Uncertainty relation for angle from a quantum-hydrodynamical perspective

TL;DR

This paper derives uncertainty relations for angle in quantum mechanics using a hydrodynamical approach, avoiding the need for an angle operator, and reveals state-dependent bounds that allow finite angular uncertainties.

Contribution

It introduces a quantum hydrodynamical method via stochastic variational approach to derive angle uncertainty relations without defining an angle operator.

Findings

Uncertainty bounds can be smaller than /2 in certain states.

The approach applies to arbitrary canonical coordinates.

Finite angular uncertainty is possible in eigenstates of angular momentum.

Abstract

We revisit the problem of the uncertainty relation for angle by using quantum hydrodynamics formulated in the stochastic variational method (SVM), where we need not define the angle operator. We derive both the Kennard and Robertson-Schroedinger inequalities for canonical variables in polar coordinates. The inequalities have state-dependent minimum values which can be smaller than \hbar/2 and then permit a finite uncertainty of angle for the eigenstate of the angular momentum. The present approach provides a useful methodology to study quantum behaviors in arbitrary canonical coordinates.