Uncertainty relation for angle from a quantum-hydrodynamical perspective

We revisit the problem of the uncertainty relation for angle by using quantum hydrodynamics formulated in terms of the stochastic variational method (SVM), where we need not define the angle operator. We derive both the Kennard and Robertson–Schrödinger inequalities for canonical variables in polar coordinates. The inequalities have state-dependent minimum values which can be smaller than ħ∕2 and then permit a finite uncertainty of angle for the eigenstate of the angular momentum. The present approach provides a useful methodology to study quantum behaviors in arbitrary coordinates. •Uncertainty relations for arbitrary coordinates are obtained.•Our approach does not introduce operators of canonical variables.•The well-known angular uncertainty paradox is resolved.