Position in Models of Quantum Mechanics with a Minimal Length

Pasquale Bosso

doi:10.3390/ecu2021-09275

Abstract

Candidate theories of quantum gravity predict the presence of a minimal measurable length at high energies. Such feature is in contrast with the Heisenberg Uncertainty Principle. Therefore, phenomenological approaches to quantum gravity introduced models spelled as modifications of quantum mechanics including a minimal length. The effects of such modification are expected to be relevant at large energies/small lengths. One first consequence is that position eigenstates are not included in such models due to the presence of a minimal uncertainty in position. Furthermore, depending on the particular modification of the position–momentum commutator, when such models are considered from momentum space, the position operator is changed, and a measure factor appears to let the position operator be self-adjoint. As a consequence, the (quasi-)position representation acquires numerous issues. For example, the position operator is no longer a multiplicative operator, and the momentum of a free particle does not correspond directly to its wave number. Here, we will review such issues, clarifying aspects of minimal length models, with particular reference to the representation of the position operator. Furthermore, we will show how such a (quasi-)position description of quantum mechanical models with a minimal length affects results concerning simple systems.

Highlights

Several approaches to quantum gravity, as well as gedanken experiments in black holes and high energy physics, suggest the presence of a minimal measurable length [1,2,3,4,5,6,7,8].Such minimal length may be due to structural properties of space time or to a fundamental minimal uncertainty in position
Regardless of its nature, such minimal length is in contrast with one of the cornerstones of quantum mechanics, viz. The Heisenberg uncertainty principle
A modification of quantum mechanics accounting for such minimal measurable length is often considered in phenomenological approaches to quantum gravity

Summary

Introduction

Several approaches to quantum gravity, as well as gedanken experiments in black holes and high energy physics, suggest the presence of a minimal measurable length [1,2,3,4,5,6,7,8]. Such minimal length may be due to structural properties of space time (e.g., causal dynamical triangulation, loop quantum gravity, etc.) or to a fundamental minimal uncertainty in position (e.g., string theory, re-elaborations of the Heisenberg microscope including gravity, etc.). A modification of quantum mechanics accounting for such minimal measurable length is often considered in phenomenological approaches to quantum gravity This modification is referred to as the generalized uncertainty principle (GUP). We will see how a new representation is possible, following the ideas developed in [16] and the implications of such alternative perspective

Quasi-Position Representation

Particle in a Box

Conclusions

2.References

A Unified