Applications of a Simple Formula

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New 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 pages

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