2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/93151New applications of the formula $A |ψ\rangle = \langle A \rangle |ψ\rangle + ΔA |ψ_{\perp} \rangle$ are discussed. Simple derivations of the Heisenberg uncertainty principle and of related inequalities are presented. In addition, the formula is used in an instructive paradox which clarifies a fundamental notion in quantum mechanics.Latex, 7 pagesQuantum Physicstext