Remarks on the uncertainty relations

Abstract

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 $H$.