Confidence uncertainty: position and momentum can be jointly determined with a guaranteed probability

Standard-deviation and entropic formulations of uncertainty principle capture the spread of the probability distribution but say little about the probability itself contained in a small region. We introduce the confidence uncertainty ${\mathrm{\Delta }}^{c}x({\theta }_x)$ as the minimal Lebesgue measure of the support set in which the particle is found with probability at least ${\theta }_x$, and the companion interval confidence uncertainty ${\mathrm{\Delta }}^{I}x({\theta }_x)$ which restricts the support to a single interval. We prove two complementary uncertainty inequalities. (i) For ${\theta }_x+{\theta }_p\le 1$ both confidence uncertainties can be made arbitrarily small simultaneously, so that no nontrivial product bound holds; in particular, position and momentum can be jointly localised with probability at least~$50\mathrm{\%}$. (ii) For ${\theta }_x+{\theta }_p>1$ a lower bound holds: combining Lenard’s projection inequality with the Donoho–Stark operator-norm bound we obtain ${\mathrm{\Delta }}^{c}x{\mathrm{\Delta }}^{c}p\ge 2\pi \mathrm{\hslash }(\sqrt{{\theta }_x{\theta }_p}-\sqrt{(1-{\theta }_x)(1-{\theta }_p)}{)}^{\text{\hspace{0.17em}⁣}2}$, and for the interval version we obtain the sharp implicit Landau–Pollak bound ${\mathrm{\Delta }}^{I}x{\mathrm{\Delta }}^{I}p\ge 4\mathrm{\hslash }{\lambda }_0^{-1}\text{\hspace{0.17em}⁣}((\sqrt{{\theta }_x{\theta }_p}-\sqrt{(1-{\theta }_x)(1-{\theta }_p)}{)}^2)$, where ${\lambda }_0(c)$ is the largest prolate-spheroidal eigenvalue. We support the analytical bounds with numerical evaluation of ${\lambda }_0(c)$, provide closed-form small-$c$ and large-$c$ asymptotics, compute the optimal Slepian-superposition states that saturate the interval bound, and compare the resulting product against the variance Heisenberg–Kennard, the Białynicki-Birula–Mycielski entropic, and the Donoho–Stark concentration bounds. The unified picture provides a complete phase diagram on $({\theta }_x,{\theta }_p)\in [0,1{]}^2$.