Refinement of Uncertainty Relations in Quantum Mechanics

TL;DR

This paper refines the quantum uncertainty relations by incorporating third-order commutators, providing tighter bounds for non-conjugate operators and applying these to kinetic energy, position, and angular momentum in quantum systems.

Contribution

It introduces a refined inequality for uncertainty relations that accounts for third-order commutators, extending the Heisenberg principle for non-c-number commutators.

Findings

Refined uncertainty inequality for non-conjugate operators.

Application to kinetic energy and position in 1D systems.

Analysis of angular momentum component uncertainties.

Abstract

The uncertainty relation of three quantities in quantum mechanics is estimated in terms of commutators. The Pauli matrices are used to find a contribution of third-order commutators. The resulting inequality refines the Heisenberg indeterminacy for two non-conjugate operators when their commutator is not a c-number. The inequality is applied to operators of the kinetic energy and coordinate of one-dimensional system. In addition, uncertainty of the angular momentum components is considered.