Remarks on the uncertainty relations

TL;DR

This paper explores the limitations and conditions of uncertainty relations in quantum mechanics, showing that the lower bounds can be zero and analyzing the scope of Heisenberg and Mandelstam–Tamm relations.

Contribution

It provides a rigorous analysis of uncertainty relations, revealing cases where the lower bound is zero and clarifying the applicability of time-energy uncertainty relations.

Findings

Lower bounds for product of standard deviations can be zero.

Sets of vectors can be complete where the product is non-negative.

Time-energy uncertainty relations have limited validity.

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 $H$.