Fractional Statistics in Quantum Mechanics

Charles B. Hanna

Chihong Chou

Yong-Shi Wu

Nick Laskin

A path integral approach to quantum physics has been developed. Fractional path integrals over the paths of the Levy flights are defined. It is shown that if the fractality of the Brownian trajectories leads to standard quantum and statistical mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics and fractional statistical mechanics. The fractional quantum and statistical mechanics have been developed via our fractional path integral approach. A fractional generalization of the Schrodinger equation has been found. A relationship between the energy and the momentum of the nonrelativistic quantum-mechanical particle has been established. The equation for the fractional plane wave function has been obtained. We have derived a free particle quantum-mechanical kernel using Fox's H function. A fractional generalization of the Heisenberg uncertainty relation has been established. Fractional statistical mechanics has been developed via the path integral approach. A fractional generalization of the motion equation for the density matrix has been found. The density matrix of a free particle has been expressed in terms of the Fox's H function. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum and statistical mechanics.

Avinash Khare

Zineb Korichi, Mohammed Tayeb Meftah

In this work, we focus our study on some systems in statistical mechanics by using a recent developpment of the fractional classical and quantum mechanics. At first, we present the thermodynamical properties of N classical and quantum oscillators in 3-dimensional space. In each case, the Hamiltonian, describing the system, exhibits fractional powers. Secondly, in the context of the fractional quantum mechanics, we calculate the thermodynamical properties for the systems: the trapped ideal Bose and Fermi gases. For the trapped ideal Bose system, Bose- Einstein condensation and the thermodynamics properties are discussed and plotted with respect to the fractional powers and the temperature. It is shown that the trapped ideal Bose gas exhibits a phase transition at a temperature TC for which the specific heat presents a discontinuity depending on the fractional powers. All calculations are performed in 3-dimensional space.