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arXiv:1703.05238 · Europhysics Letters (EPL)

Non-exponential and oscillatory decays in quantum mechanics

Murray Peshkin, Alexander Volya, Vladimir Zelevinsky

Published Aug 1, 2014• 67 citations
arXivHTMLPDF

Abstract

The quantum-mechanical theory of the decay of unstable states is revisited. We show that the decay is non-exponential both in the short-time and long-time limits using a more physical definition of the decay rate than the one usually used. We report results of numerical studies based on Winter's model that may elucidate qualitative features of exponential and non-exponential decay more generally. The main exponential stage is related to the formation of a radiating state that maintains the shape of its wave function with exponentially diminishing normalization. We discuss situations where the radioactive decay displays several exponents. The transient stages between different regimes are typically accompanied by interference of various contributions and resulting oscillations in the decay curve. The decay curve can be fully oscillatory in a two-flavor generalization of Winter's model with some values of the parameters. We consider the implications of that result for models of the oscillations reported by GSI.

DOI
10.1209/0295-5075/107/40001

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•Jan 1, 2015•1 citation
The quantum-mechanical theory of the decay of unstable states is revisited. We show that the decay is non-exponential both in the short-time and long-time limits using a more physical definition of the decay rate than the one usually used. We report results of numerical studies based on Winter's model that may elucidate qualitative features of exponential and non-exponential decay more generally. The main exponential stage is related to the formation of a radiating state that maintains the shape of its wave function with exponentially diminishing normalization. We discuss situations where the radioactive decay displays several exponents. The transient stages between different regimes are typically accompanied by interference of various contributions and resulting oscillations in the decay curve. The decay curve can be fully oscillatory in a two-flavor generalization of Winter's model with some values of the parameters. We consider the implications of that result for models of the oscillations reported by GSI.
Results of theoretical studies of the quantum unstable systems caused that there are rather widespread belief that a universal feature of the quantum decay process is the presence of three time regimes of the decay process: the early time (initial) leading to the Quantum Zeno (or Anti Zeno) Effects, “exponential” (or “canonical”) described by the decay law of the exponential form, and late time characterized by the decay law having inverse-power law form. Based on the fundamental principles of the quantum theory we give the proof that there is no time interval in which the survival probability (decay law) could be a decreasing function of time of the purely exponential form but even at the “exponential” regime the decay curve is oscillatory modulated with a smaller or a large amplitude of oscillations depending on parameters of the model considered.
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.
After reviewing the description of an unstable state in the framework of Lee Hamiltonians (valid both for Quantum Mechanics (QM) and Quantum Field Theory (QFT)), we consider some theoretical aspects of non-exponential decays: the case of two decay channels, the broadening of the energy spectrum at short times, the effect of an imperfect measurement, the link to QFT, and the decay of an unstable moving particle with definite momentum. All the presented effects were not confirmed in experiments, hence are at the present stage predictions.
The most important law of radioactivity is that of the exponential decay. In the realm of quantum mechanics, however, this decay law is neither rigorous nor fundamental. The deviations from the exponential decay have been observed experimentally at the early stage of a decay process, but there is little evidence for nonexponential behavior at long times. Yet such long-term nonexponentiality is expected theoretically to probe the nonresonant background components of the initial wave function which preserve the structural interference and the memory of how the state was created. In this paper, we propose new observables that can be used for experimental investigations of the post-exponential decay regime, including the decay of threshold resonances, particle correlations in three-body decays, and interference between near-lying resonances. While the specific examples presented in this work pertain to atomic nuclei, the properties of nonexponential decay are generic, i.e., they apply to other many-body open quantum systems, such as hadrons, atoms, molecules, and nanostructures.
The quantum-mechanical time evolution of an isolated \ensuremath{\beta}-unstable nuclear state in the context of certain models predicts that the square of the amplitude of the initial undecayed state becomes an approximately exponential function of time on a timescale on the order of ${10}^{\ensuremath{-}22}--{10}^{\ensuremath{-}21}\phantom{\rule{0.16em}{0ex}}\mathrm{s}$. It was argued that a measurement process required to distinguish between the parent and the daughter nuclear states in such a short time would destroy the characteristics of the long-lived \ensuremath{\beta}-unstable nuclear state, thus fundamentally restricting the observability of \ensuremath{\beta} decay on a short timescale. Since the interaction of the nuclear state with the environment is almost inevitable, we have obtained the timescale of initial non-exponential decay for the nuclear \ensuremath{\beta} decay from an estimation of quantum decoherence time considering the atom of the decaying nucleus as a quantum detector. It has been found from such considerations that a decoherence timescale of \ensuremath{\beta} decay is on the order of ${10}^{\ensuremath{-}16}--{10}^{\ensuremath{-}15}\phantom{\rule{0.16em}{0ex}}\mathrm{s}$ and the decay should remain reversible and non-exponential on this timescale. The possibilities of observing the effect of non-exponential decay in nuclear systems have been discussed.
We study the survival probability of moving relativistic unstable particles and we found the integral representation of the survival amplitude for such particles moving with constant momentum ~ 6 0. Using this representation of the amplitude we found decay curves of such particles for the quantum mechanical models considered. These model studies show that late time deviations of the survival probability of these particles from the exponential form of the decay law, that is the transition times region between exponential and non-exponential form of the survival probability, occur much earlier than it follows from the classical standard approach resolving itself into replacing time t by t/γ (where γ is the relativistic Lorentz factor) in the formula for the survival probability and that the survival probabilities tend to zero as t → ∞ much slower than one would expect using classical time dilation relation. Here we show also that for some physically admissible models of unstable states the properly computed decay curves of the moving particles have analogous fluctuating form as the decay curves obtained in the GSI experiment.