Generalized uncertainty relations in spherical coordinates

TL;DR

This paper extends the Heisenberg-Robertson uncertainty relation to spherical coordinates, incorporating boundary conditions and surface terms, and analyzes their effects on quantum uncertainties and potentials.

Contribution

It introduces a generalized uncertainty relation in spherical coordinates, accounting for boundary conditions and surface terms, which differ from traditional approaches.

Findings

Extra surface terms are derived for various potentials.

Boundary conditions significantly affect uncertainty relations.

Differences are identified between the new approach and traditional product-based methods.

Abstract

Following to the Weil method we generalize the Heisenberg-Robertson uncertainty relation for arbitrary two operators. Consideration is made in spherical coordinates, where the distant variable is restricted from one side, . By this reason accounting of suitable boundary condition at the origin for radial wave functions and operators is necessary. Therefore, there arise extra surface terms in comparison with traditional approaches. These extra terms are calculated for various solvable potentials and their influence is investigated. At last, the time-energy uncertainty relations are also analysed. Some differences between our approach and that, in which a direct product for separate variances were considered are discussed.