We consider particles prepared by a single slit diffraction experiment. For those particles the standard deviation σ p of the momentum is discussed. We find out that σ p =∞ is not an exception but a rather typical case. A necessary and sufficient condition for σ p <∞ is given. Finally, the inequality σ p Δx≥π ℏ is derived and it is shown that this bound cannot be improved.

We consider the Heisenberg uncertainty principle of position and momentum in 3-dimensional spaces of constant curvature K. The uncertainty of position is defined coordinate independent by the geodesic radius of spherical domains in which the particle is localized after a von Neumann–Lüders projection. By applying mathematical standard results from spectral analysis on manifolds, we obtain the largest lower bound of the momentum deviation in terms of the geodesic radius and K. For hyperbolic spaces, we also obtain a global lower bound (...) \, which is non-zero and independent of the uncertainty in position. Finally, the lower bound for the Schwarzschild radius of a static black hole is derived and given by \, where \ is the Planck length. (shrink)