Hüseyin Ertik

Time fractional development of quantum systems

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 (nano) systems, time fractional forms of motion are obtained for a Schrodinger 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 α, which is a measure of the fractality of time. In the case α=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 α is relevant to time. It is understood that the fractionalization of...

A FRACTIONAL CALCULUS APPROACH TO INVESTIGATE THE ALPHA DECAY PROCESSES

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.

Investigation of the Bose?Einstein condensation based on fractality using fractional mathematics

Although atomic Bose gases are investigated in the dilute gas regime, the physical properties of the Bose–Einstein condensates are affected by interparticle interactions and the fractal nature of the space where the Bose systems are evolving. Theoretical predictions of the traditional Bose–Einstein thermostatistics do not account for the deviations from the experimental results, which are related to internal energy, specific heat, transition temperature, etc. On the other hand, in this study, fractional calculus is introduced where effects of the fractality of space are taken into account. Meanwhile, the order of the fractional derivative α is handled as a measure of the fractality of space. In this content, some improvements which take into account the effects of the fractal nature of the system are made in the standard physical results of the Bose–Einstein condensation phenomena. As an example, for the dilute atomic gas 7Li, the measured transition temperature of Bose–Einstein condensation could be obtained for the value of α ≈ 0.976, and one could predict that the interparticle interactions have a weak attractive nature consistent with experiment (Bradley et al 1995 Phys. Rev. Lett. 75 1687). Thus, a fractional mathematical theory is established in coherence with experimental results of Bose–Einstein condensation.

FRACTIONAL MATHEMATICAL INVESTIGATION OF BOSE–EINSTEIN CONDENSATION IN DILUTE 87Rb, 23Na AND 7Li ATOMIC GASES

Although atomic Bose gases are experimentally investigated in the dilute regime, interparticle interactions play an important role on the transition temperatures of Bose–Einstein condensation. In this study, Bose–Einstein condensation is handled using fractional calculus for a Bose gas consisting of interacting bosons which are trapped in a three-dimensional harmonic oscillator. In this frame, in order to introduce the nonextensive effect, fractionally generalized Bose–Einstein distribution function which features Mittag–Leffler function is adopted. The dependence of the transition temperature of Bose–Einstein condensation on α (a measure of fractality of space) has been established. The transition temperatures for the dilute 87Rb, 23Na and 7Li atomic gases have been obtained in consistent with experimental data and the nature of the interactions in the Bose–Einstein condensate has been enlightened. In the course of our investigations, we have arrived to the conclusion that for α 1 repulsive interactions are predominant.

Half-lives of spherical proton emitters within the framework of fractional calculus

In this paper, the half-life values of spherical proton emitters such as Sb, Tm, Lu, Ta, Re, Ir, Au, Tl and Bi have been calculated within the framework of fractional calculus. Nuclear decay equation, related to this phenomenon, has been resolved by using Caputo fractional derivative. The order of fractional derivative μ being considered is 0 < μ ≤ 1, and characterizes the fractality of time. Half-life values have been calculated equivalent with empirical ones. The dependence of fractional derivative order μ on the nuclear structure has also been investigated.