About Schrödinger Equation on Fractals Curves Imbedding in R 3

Abstract

In this paper we have generalized the quantum mechanics on fractal time-space. The time is changing on Cantor-set like but space is considered as fractal curve like Von-Koch curve. The Feynman path method in quantum mechanics has been suggested on fractal curve. Using F α -calculus and Feynman path method we found the Schrëdinger on fractal time-space. The Hamiltonian operator and momentum operator has been derived. More, the continuity equation and the probability density is given in generalized formulation.

Charged Particle Oscillations in Transient Plasmas Generated by Nanosecond Laser Ablation on Mg Target

“…In such a context, fractal theories of motion [31][32][33][34] become functional for describing the dynamics of ablation plasma. Significant advancements for the implementation of a fractal paradigm to a classical physical system has been done in a series of papers [35][36][37], especially for the problem of the Schrodinger representation of fractal type [38][39][40][41]. The fundamental hypothesis of our model is that the dynamics of all ablation plasma entities can be characterized by fractal motion curves.…”

“…kT e mv A , λ S = kT e mv S , (39) where k is the Boltzmann constant, T e is the electron temperature, and m is the atomic mass of the ions. v A is the ion−electron collision frequency at Coulomb-scale resolutions, and v S is the collision frequency at thermal-scale resolutions, causing Equations (38) with (39) to become…”

Non-local Integrals and Derivatives on Fractal Sets with Applications

“…The F α -calculus is generalized and applied in physics as a new and useful tool for modelling processes on the fractals. Newtonian mechanics and Schrödinger equation on the fractal sets and curves are given [15,16,17]. The gauge integral is utilized to generalized the F α -calculus for the unbound and singular function [18].…”

Diffusion on Middle-ξ Cantor Sets

“…Local fractional derivatives were suggested and applied from a physics perspective [ 33 , 34 ] in a formalism called -calculus (or -C), which has also been generalized for unbounded and singular functions [ 35 ]. Schrödinger equations on fractal curves were derived using -C and Feynman path methods [ 36 ]. A mathematical model of diffraction was given for fractal sets [ 37 ].…”