Fractal Schr\"{o}dinger Equation: Implications for Fractal Sets

TL;DR

This paper explores the integration of fractal calculus with quantum mechanics by introducing the Fractal Schrödinger Equation, analyzing its effects on quantum systems within fractal geometries, and presenting new solutions and probability densities.

Contribution

It introduces the Fractal Schrödinger Equation and applies it to various quantum systems, providing new insights into fractal quantum mechanics and extending traditional models.

Findings

Derived a general solution for the time-dependent fractal Schrödinger equation

Analyzed probability densities of the hydrogen atom within fractal dimensions

Applied fractal calculus to simple harmonic motion and introduced a fractal probability density function

Abstract

This paper delves into the world of fractal calculus, investigating its implications for fractal sets. It introduces the Fractal Schr\"{o}dinger Equation and provides insights into its consequences. The study presents a General Solution for the Time-Dependent Schr\"{o}dinger Equation, unveiling its core aspects. Exploring quantum mechanics in the context of fractals, the paper analyzes the Probability Density of the Radial Hydrogen Atom, unveiling its behavior within fractal dimensions. The investigation extends to deciphering the intricate Energy Levels of the Hydrogen Atom, uncovering the interplay of quantum mechanics and fractal geometry. Innovatively, the research applies the Fractal Schr\"{o}dinger Equation to Simple Harmonic Motion, leading to the introduction of the Fractal Probability Density Function for the Harmonic Oscillator. The paper employs a series of illustrative figures that enhance the comprehension of the findings. By intertwining quantum mechanics and fractal mathematics, this research paves the way for deeper insights into their relationship.