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arXiv:1010.5926

Nonperturbative Analysis of a Quantum Mechanical Model for Unstable Particles

U. Aglietti, Paolo Maria Santini

Published Oct 28, 2010• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1010.5926

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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.
It is well known, both theoretically and experimentally, that the survival probability for an unstable quantum state, formed at t = 0 , is not a simple exponential function, even if the latter is a good approximation for intermediate times. Typically, unstable quantum states/particles can decay in more than a single decay channel. In this work, the general expression for the probability that an unstable state decays into a certain i -th channel between the initial time t = 0 and an arbitrary t > 0 is provided, both for nonrelativistic quantum states and for relativistic particles . These partial decay probabilities are also not exponential and their ratio turns out to be not a simple constant, as it would be in the exponential limit. Quite remarkably, these deviations may last relatively long, thus making them potentially interesting in applications. Thus, multichannel decays represent a promising and yet unexplored framework to search for deviations from the exponential decay law in quantum mechanical systems, such as quantum tunneling, and in the context of particle decays .
The time evolution of quantum states for unstable particles can be conveniently divided into three domains: the very short time where Zeno's paradox is relevant, the intermediate interval where the exponential decay holds more or less, and the very long time where the decay is governed by a power law. In this work, we reexamine several questions relating to the deviations from the simple exponential decay law. On the basis of general considerations, we demonstrate that deviations from exponential decay near t=0t=0t=0 are inevitable. We formulate general resonance models for the decay. From analytic solutions to specific narrow-width models, we estimate the time parameters T1{T}_{1}T1​ and T2{T}_{2}T2​ separating the three domains. The parameter T1{T}_{1}T1​ is found to be much much less than the lifetime ${\ensuremath{\Gamma}}^{\ensuremath{-}1}$, while T2{T}_{2}T2​ is much greater than the lifetime. For instance, for the charged pion decay, ${T}_{1}\ensuremath{\sim}\frac{{10}^{\ensuremath{-}14}}{\ensuremath{\Gamma}}$ and ${T}_{2}\ensuremath{\sim}\frac{190}{\ensuremath{\Gamma}}$. A resolution of Zeno's paradox provided by the present consideration and its limitaions are discussed.
The time behavior of a wave packet, transmitted through a potential barrier, is investigated for all times assuming it to be initially located outside the asymmetric barrier. The survival probability of the initial state, which characterizes the decay of the unstable quantum system in three time domains, is discussed. The presence of initial, exponential, and long-time regimes is a general feature of the decay process. We analytically found the three crucial points of time that separate these regimes. These characteristic times are set by simple formulae, including momentum distribution. It is then shown that the radius of convergence of the cumulant expansion for the survival probability is always small. A long-time behavior of survival probability is described by the asymptotic series whose coefficients decrease with increasing particle energy but at different rates. To be specific, we have examined the exactly solvable model of tunneling through a finite range triangular barrier.
While exponential decay is ubiquitous in nature, deviations at both short and long times are dictated by quantum mechanics. Nonexponential decay is known to arise due to the possibility of reconstructing the initial state from the decaying products. We discuss the quantum decay dynamics by tunneling of a many-particle system, characterizing the long-time nonexponential behavior of the nonescape and survival probabilities. The effects of contact interactions and quantum statistics are described. It is found that, whereas for noninteracting bosons the long-time decay follows a power law with an exponent linear in the number of particles NNN, the exponent becomes quadratic in NNN in the fermionic case. The same results apply to strongly interacting many-body systems related by the generalized Bose-Fermi duality. The faster fermionic decay can be traced back to the effective hard-core interactions between particles, which are as well the decaying products, and exhibit spatial antibunching which hinders the reconstruction of the initial unstable state. The results are illustrated with a paradigmatic model of quantum decay from a trap allowing leaks by tunneling, whose dynamics is described exactly by means of an expansion in resonant states.
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.
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.