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arXiv:quant-ph/0207066 · Journal of Physics A Mathematical and General

The various power decays of the survival probability at long times for a free quantum particle

Manabu Miyamoto

Published Aug 7, 2002• 12 citations
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Abstract

The long time behaviour of the survival probability of the initial state and its dependence on the initial states are considered, for the one-dimensional free quantum particle. We derive the asymptotic expansion of the time evolution operator at long times, in terms of the integral operators. This enables us to obtain the asymptotic formula for the survival probability of the initial state ψ( x ), which is assumed to decrease sufficiently rapidly at large | x |. We then show that the behaviour of the survival probability at long times is determined by that of the initial state ψ at zero momentum k = 0. Indeed, it is proved that the survival probability can exhibit the various power decays such as t −2 m −1 for an arbitrary non-negative integer m as t → ∞, corresponding to the initial states with the condition ( k ) = O ( k m ) as k → 0.

DOI
10.1088/0305-4470/35/33/312

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The long time behaviour of the survival probability of the initial state and its dependence on the initial states are considered, for the one-dimensional free quantum particle. We derive the asymptotic expansion of the time evolution operator at long times, in terms of the integral operators. This enables us to obtain the asymptotic formula for the survival probability of the initial state ψ( x ), which is assumed to decrease sufficiently rapidly at large | x |. We then show that the behaviour of the survival probability at long times is determined by that of the initial state ψ at zero momentum k = 0. Indeed, it is proved that the survival probability can exhibit the various power decays such as t −2 m −1 for an arbitrary non-negative integer m as t → ∞, corresponding to the initial states with the condition ( k ) = O ( k m ) as k → 0.
The power-law decrease of the survival probability S ( t ) at long times for the one-dimensional free-particle system is shown to be characterized by the small-momentum behavior of the initial wave packet. We consider and compare the result based on the resolvent analysis and that on the asymptotic method of the Fourier integral, where the effect of a specialty of a one-dimensional free resolvent and that of the discontinuity of initial states at zero momentum on the behavior of S ( t ) are pointed out.
The behavior of both the survival S(t)S(t)S(t) and the nonescape P(t)P(t)P(t) probabilities at long times for the one-dimensional free-particle system is shown to be closely connected to that of the initial wave packet at small momentum. We prove that both S(t)S(t)S(t) and P(t)P(t)P(t) asymptotically exhibit the same power-law decrease at long times, when the initial wave packet in momentum representation behaves as km{k}^{m}km with m=0m=0m=0 or 111 at small momentum. On the other hand, if the integer m becomes greater than 1,1,1, S(t)S(t)S(t) and P(t)P(t)P(t) decrease in different power laws at long times.
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.
We study the survival probability of moving relativistic unstable particles with definite momentum p⃗≠0\vec{p} \neq 0p​=0. The amplitude of the survival probability of these particles is calculated using its integral representation. We found decay curves of such particles for the quantum mechanical models considered. These model studies show that late time deviations of the survival probability of these particles from the exponential form of the decay law, that is the transition times region between exponential and non-expo\-nen\-tial form of the survival probability, should occur much earlier than it follows from the classical standard approach resolving itself into replacing time ttt by t/γt/γt/γ (where γγγ is the relativistic Lorentz factor) in the formula for the survival probability and that the survival probabilities should tend to zero as t→∞t\rightarrow \inftyt→∞ much slower than one would expect using classical time dilation relation. Here we show also that for some physically admissible models of unstable states the computed decay curves of the moving particles have fluctuating form at relatively short times including times of order of the lifetime.
The time evolution of quantum decay of an arbitrary state initially located within an interaction region of finite range is investigated. We compare the survival S(t)S(t)S(t) and nonescape P(t)P(t)P(t) probabilities. Our approach considers the full resonant spectra of the system using a novel representation of the time-dependent Green function. It is shown that, for an initial state near a resonance, S(t)S(t)S(t) and P(t)P(t)P(t) exhibit at long times a different behavior; not only the onset to a power law decay occurs at a different time, but also instead of the well known $S(t)\ensuremath{\sim}{t}^{\ensuremath{-}3}$ we obtain $P(t)\ensuremath{\sim}{t}^{\ensuremath{-}1}$.
The decaying behavior of both the survival S(t) and total P(t) probabilities for unstable multilevel systems at long times is investigated by using the N-level Friedrichs model. The long-time asymptotic-forms of both S(t) and P(t) are obtained for an arbitrary initial-state extending over the unstable levels. It is then clarified how the asymptotic forms depend on the initial population in unstable levels. In particular, a special initial state that maximizes the asymptotic form of both S(t) and P(t) is found. On the other hand, the initial states eliminating the first term of their asymptotic expnasions also exist, which implies that a faster decay rather than expected can be realized. This faster decay for S(t) is numerically confirmed by considering the spontaneous emission process for the hydrogen atom interacting with the electromagnetic field. It is demonstrated that the t-4-decay and a faster decay are realized depending on the initial states, where the latter is estimated as t-8.