  The uncertainty relations for angle and angular momentum are revisited. We
use the exponential of the angle instead of the angle itself and adopt
dispersion as a natural measure of resolution. We find states that minimize the
uncertainty product under the constraint of a given uncertainty in angle or in
angular momentum. These states are described in terms of Mathieu wave functions
and may be approximated by a von Mises distribution, which is the closest
analogous of the Gaussian on the unit circle. We report experimental results
using beam optics that confirm our predictions.
