The generalized uncertainty principle within the ordinary framework of quantum mechanics

TL;DR

This paper demonstrates that a generalized uncertainty principle, arising from deformed commutation relations, can be integrated into standard quantum mechanics without altering its fundamental formalism.

Contribution

It shows how to incorporate a generalized uncertainty principle into ordinary quantum mechanics by embedding the deformed algebra within its existing framework.

Findings

Deformed algebra can be embedded into standard quantum mechanics.

The generalized uncertainty principle does not require a new formalism.

Minimum position uncertainty is related to the Planck length.

Abstract

A proper deformation of the underlying coordinate and momentum commutation relations in quantum mechanics provides a phenomenological approach to account for the influence of gravity on small scales. Introducing the squared momentum term results in a generalized uncertainty principle, which limits the minimum uncertainty in particle position to the Planck length. However, such a deformation of the commutator significantly changes the formalism, making it separate from the canonical formalism of quantum mechanics. In this study, it is shown that the deformed algebra of position and momentum operators can be incorporated into the framework of ordinary quantum mechanics.