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The decay law can have an irregular character

Exner, P, Fraas, M

Published Mar 8, 2006• 12 citations
OpenAlex

Abstract

Within a well-known decay model describing a particle confined initially within a spherical $\\delta$ potential shell, we consider the situation when the undecayed state has an unusual energy distribution decaying slowly as $k\\to\\infty$; the simplest example corresponds to a wave function constant within the shell. We show that the non-decay probability as a function of time behaves then in a highly irregular, most likely fractal way.

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A simple model of a decaying quantum mechanical state

A Ludviksson

A model featuring a one-dimensional particle in a tilted potential which may be trapped in a delta type potential well is considered. The time dependence of a state where the particle is trapped and where it decays via the quantum mechanical tunnel effect is calculated explicitly. Since the background potential decreases without limit (in one direction), the decay is purely exponential for long times and the decay rate is equal to the imaginary part of the eigenenergy in a Schrodinger equation that is solved for a quasi-stationary state.

Journal of Physics A Mathematical and General•Oct 1, 1987•30 citations
Within a well-known decay model describing a particle confined initially within a spherical delta\\deltadelta potential shell, we consider the situation when the undecayed state has an unusual energy distribution decaying slowly as ktoinftyk\\to\\inftyktoinfty; the simplest example corresponds to a wave function constant within the shell. We show that the non-decay probability as a function of time behaves then in a highly irregular, most likely fractal way.
We derive an analytical expression for the propagation of a quantum particle at asymptotic long distances and times from the potential where the particle was initially confined. We obtain that the particle is described by an evolving resonance in time domain that possesses a peaked shape characterized by the resonance parameters of the dominant resonance coefficient of the decaying wave function. The above situation corresponds to an unexplored postexponential regime that represents the ultimate fate of a decaying particle.
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.
We present a detailed non-perturbative analysis of the time-evolution of a well-known quantum-mechanical system - a particle between potential walls - describing the decay of unstable states. For sufficiently high barriers, corresponding to unstable particles with large lifetimes, we find an exponential decay for intermediate times, turning into an asymptotic power decay. We explicitly compute such power terms in time as a function of the coupling in the model. The same behavior is obtained with a repulsive as well as with an attractive potential, the latter case not being related to any tunnelling effect.
A model featuring a one-dimensional particle in a tilted potential which may be trapped in a delta type potential well is considered. The time dependence of a state where the particle is trapped and where it decays via the quantum mechanical tunnel effect is calculated explicitly. Since the background potential decreases without limit (in one direction), the decay is purely exponential for long times and the decay rate is equal to the imaginary part of the eigenenergy in a Schrodinger equation that is solved for a quasi-stationary state.