Position in Minimal Length Quantum Mechanics

Pasquale Bosso

doi:10.3390/universe8010017

Abstract

Several approaches to quantum gravity imply the presence of a minimal measurable length at high energies. This is in tension with the Heisenberg Uncertainty Principle. Such a contrast is then considered in phenomenological approaches to quantum gravity by introducing a minimal length in quantum mechanics via the Generalized Uncertainty Principle. Several features of the standard theory are affected by such a modification. For example, position eigenstates are no longer included in models of quantum mechanics with a minimal length. Furthermore, while the momentum-space description can still be realized in a relatively straightforward way, the (quasi-)position representation acquires numerous issues. Here, we will review such issues, clarifying aspects regarding models with a minimal length. Finally, we will consider the effects of such models on simple quantum mechanical systems.

Highlights

Candidate theories of quantum gravity and thought experiments in black hole physics predict the existance of a minimal measurable length [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]
Generalized Uncertainty Principle (GUP) can be introduced by modifying the bound for the product of the uncertainties in position and momentum, making such a bound momentum dependent [12,19,20]; or it can be cast by modifying the commutation relation between position and momentum, as a particular subgroup of models of Generalized Uncertainty Relations (GUR) [8,21,22,23]
Phenomenological studies of a minimal measurable length in quantum mechanics are motivated by a large fraction of quantum approaches to gravity and black hole physics

Summary

Introduction

Candidate theories of quantum gravity and thought experiments in black hole physics predict the existance of a minimal measurable length [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. From a phenomenological perspective, considered in this work, one of the most interesting aspects is that it is in strong contrast with the Heisenberg Uncertainty Principle. Such a tension led to the low-energy implementation of a minimal measurable length in quantum mechanics via adequate modifications. Models that include such a minimal length are generically referred to as Generalized Uncertainty Principle (GUP). Models with more dimensions have been considered in the literature, showing interesting features [18]

Quasi-Position Representation

Examples

Particle in a Box

Potential Barrier

Conclusions