Time fractional development of quantum systems
Articles > Article · 2010
Authors: Huseyin ERTİK, Dogan DEMIRHAN, Hüseyin ŞİRİN, FEVZİ BÜYÜKKILIÇ
Journal of Mathematical Physics, Journal Indexed in SCI Expanded, International, WoS, 2010, vol 51
DOI: 10.1063/1.3464492
Summary
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 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 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 time gives rise to energy fluctuations of the quantum (nano) systems.