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 $ΔA$ and $ΔB$ calculated for these vectors is zero: $ΔA\,\cdot\,ΔB \geq 0$. We show also that for some pairs of non--commuting observables the sets of vectors for which $ΔA\,\cdot\,ΔB \geq 0$ can be complete (total). The Heisenberg, $Δt \,\cdot\, ΔE \geq \hbar/2$, and Mandelstam--Tamm (MT), $τ_{A}\,\cdot \,ΔE \geq \hbar/2$, time--energy uncertainty relations ($τ_{A}$ is the characteristic time for the observable $A$) are analyzed too. We show that the interpretation $τ_{A} = \infty$ 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 $H$.

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