About maximally localized states in quantum mechanics

TL;DR

This paper investigates the existence of a minimal length in quantum mechanics under generalized commutation relations, revealing limitations of traditional localization methods and proposing a constrained variational approach.

Contribution

It demonstrates the failure of squeezed states for maximal localization and introduces a new variational principle for such states under generalized relations.

Findings

Maximal localization via squeezed states generally fails.

A constrained variational principle is necessary for localization.

Minimal length emerges under certain generalized commutation relations.

Abstract

We analyze the emergence of a minimal length for a large class of generalized commutation relations, preserving commutation of the position operators and translation invariance as well as rotation invariance (in dimension higher than one). We show that the construction of the maximally localized states based on squeezed states generally fails. Rather, one must resort to a constrained variational principle.