  so that inequality  (16)  implies (by choosing \(\alpha,\beta\) optimal) that \( \Delta_\psi \bQ \Delta_\psi \bP \geq 0.12 \hbar \). This, obviously, is not the best lower bound for the product of standard deviations, but the important point is here that the Landau-Pollak inequality  (16)  in terms of bulk widths implies the existence of a lower bound on the product of standard deviations, while conversely, the Heisenberg-Kennard equality  (9)  does not imply any bound on the product of bulk widths. A generalization of this approach to non-commuting observables in a finite-dimensional Hilbert space is discussed in Uffink 1990.