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Efficient quantum trajectory representation of wavefunctions evolving in imaginary time J. Chem. Phys. 135, 034104 (2011); 10.1063/1.3610165Addendum to: "The one-dimensional harmonic oscillator in the presence of a dipole-like interaction" [Am.In this study, the effect of time fractionalization on the development of quantum systems is taken under consideration by making use of fractional calculus. In this context, a Mittag-Leffler function is introduced as an important mathematical tool in the generalization of the evolution operator. In order to investigate the time fractional evolution of the quantum ͑nano͒ systems, time fractional forms of motion are obtained for a Schrödinger equation and a Heisenberg equation. As an application of the concomitant formalism, the wave functions, energy eigenvalues, and probability densities of the potential well and harmonic oscillator are time fractionally obtained via the fractional derivative order ␣, which is a measure of the fractality of time. In the case ␣ = 1, where time becomes homogenous and continuous, traditional physical conclusions are recovered. Since energy and time are conjugate to each other, the fractional derivative order ␣ is relevant to time. It is understood that the fractionalization of time gives rise to energy fluctuations of the quantum ͑nano͒ systems.

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In this study, the nuclear decay equation is taken under consideration by making use of fractional calculus. In this context, the first-order time derivative is changed to a Caputo fractional derivative hence, the resulting equation is the time fractional nuclear decay equation. The solution of this equation is obtained in terms of Mittag–Leffler function which plays an important role to study the non-Markovian feature of physical processes. As an application of this time fractional formalism, alpha decay half-life values have been calculated for Pb , Po , Rn , Ra , Th and U isotopes. Consequently, the theoretical half-life values have been obtained in consistent with the experimental data. The dependence of the order of fractional derivative μ being a measure of fractality of time, on the nuclear structure has been established. In the investigations carried out, we have arrived to the conclusion that for the μ values which are closed to one, where time becomes homogenous and continuous, the shell closure effects are predominant and that the fractional derivative order μ (i.e., fractality of time) and nuclear structure are closely related to each other.