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

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.