# Remarks on the uncertainty relations

Source: [Remarks on the uncertainty relations](<https://www.sota2.com/research/paper/68fa3c231e209131ce8d8587>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T00:58:50.667Z

Authors: K. Urbanowski

Published: 2018-10-27T12:50:27.000Z

## 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 $H$.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1810.11462>)

[PDF](<https://arxiv.org/pdf/1810.11462.pdf>)