Results of theoretical studies and numerical calculations presented in the literature suggest that the survival probability P0(t) has the exponential form starting from times much smaller than the lifetime τ up to times t ⪢τ and that P0(t) exhibits inverse power–law behavior at the late time region for times longer than the so–called crossover time T ⪢ τ (The crossover time T is the time when the late time deviations of P0(t) from the exponential form begin to dominate). More detailed analysis of the problem shows that in fact the survival probability P0(t) can not take the pure exponential form at any time interval including times smaller than the lifetime τ or of the order of τ and it has has an oscillating form. We also study the survival probability of moving relativistic unstable particles with definite momentum . These studies show that late time deviations of the survival probability of these particles from the exponential–like form of the decay law, that is the transition times region between exponential–like and non-exponential form of the survival probability, should occur much earlier than it follows from the classical standard considerations.
We modify the theory of the Quantum Zeno Effect to make it consistent with the postulates of quantum mechanics. This modification allows one, throughout a sequence of observations of an excited system, to address the nature of the observable and thereby to distinguish survival from non-decay, which is necessary whenever excited states are degenerate. As a consequence, one can determine which types of measurements can possibly inhibit the exponential decay of the system. We find that continuous monitoring taken as the limit of a sequence of ideal measurements will only inhibit decay in special cases, such as in well-controlled experiments. Uncontrolled monitoring of an unstable system, however, can cause exponentially decreasing non-decay probability at all times. Furthermore, calculating the decay rate for a general sequence of observations leads to a straightforward derivation of Fermi's Golden Rule, that avoids many of the conceptual difficulties normally encountered. When multiple decay channels are available, the derivation reveals how the total decay rate naturally partitions into a sum of the decay rates for the various channels, in agreement with observations. Continuous and unavoidable monitoring of an excited system by an uncontrolled environment may therefore be a mechanism by which to explain the exponential decay law.
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 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 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 model considered, 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.
The exponential decay of an unstable quantum state, as described by standard Markovian theories such as Fermi's Golden Rule, assumes a simple, structureless environment. However, in complex environments characterized by disorder, long-range interactions, or strong fluctuations, local decay rates fluctuate, leading to non-Markovian dynamics and power-law ``long-time tails.'' In this paper, we apply the recently proposed \textit{hyperstatistics} framework to the problem of quantum decay in such complex environments. By considering a
γ-distribution of local decay rates across mesoscopic domains, we derive a macroscopic survival probability governed by a
q-exponential function. We then use the
q-generalized Gamma function, defined via the Mellin transform of the
q-exponential, to calculate the moments of the decay-time distribution. We show that the mean quantum lifetime is
⟨t⟩=∫0∞P(t)dt=[⟨Γ⟩(2−q)]−1, and is therefore finite for
q<2. The convergence of the second moment, and hence of the lifetime variance, instead requires the stricter condition
q<3/2. These results provide a refined physical interpretation: for
1<q<3/2 both the mean lifetime and its variance are finite; for
3/2≤q<2 the mean lifetime remains finite but lifetime fluctuations become infinitely broad; and for
q≥2 the mean lifetime itself diverges, corresponding to a truly trapped, localized, or Griffiths-like regime.
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) of the state considered is found. Our attention is focused on the late time properties of
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
t goes to the infinity more slowly than any exponential function of
t. This effect can be described using a time-dependent decay rate
γ(t) and then the Khalfin result means that this
γ(t) is not a constant but at late times it tends to zero as
t goes to the infinity. It appears that the energy
E(t) of the unstable state behaves similarly: It tends to the minimal energy
Emin of the system as
t→∞. Within the model considered we find two first leading time dependent elements of late time asymptotic expansions of
E(t) and
γ(t). We discuss also possible implications of such a late time asymptotic properties of
E(t) and
γ(t) and cases where these properties may manifest themselves.