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We rederive uncertainty relations for the angular position and momentum of aparticle on a circle by employing the exponential of the angle instead of theangle itself, which leads to circular variance as a natural measure ofresolution. Intelligent states minimizing the uncertainty product under theconstraint of a given uncertainty in angle or in angular momentum turn out tobe given by Mathieu wave functions. We also discuss a number of physicallyfeasible approximations to these optimal states. The theory is applied to theorbital angular momentum of a beam of photons and verified in an experimentthat employs computer-controlled spatial light modulators both at the statepreparation and analyzing stages.

Experimental test of uncertainty relations for quantum mechanics on a circle