Quantum decay law: critical times and the equivalence of approaches

Abstract

Methods based on the use of Green's functions or the 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 to understand 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, t c , from the exponential to the inverse power law behaviour at large times. Evaluating τ c = Γt c 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.

Transformation of intermediate times in the decays of moving unstable quantum systems via the exponential modes

“…has not yet become the inverse power law which is determined, over very long times, by the threshold of the MDD [5,11,[17][18][19][20][21][22]. Over the intermediate times, the modulus of the survival amplitude at rest is approximated with a finite number of exponential modes,…”

“…Usually, the long-time deviations from the exponential decays are determined by the threshold and by the integral and analytical properties of the mass distribution density (MDD). See [5,[11][12][13][14][15][16][17][18][19][20][21][22] for details, to name but a few.…”

“…The long-time inverse-power-law decays have been detected experimentally by measuring the luminescence decays of dissolved organic materials [27]. However, long-time deviations from the exponential decays into inverse power laws are difficult to detect in experiments of particle and nuclear physics [21]. Many unstable decaying particles are generated in astrophysical and cosmological processes.…”