Heisenberg position-momentum uncertainty relation beyond central potentials

TL;DR

This paper extends the Heisenberg uncertainty relation for particles in central potentials to include states not eigenstates of angular momentum, deriving a new lower bound involving angular momentum mean and variance.

Contribution

It introduces a generalized uncertainty relation incorporating the mean and variance of angular momentum squared for a broader class of quantum states.

Findings

Derived a new lower bound for the uncertainty relation.

Extended previous results to non-eigenstates of angular momentum.

Provides a more comprehensive understanding of quantum uncertainties in central potentials.

Abstract

Recently it was shown in [New J. Phys. 8, 330 (2006)] that the three dimensional Heisenberg uncertainty principle might be signifficantly sharpened if the relevant quantum state describes the particle in a central potential. I extend that result to the case of states which are not the eigenstates of the square of the angular momentum operator. I derive a new lower bound which involves the mean value and the variance of the $\hat{L}^2$ operator.