The Extended Uncertainty Principle from an Operational Viewpoint

Abstract: We revisit the traditional inequalities of the Extended Uncertainty Principle (EUP) from the perspective of operational quantum mechanics. Instead of relying on purely wavefunction-based measures (e.g. the standard deviation $\sigma_x$), we introduce a apparatus-centred definition of positional uncertainty, such as a finite slit width $\Delta x$ or a spherical region of radius $R$. This choice anchors the theory directly in realistic measurement protocols and avoids ambiguities arising from wavefunction tails or boundary conditions. Using hermitian EUP momentum operators, we establish a rigorous new lower bound on the uncertainty product $\sigma_p \Delta x$. In particular, our first theorem shows that $\sigma_p \Delta x \ge \pi \hbar\,\Phi_\alpha(\Delta x/2)$, where $\Phi_\alpha(\cdot)$ encodes EUP corrections via the real parameter $\alpha \ge 0$. In the limit $\alpha \to 0$ one recovers the canonical momentum operator and the ordinary quantum mechanical inequality $\sigma_p \Delta x \ge \pi \hbar$. Extending these ideas to three (and $d$) dimensions, we also derive an analogous lower bound for systems confined in higher-dimensional spherical regions of radius $R$.