The 'most classical' states of Euclidean invariant elementary quantum mechanical systems

TL;DR

This paper characterizes all states in Euclidean invariant quantum systems where the uncertainty relations are exactly saturated, revealing conditions under which such states exist for various momentum and angular momentum components.

Contribution

It provides a complete classification of states saturating uncertainty relations in Euclidean invariant systems, highlighting the dependence on the choice of generators and geometric configurations.

Findings

States exist for orthogonal linear and angular momentum components.

States exist for linear momentum and center-of-mass components only at zero or acute angles.

No such states exist for any pair of center-of-mass components.

Abstract

Complex techniques of general relativity are used to determine \emph{all} the states in the two and three dimensional momentum spaces in which the equality holds in the uncertainty relations for the non-commuting basic observables of Euclidean invariant elementary quantum mechanical systems, even with non-zero intrinsic spin. It is shown that while there is a 1-parameter family of such states for any two components of the angular momentum vector operator with any angle between them, such states exist for a component of the linear and angular momenta \emph{only if} these components are orthogonal to each other and hence the problem is reduced to the two-dimensional Euclidean invariant case. We also show that the analogous states exist for a component of the linear momentum and of the centre-of-mass vector \emph{only if} the angle between them is zero or an acute angle. \emph{No} such state (represented by a square integrable and differentiable wave function) can exist for \emph{any} pair of components of the centre-of-mass vector operator. Therefore, the existence of such states depends not only on the Lie algebra, but on the choice for its generators as well.