Maximally Localized States in Quantum Mechanics with a Modified Commutation Relation to All Orders

TL;DR

This paper develops maximally localized states in quantum mechanics incorporating a all-orders modified commutation relation, showing minimal length acts as a natural regulator and affects phenomena like the Casimir Effect.

Contribution

It constructs maximally localized states considering an all-orders modified commutation relation, extending previous first-order approaches and exploring physical implications.

Findings

Minimal length acts as a natural regulator eliminating infinities.

Calculated first correction to Casimir Effect due to minimal length.

Discussed implications for the concept of position measurement.

Abstract

We construct the states of maximal localization taking into account a modification of the commutation relation between position and momentum operators to all orders of the minimum length parameter. To first order, the algebra we use reproduces the one proposed by Kempft, Mangano and Mann. It is emphasized that a minimal length acts as a natural regulator for the theory, thus eliminating the otherwise ever appearing infinities. So, we use our results to calculate the first correction to the Casimir Effect due to the minimal length. We also discuss some of the physical consequences of the existence of a minimal length, culminating in a proposal to reformulate the very concept of "position measurement".