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arXiv:1801.02350 · Acta Physica Polonica B

Analysis of the Breakdown of Exponential Decays of Resonances

Adam Wyrzykowski

Published Jan 1, 2020• 5 citations
arXivHTMLPDF

Abstract

In the paper, a simple model of alpha decay with Dirac delta potential is studied. The model leads to breakdown of the exponential decay and to power law behavior at asymptotic times. Time dependence of the survival probability of the particle in the potential well is analyzed numerically with two methods: integration of Green's function representation and numerical solution of the time-dependent Schrödinger equation. In particular, finite depth potential wells and behavior between the exponential and power law regimes, which are situations that could not be described in detail analytically, are studied. The numerical results confirm power law with exponent n = 3 after the turnover into the non-exponential decay regime. Moreover, the constructive and destructive interference is observed in the intermediate stage of the process. The simple alpha decay model is compared to the results of Rothe- Hintschich-Monkman experiment which was the first experimental proof of violation of the exponential law.

DOI
10.5506/aphyspolb.51.2015

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In the paper, a simple model of alpha decay with Dirac delta potential is studied. The model leads to breakdown of the exponential decay and to power law behavior at asymptotic times. Time dependence of the survival probability of the particle in the potential well is analyzed numerically with two methods: integration of Green's function representation and numerical solution of the time-dependent Schrödinger equation. In particular, finite depth potential wells and behavior between the exponential and power law regimes, which are situations that could not be described in detail analytically, are studied. The numerical results confirm power law with exponent n = 3 after the turnover into the non-exponential decay regime. Moreover, the constructive and destructive interference is observed in the intermediate stage of the process. The simple alpha decay model is compared to the results of Rothe- Hintschich-Monkman experiment which was the first experimental proof of violation of the exponential law.
A simple model of alpha decay with the Dirac delta potential is studied.The model leads to breakdown of the exponential decay and to the power law behavior at asymptotic times.Time dependence of the survival probability of the particle in the potential well is analyzed numerically with two methods: integration of the Green's function representation and numerical solution of the time-dependent Schrödinger equation.The numerical results confirm power law with exponent n = 3 after the turnover into the non-exponential decay regime.Moreover, oscillations of non-escape probability are observed in the intermediate stage of the process.The simple alpha-decay model is compared to the results of the Rothe-Hintschich-Monkman experiment which was the first experimental proof of violation of the exponential law.
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.
We study resonances and time decay in a quantum waveguide with two semitransparent barriers. The model is first analyzed as a one-dimensional Schrödinger operator with two delta interactions and then extended to a two-dimensional strip, where transverse Dirichlet modes reduce the problem, channel by channel, to the one-dimensional setting. We derive explicit scattering amplitudes and characterize resonance poles by analytic continuation. In the strong-barrier regime ρ=α−1→0+, the real parts of the resonances approach the eigenvalues of the decoupled cavity, while the imaginary parts are negative and of order α−2, so the lifetimes grow like α2. We also study the time evolution on short, intermediate, and long time scales: the survival probability is quadratic at short times, in accordance with the quantum Zeno effect; the intermediate regime is resonance-dominated and approximately exponential; and the long-time behavior is governed by the low-energy threshold and becomes algebraic. Numerical simulations complement the analysis by confirming these regimes and by showing the breakdown of exponential decay. They also provide an empirical fit for the breakdown time tbr in terms of the barrier strength α and the distance d1 between the barriers.
The departure at large times from exponential decay in the case of resonance wavefunctions is mathematically demonstrated. Then, exact, analytical solutions to the time-dependent Schrödinger equation in one dimension are developed for a time-independent potential consisting of an infinite wall and a repulsive delta function. The exact solutions are obtained by means of a superposition of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions for which the resulting integrals can be evaluated exactly. Square-integrability and the boundary conditions are satisfied. The simplest of the obtained solutions is presented and the probability for the particle to be found inside the potential well as a function of time is calculated. The system exhibits non-exponential decay for all times; the probability decreases at large times as . Other exact solutions found exhibit power law behavior at large times. The results are generalized to all normalizable solutions to this problem. Additionally, numerical solutions are obtained using the staggered leap-frog algorithm for select potentials exhibiting the prevalence of non-exponential decay at short times.
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Recently we outlined an analytical method of solving the time-dependent Schr\"odinger equation for a model which simulates a decaying quantum system such as an $\ensuremath{\alpha}$-decaying nucleus. A particle in the model is initially confined around the origin and leaks out, tunneling through a potential barrier. The solution can be expressed as a linear combination of the Moshinsky functions, each of which is associated with a pole of the scattering S\mathbf{S}S matrix of the model. In this paper we give a full account of the method with a few explicit examples. We examine deviations from the exponential decay law at very large times. We comment on a recent controversy regarding the t dependence of the survival and nonescape probabilities when t is very large.