The Fock-Krylov formalism for the calculation of survival probabilities of unstable states is revisited paying particular attention to the mathematical constraints on the density of states, the Fourier transform of which gives the survival amplitude. We show that it is not possible to construct a density of states corresponding to a purely exponential survival amplitude. The survival probability
P(t) and the autocorrelation function of the density of states are shown to form a pair of cosine Fourier transforms. This result is a particular case of the Wiener-Khinchin theorem and forces
P(t) to be an even function of time which in turn forces the density of states to contain a form factor which vanishes at large energies. Subtle features of the transition regions from the nonexponential to the exponential at small times and the exponential to the power-law decay at large times are discussed by expressing
P(t) as a function of the number of oscillations
n performed by it. The transition at short times is shown to occur when the survival probability has completed one oscillation. The number of oscillations depends on the properties of the resonant state and a complete description of the evolution of the unstable state is provided by determining the limits on the number of oscillations in each region.
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.
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
t goes to the infinity more slowly than any exponential function of
t. This effect can be described in terms of time-dependent decay rate
γ(t) and then the Khalfin result means that this
γ(t) is not a constant for long times but that it tends to zero as
t 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
t than the energy of this state for
t 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. It is shown that these corrections decrease to
E=Emin<Eϕ as
t→∞, where
Eϕ is the energy of the system in the state
∣ϕ> measured at times
t∼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 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.
\nWe study the properties of the survival probability of an unstable quantum state described by a Lee Hamiltonian. This theoretical approach resembles closely Quantum Field Theory (QFT): one can introduce in a rather simple framework the concept of propagator and Feynman rules, Within this context, we re-derive (in a detailed and didactical way) the well-known result according to which the amplitude of the survival probability is the Fourier transform of the energy distribution (or spectral function) of the unstable state (in turn, the energy distribution is proportional to the imaginary part of the propagator of the unstable state). Typically, the survival probability amplitude is the starting point of many studies of non-exponential decays. This work represents a further step toward the evaluation of the survival probability amplitude in genuine relativistic QFT. However, although many similarities exist, QFT presents some differences w.r.t. the Lee Hamiltonian which should be studied in the future.\n
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.