Remarks on the uncertainty relations

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 $\mathrm{\Delta }A$ and $\mathrm{\Delta }B$ calculated for these vectors is zero: $\mathrm{\Delta }A\cdot \mathrm{\Delta }B\ge 0$. We show also that for some pairs of non–commuting observables the sets of vectors for which $\mathrm{\Delta }A\cdot \mathrm{\Delta }B\ge 0$ can be complete (total). The Heisenberg, $\mathrm{\Delta }t\cdot \mathrm{\Delta }E\ge \mathrm{\hslash }\mathrm{/}2$, and Mandelstam–Tamm (MT), ${\tau }_A\cdot \mathrm{\Delta }E\ge \mathrm{\hslash }\mathrm{/}2$, time–energy uncertainty relations (${\tau }_A$ is the characteristic time for the observable $A$) are analyzed too. We show that the interpretation ${\tau }_A=\mathrm{\infty }$ 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 $H$.