  We analyze general uncertainty relations and we show that there can exist
such pairs of non--commuting observables $A$ and $B$ and such vectors that the
lower bound for the product of standard deviations $\Delta A$ and $\Delta B$
calculated for these vectors is zero: $\Delta A\,\cdot\,\Delta B \geq 0$. We
show also that for some pairs of non--commuting observables the sets of vectors
for which $\Delta A\,\cdot\,\Delta B \geq 0$ can be complete (total). The
Heisenberg, $\Delta t \,\cdot\, \Delta E \geq \hbar/2$, and Mandelstam--Tamm
(MT), $ \tau_{A}\,\cdot \,\Delta E \geq \hbar/2$, time--energy uncertainty
relations ($\tau_{A}$ is the characteristic time for the observable $A$) are
analyzed too. We show that the interpretation $\tau_{A} = \infty$ for
eigenvectors of a Hamiltonian $H$ does not follow from the rigorous analysis of
MT relation. We show also that contrary to the position--momentum uncertainty
relation, the validity of the MT relation is limited: It does not hold on
complete sets of eigenvectors of $A$ and $H$.

Quantum Physics;General Relativity and Quantum Cosmology;High Energy Physics - Phenomenology;Popular Physics