Schr\"odinger Equation on Fractals Curves Imbedding in $R^3$

TL;DR

This paper extends quantum mechanics to fractal space-time, deriving the Schrödinger equation, Hamiltonian, and probability density on fractal curves using $F^{ ext{alpha}}$-calculus and Feynman path methods.

Contribution

It introduces a generalized Schrödinger equation on fractal curves in space-time, combining $F^{ ext{alpha}}$-calculus with Feynman path integrals for the first time.

Findings

Derived Schrödinger equation on fractal curves.

Formulated Hamiltonian and momentum operators in fractal space.

Presented a generalized continuity equation and probability density.

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 $F^{\alpha}$-calculus and Feynman path method we found the Schr\"{e}dinger 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.