Aspects of the Coarse-Grained-Based Approach to a Low-Relativistic Fractional Schr\"odinger Equation

TL;DR

This paper develops a coarse-grained fractional Schrödinger equation incorporating relativistic effects, deriving related quantum and classical laws, and analyzing the impact of relativistic corrections on quantum potential and energy-momentum relations.

Contribution

It introduces a novel coarse-grained formulation of the fractional Schrödinger equation with relativistic corrections and derives related classical and quantum relations.

Findings

Derived a fractional Newtonian law with quantum potential.

Formulated the fractional counterparts of De Broglie's energy and momentum.

Analyzed the contribution of relativistic corrections to the quantum potential.

Abstract

The main goal of this paper is to set up the coarse-grained formulation of a fractional Schr\"odinger equation that incorporates a higher (spatial) derivative term which accounts for relativistic effects at a lowest order. The corresponding continuity equation is worked out and we also identify the contribution of the relativistic correction the quantum potential in the coarse-grained treatment. As a consequence, in the classical regime, we derive the sort of fractional Newtonian law with the quantum potential included and the fractional conterparts of the De Broglies's energy and momentum relations.