A Link Between Relativistic Rest Energy and Fractionary Momentum Operators of Order 1/2

TL;DR

This paper explores a fractional wave equation in quantum mechanics, linking relativistic rest energy with fractional momentum operators, and investigates solutions in potential wells with implications for particle energy and proton radius.

Contribution

It introduces a fractional wave equation with order 1/2, connecting relativistic energy to fractional momentum, and analyzes solutions in potential wells with physical implications.

Findings

Derived normalizable solutions resembling wave packets.

Matched damping coefficient with Yukawa potential, leading to energy offset E=mc^2/2.

Linked wave damping to proton radius for proton-mass particles.

Abstract

The solution of a causal fractionary wave equation in an infinite potential well was obtained. First, the so-called "free particle" case was solved, giving as normalizable solutions a superposition of damped oscillations similar to a wave packet. From this results, the infinite potential well case was then solved. The damping coefficient of the equation obtained was matched with the exponent appearing in the Yucawa potential or "screened" Coulomb potential. When this matching was forced, the particle aquires an offset energy of E = mc^2/2 which then can be increased by each energy level. The expontential damping of the wave solutions in the box was found to be closely related with the radius of the proton when the particle has a mass equal to the mass of the proton. Lastly the fractionary wave equation was expressed in spherical coordinates and remains to be solved through analytical or numerical methods.