Wave-particle duality as an uncertainty relation for the average confidence width

Abstract

We introduce the average confidence width $Δ_a x=\int_0^1 Δ_c x (θ_x) d θ_x$: the confidence width $Δ_c x(θ_x)$ -- the smallest position interval carrying a fraction $θ_x$ of the probability -- averaged over all levels. It is the first moment of the decreasing rearrangement of $|ψ|^2$, an $L^1$ mean-absolute-deviation measure of localization, so the product $Δ_{a} x\,Δ_{a} p$ is dilation invariant and obeys $Δ_{a} x\,Δ_{a} p\ge c\,\hbar$. Reading $1/Δ_{a} x$ as a particle character and $1/Δ_{a} p$ as a wave character, this lower bound on combined spread is identically an upper bound on combined particle-and-wave character: uncertainty and wave-particle duality are two faces of one inequality. A mean-entropy argument with the Bialynicki-Birula-Mycielski relation gives the rigorous $c\geπ/e$, while the achievable constant $c^\ast$ is set by the ground state of the Fourier-invariant operator $|x|+|p|$, $c^\ast\le E_0^2\approx 1.217$. Hence $π/e\le c^\ast\le E_0^2<4/π$: the optimal state is sub-Gaussian, so the Gaussian -- optimal for the Heisenberg and entropic relations -- is not the duality optimum.