The decay law, $N={N}_{0}{e}^{\ensuremath{-}\ensuremath{\lambda}t}$, is based on the assumption that the probability
w of an atom disintegrating in unit time is constant. The question arises whether this assumption is necessary. It might happen that atoms of different ages have different decay probabilities; and
w might depend on age in such a way that exponential decay curves would result. A theorem of Wolfke, however, can be interpreted in such a manner as to show that
w cannot depend on the age of the atom. This result is generalized. Consider a series whose members
A,
B, etc., have decay probabilities
wA,
wB, etc. If all members decay exponentially, if all
w′s are smooth and continuous functions of their variables and if each
w is, at most, a function of the age of the atom, the ages of its ancestors and the ages of its descendants; then the
w′s are constant; and so in this more general case, the assumption of constant decay probability which underlies the wave mechanical theory of disintegration is necessary as well as sufficient.
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 evolution of views on the possibility of external influence on the process of radioactive decay is briefly presented. Such an effect can lead to the appearance of periods in the time series of the radioactive decay rate fluctuations, which have been the subject of intensive study in the last decade. Two mechanisms for identifying periods are considered: the study of deviations from the theoretical curve of the radioactive decay law and the study of the properties of fluctuations. It is shown that the latter method leads to a universal spectrum of periods observed not only in the time series of the radioactive decay rate fluctuations, but also in the time series of fluctuations of processes of various nature. The main object of our study are periods in the radioactive decay rate fluctuations. The presence of such periods suggests the possibility of external influence on the process of radioactive decay. Therefore, we briefly consider the evolution of views on the possibility of such an effect. To do this, we distinguish several stages. The division into stages is only partially historical, but, mainly, each stage characterizes a certain ideas that is implemented in it.
It is generally believed that physical laws, reflecting an inherent order in the universe, are ordained by nature. However, in modern physics the observer plays a central role raising questions about how an observer-centric physics can result in laws apparently worthy of a universal nature-centric physics. Over the last decade, we have found that the consistent apt quantification of algebraic and order-theoretic structures results in calculi that possess constraint equations taking the form of what are often considered to be physical laws. I review recent derivations of the formal relations among relevant variables central to special relativity, probability theory and quantum mechanics in this context by considering a problem where two observers form consistent descriptions of and make optimal inferences about a free particle that simply influences them. I show that this approach to describing such a particle based only on available information leads to the mathematics of relativistic quantum mechanics as well as a description of a free particle that reproduces many of the basic properties of a fermion. The result is an approach to foundational physics where laws derive from both consistent descriptions and optimal information-based inferences made by embedded observers.
The quantum-mechanical theory of the decay of unstable states is revisited. We show that the decay is non-exponential both in the short-time and long-time limits using a more physical definition of the decay rate than the one usually used. We report results of numerical studies based on Winter's model that may elucidate qualitative features of exponential and non-exponential decay more generally. The main exponential stage is related to the formation of a radiating state that maintains the shape of its wave function with exponentially diminishing normalization. We discuss situations where the radioactive decay displays several exponents. The transient stages between different regimes are typically accompanied by interference of various contributions and resulting oscillations in the decay curve. The decay curve can be fully oscillatory in a two-flavor generalization of Winter's model with some values of the parameters. We consider the implications of that result for models of the oscillations reported by GSI.
Application of quantum mechanics to a simple model of the nucleus gives the phenomenon of radioactive disintegration. The statistical nature of the quantum mechanics gives directly disintegration as a chance phenomenon without any special hypothesis. \textsection{}1 contains a presentation of those features of quantum mechanics which are here used and gives a simple calculation of the disintegration constant. \textsection{}2 discusses the qualitative application of the model to the nucleus. \textsection{}3 presents quantitative calculations amounting to a theoretical interpretation of the Geiger-Nuttall relation between the rate of disintegration and the energy of the emitted $\ensuremath{\alpha}$-particle. In getting this relation one arrives at the rather remarkable conclusion that the law of force between emitted $\ensuremath{\alpha}$-particle and the rest of the nucleus is substantially the same in all the atoms even where the decay rates stand in the ratio
1022. \textsection{}4 calls attention to the natural way in which the paradoxical results of Rutherford and Chadwick on the scattering of fast $\ensuremath{\alpha}$-particles by uranium receive explanation with the model here used. \textsection{}5 discusses certain limitations inherent in the methods employed.