Position Uncertainty Measures on the Sphere

TL;DR

This paper introduces new position uncertainty measures for particles on the sphere, based on angle operator variances, which depend only on the state and satisfy uncertainty relations.

Contribution

It proposes novel position uncertainty measures on the sphere that are state-dependent and obey Schrödinger--Robertson uncertainty relations, with complementary operators identified.

Findings

Measures depend solely on the particle's state

Uncertainty relations of Schrödinger--Robertson type are satisfied

Complementary Hermitian operators with continuous spectrum are identified

Abstract

Position uncertainty (delocalization) measures for a particle on the sphere are proposed and illustrated on several examples of states. The new measures are constructed using suitably the standard multiplication angle operator variances. They are shown to depend solely on the state of the particle and to obey uncertainty relations of the Schroedinger--Robertson type. A set of Hermitian operators with continuous spectrum is pointed out the variances of which are complementary to the longitudinal angle uncertainty measure.