 3 citations
Publisher

Abstract

There is given a generalization of the theorem of Misra and Sinha [Helv. Phys. Acta 50, 99 (1977)], which shows that in the decay process the regeneration of the undecayed unstable states from the decay products cannot proceed too slowly at arbitrarily short time and, correspondingly, the deviation from exponential decay at arbitrarily short time cannot be too small either. The generalization presented here is shown to be valid in the region where the theorem of Misra and Sinha cannot be applied.

DOI
10.1063/1.527889

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There is given a generalization of the theorem of Misra and Sinha [Helv. Phys. Acta 50, 99 (1977)], which shows that in the decay process the regeneration of the undecayed unstable states from the decay products cannot proceed too slowly at arbitrarily short time and, correspondingly, the deviation from exponential decay at arbitrarily short time cannot be too small either. The generalization presented here is shown to be valid in the region where the theorem of Misra and Sinha cannot be applied.
Consider an unstable quantum system that has been found undecayed at an instant s and denote by R(t,s) the rate of its regeneration into an original (undecayed) state at a later instant t+s. It is proved that the reduced evolution is a semigroup, i.e., there is no regeneration at all, provided R(t,s)1/2 can be estimated by a sufficiently regular function that is nondecreasing in s and has zero derivative with respect to t at t=0 for every s. This generalizes the theorem of Misra and Sinha [Helv. Phys. Acta 45, 619 (1972)] in a different direction than in a recent paper by Nishioka [J. Math. Phys. 29, 1860 (1988)].
It is assumed that any decay process is described in a Hilbert space $\Cal H=\oplus \Cal H_D$, where $\Cal H_A$ corresponds to an unstable particle and $\Cal H_D$ to its decay products. It is shown that the generalized Weisskopf-Wigner condition (which guarantees an exponential decay law of the given unstable particle) has a close relation to the irreversibility of decay processes as well as of collision ones described in the same space $\Cal H$. If unitarity is added the principal structure of $\Cal H$ is identical with that onwhich the scattering theory of Lax and Phillips is based.
Results presented in a recent paper "Which is the Quantum Decay Law of Relativistic particles?", arXiv: 1412.3346v2 [quant--ph]], are analyzed. We show that approximations used therein to derive the main final formula for the survival probability of finding a moving unstable particle to be undecayed at time ttt force this particle to almost stop moving, that is that, in fact, the derived formula is approximately valid only for γ≅1γ\cong 1γ≅1, where γ=1/1−β2γ= 1/\sqrt{1-β^{2}}γ=1/1−β2​ and β=v/cβ= v/cβ=v/c, or in other words, for the velocity v≃0v \simeq 0v≃0.
The time evolution of quantum states for unstable particles can be conveniently divided into three domains: the very short time where Zeno's paradox is relevant, the intermediate interval where the exponential decay holds more or less, and the very long time where the decay is governed by a power law. In this work, we reexamine several questions relating to the deviations from the simple exponential decay law. On the basis of general considerations, we demonstrate that deviations from exponential decay near t=0t=0t=0 are inevitable. We formulate general resonance models for the decay. From analytic solutions to specific narrow-width models, we estimate the time parameters T1{T}_{1}T1​ and T2{T}_{2}T2​ separating the three domains. The parameter T1{T}_{1}T1​ is found to be much much less than the lifetime ${\ensuremath{\Gamma}}^{\ensuremath{-}1}$, while T2{T}_{2}T2​ is much greater than the lifetime. For instance, for the charged pion decay, ${T}_{1}\ensuremath{\sim}\frac{{10}^{\ensuremath{-}14}}{\ensuremath{\Gamma}}$ and ${T}_{2}\ensuremath{\sim}\frac{190}{\ensuremath{\Gamma}}$. A resolution of Zeno's paradox provided by the present consideration and its limitaions are discussed.
It is well known, both theoretically and experimentally, that the survival probability for an unstable quantum state, formed at t = 0 , is not a simple exponential function, even if the latter is a good approximation for intermediate times. Typically, unstable quantum states/particles can decay in more than a single decay channel. In this work, the general expression for the probability that an unstable state decays into a certain i -th channel between the initial time t = 0 and an arbitrary t > 0 is provided, both for nonrelativistic quantum states and for relativistic particles . These partial decay probabilities are also not exponential and their ratio turns out to be not a simple constant, as it would be in the exponential limit. Quite remarkably, these deviations may last relatively long, thus making them potentially interesting in applications. Thus, multichannel decays represent a promising and yet unexplored framework to search for deviations from the exponential decay law in quantum mechanical systems, such as quantum tunneling, and in the context of particle decays .