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Decay law of unstable states

Jan Mostowski, K. Wódkiewicz

Published Jan 1, 1973• 1 citation
OpenAlex

Abstract

An outline of the Wigner-Weisskopf theory of unstable states is presented. Numerical analysis of the decay law is given, deviations from the exponential decay law are studied in some detail. The interpretation of the nonexponential decay law is also briefly discussed. (auth)

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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.
A multiple repeated measurement of the unstable state evolving in time is discussed. We show that there is no decay in the limit of continuous measurements and there is a purely exponential decay law in other cases. We also give formulae for the effective decay width in the cases of very short, intermediate, relatively long and very long time intervals separating successive measurements. Our results are compared with the Weisskopf-Wigner formula for weak decays. Based on our model of a measurement, we show that the decay width does not depend on properties of the real detectors in practice.
We study the decay process of an unstable quantum system, especially the deviation from the exponential decay law. We show that the exponential period no longer exists in the case of the sss-wave decay with small QQQ value, where the QQQ value is the difference between the energy of the initially prepared state and the minimum energy of the continuous eigenstates in the system. We also derive the quantitative condition that this kind of decay process takes place and discuss what kind of system is suitable to observe the decay.
Unstable state decay without Exponential Law - small QQQ value s-wave (sQs) decay -
We study the decay process of an unstable quantum system, especially the deviation from the exponential decay law. We show that the exponential period no longer exists in the case of the s-wave decay with small QQQ value. We also derive the quantitative condition that this kind of the decay process takes place and discuss what kind of system is suitable to observe the decay.
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 .
We study the deviations from the usual exponential decay law for quantum mechanical systems. We show that simple and physically motivated deviations from the Breit-Wigner energy distribution of the unstable state are sufficient to generate peculiar deviations from the exponential decay law. Denoting with p(t)p(t)p(t) the survival probability, its derivative h(t)h(t)h(t) shows typically an oscillating behavior on top of the usual exponential function. We argue that this can be a viable explanation of the observed experimental results at GSI Darmstadt, where the function h(t)h(t)h(t) has been experimentally measured for electron capture decays of Hydrogen-like ions. Moreover, if our interpretation is correct, we predict that by measuring h(t)h(t)h(t) at times close to the initial one, the number of decays per second rapidly drops to zero.