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Long time properties of the evolution of an unstable

K. Urbanowski

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

An eect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region is considered. In 1957 Khalfin proved that this amplitude tends to zero as t ! 1 more slowly than any exponential function oft. This can be described in terms of the time-dependent decay rate (t) which, when considered with the Khalfin result, means that this (t) is not a constant for large t but that it tends to zero as t!1. We find that a similar conclusion can be drawn for a large class of models of unstable states for a quantity, which can be interpreted as the ”instantaneous energy” of the unstable state. This energy should be much smaller for suitably larger values of t than when t is of the order of the lifetime of the considered state. Within a given model we show that the energy corrections in the long (t ! 1) and relatively short (lifetime of the state) time regions, are dierent. This is a purely quantum mechanical eect. It is hypothesized that there is a possibility to detect this eect by analyzing the spectra of distant astrophysical objects. The above property of unstable states may influence the measured values of astrophysical and cosmological parameters.

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Journal of Nuclear and Particle Physics•Aug 31, 2012•2 citations
An eect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region is considered. In 1957 Khalfin proved that this amplitude tends to zero as t ! 1 more slowly than any exponential function oft. This can be described in terms of the time-dependent decay rate (t) which, when considered with the Khalfin result, means that this (t) is not a constant for large t but that it tends to zero as t!1. We find that a similar conclusion can be drawn for a large class of models of unstable states for a quantity, which can be interpreted as the ”instantaneous energy” of the unstable state. This energy should be much smaller for suitably larger values of t than when t is of the order of the lifetime of the considered state. Within a given model we show that the energy corrections in the long (t ! 1) and relatively short (lifetime of the state) time regions, are dierent. This is a purely quantum mechanical eect. It is hypothesized that there is a possibility to detect this eect by analyzing the spectra of distant astrophysical objects. The above property of unstable states may influence the measured values of astrophysical and cosmological parameters.
Abstract An effect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region is considered. In 1957 Khalfin proved that this amplitude tends to zero as t → ∞ more slowly than any exponential function of t. This can be described in terms of the time-dependent decay rate γ(t) which, when considered with the Khalfin result, means that this γ(t) is not a constant for large t but that it tends to zero as t → ∞. We find that a similar conclusion can be drawn for a large class of models of unstable states for a quantity, which can be interpreted as the “instantaneous energy” of the unstable state. This energy should be much smaller for suitably larger values of t than when t is of the order of the lifetime of the considered state. Within a given model we show that the energy corrections in the long (t → ∞) and relatively short (lifetime of the state) time regions, are different. This is a purely quantum mechanical effect. It is hypothesized that there is a possibility to detect this effect by analyzing the spectra of distant astrophysical objects. The above property of unstable states may influence the measured values of astrophysical and cosmological parameters.
We consider an effect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region: In 1957 Khalfin proved that this amplitude tends to zero as ttt goes to the infinity more slowly than any exponential function of ttt. This effect can be described in terms of time-dependent decay rate γ(t)γ(t)γ(t) and then the Khalfin result means that this γ(t)γ(t)γ(t) is not a constant for long times but that it tends to zero as ttt goes to the infinity. It appears that a similar conclusion can be drawn for the energy of the unstable state for a large class of models of unstable particles: This energy should be much smaller for suitably long times ttt than the energy of this state for ttt of the order of the lifetime of the considered state. Within a given model we show that the energy corrections in the long (t→∞t \to \inftyt→∞) and relatively short (lifetime of the state) time regions, are different. It is shown that these corrections decrease to E=Emin<Eϕ{\cal E} = {\cal E}_{min} < {\cal E}_ϕE=Emin​<Eϕ​ as t→∞t \to \inftyt→∞, where Eϕ{\cal E}_ϕEϕ​ is the energy of the system in the state ∣ϕ>|ϕ>∣ϕ> measured at times t∼τϕ=ℏγt \sim τ_ϕ= \frac{\hbar}γt∼τϕ​=γℏ​. This is a purely quantum mechanical effect. It is hypothesized that there is a possibility to detect this effect by analyzing the spectra of distant astrophysical objects. The above property of unstable states may influence the measured values of astrophysical and cosmological parameters.
An effect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region is considered. We find that the instantaneous energy of the unstable state for a large class of models of unstable states tends to the minimal energy of the system Emin{\cal E}_{min}Emin​ as t→∞t\rightarrow\inftyt→∞ which is much smaller than the energy of this state for ttt of the order of the lifetime of the considered state. Analyzing the transition time region between exponential and non-exponential form of the survival amplitude we find that the instantaneous energy of the considered unstable state can take large values, much larger than the energy of this state for ttt from the exponential time region. Taking into account results obtained for a model considered, it is hypothesized that this purely quantum mechanical effect may be responsible for the properties of broad resonances such as σσσ meson as well as having astrophysical and cosmological consequences.
An effect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region is considered.We find that the instantaneous energy of the unstable state for a large class of models of unstable states tends to the minimal energy of the system E min as t → ∞ which is much smaller than the energy of this state for t of the order of the lifetime of the considered state.Analyzing the transition time region between the exponential and non-exponential form of the survival amplitude, we find that the instantaneous energy of the considered unstable state can take large values, much larger than the energy of this state for t from the exponential time region.Taking into account results obtained for a considered model, it is hypothesized that this purely quantum mechanical effect may be responsible for the properties of broad resonances such as σ meson as well as having astrophysical and cosmological consequences Key words: decay law, non exponential decay
We consider a model of a unstable state defined by the truncated Breit-Wigner energy density distribution function. An analytical form of the survival amplitude a(t)a(t)a(t) of the state considered is found. Our attention is focused on the late time properties of a(t)a(t)a(t) and on effects generated by the non--exponential behavior of this amplitude in the late time region: In 1957 Khalfin proved that this amplitude tends to zero as ttt goes to the infinity more slowly than any exponential function of ttt. This effect can be described using a time-dependent decay rate γ(t)γ(t)γ(t) and then the Khalfin result means that this γ(t)γ(t)γ(t) is not a constant but at late times it tends to zero as ttt goes to the infinity. It appears that the energy E(t)E(t)E(t) of the unstable state behaves similarly: It tends to the minimal energy EminE_{min}Emin​ of the system as t→∞t \to \inftyt→∞. Within the model considered we find two first leading time dependent elements of late time asymptotic expansions of E(t)E(t)E(t) and γ(t)γ(t)γ(t). We discuss also possible implications of such a late time asymptotic properties of E(t)E(t)E(t) and γ(t)γ(t)γ(t) and cases where these properties may manifest themselves.
Possible consequences of non-exponential behavior of the survival amplitude of an unstable particle at very late times are considered. We describe a new quantum phenomenon: Analyzing the transition time region between exponential and post-exponential form of the survival amplitude we find that the instantaneous energy of the considered unstable state can take large values, much larger than the energy of this state for from the exponential time region. We find also that the instantaneous energy of the unstable state tends to the minimal energy of the system as which is much smaller than the energy of this state for of the order of the lifetime of the considered state. Taking into account the results obtained for the model considered, it is hypothesized that this purely quantum mechanical effect may be responsible for the properties of broad resonances such as meson as well as having astrophysical consequences.