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arXiv:1809.10673 · Journal of Physics A Mathematical and Theoretical

Quantum decay law: critical times and the equivalence of approaches

David Fernández Jiménez, N. G. Kelkar

Published Dec 19, 2018• 12 citations
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Abstract

Abstract Methods based on the use of Green’s functions or Jost functions and the Fock–Krylov method are apparently very different approaches to understand the time evolution of unstable states. We show that the two former methods are equivalent up to some constants and as an outcome find an analytic expression for the energy density of states in the Fock–Krylov amplitude in terms of the coefficients introduced in the Green’s functions and the Jost functions methods. This model-independent density is further used to obtain an analytical expression for the survival amplitude and study its behaviour at large times. Using these expressions, we investigate the origin of the oscillatory behaviour of the decay law in the region of the transition from the exponential to the non-exponential at large times. With the objective of understanding the failure of nuclear and particle physics experiments in observing the non-exponential decay law predicted by quantum mechanics for large times, we derive analytical formulae for the critical transition time, , from the exponential to the inverse power law behaviour at large times. Evaluating for some particle resonances and narrow nuclear states which have been tested experimentally to verify the exponential decay law, we conclude that the large time power law in particle and nuclear decay is hard to find experimentally.

DOI
10.1088/1751-8121/aaf9f3

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Abstract Methods based on the use of Green’s functions or Jost functions and the Fock–Krylov method are apparently very different approaches to understand the time evolution of unstable states. We show that the two former methods are equivalent up to some constants and as an outcome find an analytic expression for the energy density of states in the Fock–Krylov amplitude in terms of the coefficients introduced in the Green’s functions and the Jost functions methods. This model-independent density is further used to obtain an analytical expression for the survival amplitude and study its behaviour at large times. Using these expressions, we investigate the origin of the oscillatory behaviour of the decay law in the region of the transition from the exponential to the non-exponential at large times. With the objective of understanding the failure of nuclear and particle physics experiments in observing the non-exponential decay law predicted by quantum mechanics for large times, we derive analytical formulae for the critical transition time, , from the exponential to the inverse power law behaviour at large times. Evaluating for some particle resonances and narrow nuclear states which have been tested experimentally to verify the exponential decay law, we conclude that the large time power law in particle and nuclear decay is hard to find experimentally.
The Fock-Krylov formalism for the calculation of survival probabilities of unstable states is revisited paying particular attention to the mathematical constraints on the density of states, the Fourier transform of which gives the survival amplitude. We show that it is not possible to construct a density of states corresponding to a purely exponential survival amplitude. The survival probability P(t)P(t)P(t) and the autocorrelation function of the density of states are shown to form a pair of cosine Fourier transforms. This result is a particular case of the Wiener-Khinchin theorem and forces P(t)P(t)P(t) to be an even function of time which in turn forces the density of states to contain a form factor which vanishes at large energies. Subtle features of the transition regions from the nonexponential to the exponential at small times and the exponential to the power-law decay at large times are discussed by expressing P(t)P(t)P(t) as a function of the number of oscillations nnn performed by it. The transition at short times is shown to occur when the survival probability has completed one oscillation. The number of oscillations depends on the properties of the resonant state and a complete description of the evolution of the unstable state is provided by determining the limits on the number of oscillations in each region.
The deviation of the decay law from the exponential is a well known effect of quantum mechanics . Here we analyze the relativistic survival probabilities , S ( t , p ) , where p is the momentum of the decaying particle and provide analytical expressions for S ( t , p ) in the exponential (E) as well as the nonexponential (NE) regions at small and large times. Under minimal assumptions on the spectral density function, analytical expressions for the critical times of transition from the NE to the E at small times and the E to NE at large times are derived. The dependence of the decay law on the relativistic Lorentz factor, γ = 1 / 1 − v 2 / c 2 , reveals several interesting features. In the short time regime of the decay law, the critical time, τ s t , shows a steady increase with γ , thus implying a larger NE region for particles decaying in flight. Comparing S ( t , p ) with the well known time dilation formula, e − Γ t / γ , in the exponential region, an expression for the critical γ where S ( t , p ) deviates most from e − Γ t / γ is presented. This is a purely quantum correction. Under particular conditions on the resonance parameters, there also exists a critical γ at large times which decides if the NE region shifts backward or forward in time as compared to that for a particle at rest. All the above analytical results are supported by calculations involving realistic decays of hadrons and leptons .
An effect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region is considered.We find that the instantaneous energy of the unstable state for a large class of models of unstable states tends to the minimal energy of the system E min as t → ∞ which is much smaller than the energy of this state for t of the order of the lifetime of the considered state.Analyzing the transition time region between the exponential and non-exponential form of the survival amplitude, we find that the instantaneous energy of the considered unstable state can take large values, much larger than the energy of this state for t from the exponential time region.Taking into account results obtained for a considered model, it is hypothesized that this purely quantum mechanical effect may be responsible for the properties of broad resonances such as σ meson as well as having astrophysical and cosmological consequences Key words: decay law, non exponential decay
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.
Though the classical treatment of spontaneous decay leads to an exponential decay law, it is well known that this is an approximation of the quantum mechanical result which is non-exponential at very small and large times for narrow states. The non-exponential nature at large times is, however, hard to establish from experiments. A method to recover the time evolution of unstable states from a parametrization of the amplitude fitted to data is presented. We apply the method to a realistic example of a very broad state, the σ meson and reveal that an exponential decay is not a valid approximation at any time for this state. This example derived from experiment shows the unique nature of broad resonances.
An effect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region is considered. We find that the instantaneous energy of the unstable state for a large class of models of unstable states tends to the minimal energy of the system Emin{\cal E}_{min}Emin​ as t→∞t\rightarrow\inftyt→∞ which is much smaller than the energy of this state for ttt of the order of the lifetime of the considered state. Analyzing the transition time region between exponential and non-exponential form of the survival amplitude we find that the instantaneous energy of the considered unstable state can take large values, much larger than the energy of this state for ttt from the exponential time region. Taking into account results obtained for a model considered, it is hypothesized that this purely quantum mechanical effect may be responsible for the properties of broad resonances such as σσσ meson as well as having astrophysical and cosmological consequences.