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

Complex techniques of general relativity are used to determine all the states in two- and three-dimensional momentum spaces in which the equality holds in uncertainty relations for 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 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 analogous states exist for a component of the linear momentum and of the center-of-mass vector only if the angle between them is zero or an acute angle. No such state (represented by a square integrable and differentiable wave function) can exist for any pair of components of the center-of-mass vector operator. Therefore, the existence of such states depends not only on the Lie algebra but on the choice of its generators as well.