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arXiv:quant-ph/0401145 · Physical Review A

Survival law in a potential model

Stefano De Leo, PIETRO P. ROTELLI

Published Aug 5, 2004• 7 citations
arXivHTMLPDF

Abstract

The radial equation of a simple potential model has long been known to yield an exponential survival law in a lowest-order (Breit-Wigner) approximation. We demonstrate that if the calculation is extended to fourth order, the survival law exhibits a parabolic short-time behavior which leads to the quantum Zeno effect. This model has further been studied numerically to characterize the extra exponential time parameter which compliments the lifetime. We also investigate the inverse Zeno effect.

DOI
10.1103/physreva.70.022101

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The radial equation of a simple potential model has long been known to yield an exponential survival law in a lowest-order (Breit-Wigner) approximation. We demonstrate that if the calculation is extended to fourth order, the survival law exhibits a parabolic short-time behavior which leads to the quantum Zeno effect. This model has further been studied numerically to characterize the extra exponential time parameter which compliments the lifetime. We also investigate the inverse Zeno effect.
The characteristic short-time behavior of the survival probability of a decaying state, which underlies the quantum Zeno effect (QZE), is shown to follow quite simply from the Schr\"odinger equation even when the spectrum of the Hamiltonian has no finite lower bound, provided the mean energy of the initial state is finite. The failure of the Breit-Wigner form of energy distribution to predict QZE is traced to the nonexistence of a well-defined mean energy. On the basis of a suitably modified distribution, it is explicitly demonstrated that the existence of a finite lower bound to the energy distribution only affects the asymptotic decay law. For short times, the survival probability depends quadratically on time if the variance of the distribution is also finite, not otherwise. For a sufficiently well-behaved energy distribution, the asymptotic decay must be as ${t}^{{\ensuremath{-}}^{n}}$ if the spectrum has a finite lower bound, or faster than any inverse power of t if there is no lower bound.
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.
The temporal behavior of quantum mechanical systems is reviewed. We mainly focus our attention on the time development of the so-called “survival” probability of those systems that are initially prepared in eigenstates of the unperturbed Hamiltonian, by assuming that the latter has a continuous spectrum. The exponential decay of the survival probability, familiar, for example, in radioactive decay phenomena, is representative of a purely probabilistic character of the system under consideration and is naturally expected to lead to a master equation. This behavior, however, can be found only at intermediate times, for deviations from it exist both at short and long times and can have significant consequences. After a short introduction to the long history of the research on the temporal behavior of such quantum mechanical systems, the short-time behavior and its controversial consequences when it is combined with von Neumann’s projection postulate in quantum measurement theory are critically overviewed from a dynamical point of view. We also discuss the so-called quantum Zeno effect from this standpoint. The behavior of the survival amplitude is then scrutinized by investigating the analytic properties of its Fourier and Laplace transforms. The analytic property that there is no singularity except a branch cut running along the real energy axis in the first Riemannian sheet is an important reflection of the time-reversal invariance of the dynamics governing the whole process. It is shown that the exponential behavior is due to the presence of a simple pole in the second Riemannian sheet, while the contribution of the branch point yields a power behavior for the amplitude. The exponential decay form is cancelled at short times and dominated at very long times by the branch-point contributions, which give a Gaussian behavior for the former and a power behavior for the latter. In order to realize the exponential law in quantum theory, it is essential to take into account a certain kind of macroscopic nature of the total system, since the exponential behavior is regarded as a manifestation of a complete loss of coherence of the quantum subsystem under consideration. In this respect, a few attempts at extracting the exponential decay form on the basis of quantum theory, aiming at the master equation, are briefly reviewed, including van Hove’s pioneering work and his well-known “λ2T” limit. In the attempt to further clarify the mechanism of the appearance of a purely probabilistic behavior without resort to any approximation, a solvable dynamical model is presented and extensively studied. The model describes an ultrarelativistic particle interacting with N two-level systems (called “spins”) and is shown to exhibit an exponential behavior at all times in the weak-coupling, macroscopic limit. Furthermore, it is shown that the model can even reproduce the short-time Gaussian behavior followed by the exponential law when an appropriate initial state is chosen. The analysis is exact and no approximation is involved. An interpretation for the change of the temporal behavior in quantum systems is drawn from the results obtained. Some implications for the quantum measurement problem are also discussed, in particular in connection with dissipation.
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.