Position--momentum uncertainty relation for the average confidence width
Abstract
We introduce the average confidence width Δa x=∫01Δc x (θ) dθ: the confidence width Δc x(θ)---the measure of the smallest region of position space carrying probability θ---averaged over all confidence levels. It is the first moment of the decreasing rearrangement of the position density, an L1 (mean-absolute-deviation) measure of localization, and Δa x Δa p is dilation invariant. We study the uncertainty relation Δa x Δa p≥ c∗ℏ for pure and mixed states. A mean--entropy argument with the Bialynicki-Birula--Mycielski relation gives c∗≥π/e, while the ground state ψ0 of the Fourier-invariant operator |x|+|p|, with E0=1.1040744, gives c∗≤ E02≈1.2190<4/π: Gaussian states, which saturate the Heisenberg--Kennard and entropic relations, are not optimal here. We prove that ψ0 is nondegenerate, nodeless, even, self-dual, and symmetric decreasing, with algebraic tails, and that the sharp hybrid relations Δa x · 2 ⟨ |p−p0| ⟩ ≥ E02ℏ and 2⟨|x−x0| ⟩ ·Δa p≥ E02ℏ hold for all states, so Δa x Δa p≥ E02ℏ whenever either density is symmetric unimodal; the second variation of Δa x Δa p at ψ0 is positive definite apart from the symmetry directions. Extensive numerical minimization, including coherent states, never finds a product below E02ℏ and consistently returns ψ0, supporting the conjecture c∗=E02. In two dimensions the relation is sharp and saturated by Gaussians, identifying the dimension as the origin of the non-Gaussian optimizer in one dimension. Finally, we discuss other complementary pairs and the relation of the width to interferometric predictability and visibility.
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BibTeX
@misc{wu2026positionmomentum,
title = {{Position--momentum uncertainty relation for the average confidence width}},
author = {Shengjun Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.31443},
eprint = {2606.31443},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.31443},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Position--momentum uncertainty relation for the average confidence width
AU - Shengjun Wu
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2606.31443
UR - https://arxiv.org/abs/2606.31443
AB - We introduce the average confidence width Δ_a x=∫_0^1Δ_c x (θ) dθ: the confidence width Δ_c x(θ)---the measure of the smallest region of position space carrying probability θ---averaged over all confidence levels. It is the first moment of the decreasing rearrangement of the position density, an L^1 (mean-absolute-deviation) measure of localization, and Δ_a x Δ_a p is dilation invariant. We study the uncertainty relation Δ_a x Δ_a p≥ c^∗ℏ for pure and mixed states. A mean--entropy argument with the Bialynicki-Birula--Mycielski relation gives c^∗≥π/e, while the ground state ψ_0 of the Fourier-invariant operator |x|+|p|, with E_0=1.1040744, gives c^∗≤ E_0^2≈1.2190<4/π: Gaussian states, which saturate the Heisenberg--Kennard and entropic relations, are not optimal here. We prove that ψ_0 is nondegenerate, nodeless, even, self-dual, and symmetric decreasing, with algebraic tails, and that the sharp hybrid relations Δ_a x · 2 ⟨ |p−p_0| ⟩ ≥ E_0^2ℏ and 2⟨|x−x_0| ⟩ ·Δ_a p≥ E_0^2ℏ hold for all states, so Δ_a x Δ_a p≥ E_0^2ℏ whenever either density is symmetric unimodal; the second variation of Δ_a x Δ_a p at ψ_0 is positive definite apart from the symmetry directions. Extensive numerical minimization, including coherent states, never finds a product below E_0^2ℏ and consistently returns ψ_0, supporting the conjecture c^∗=E_0^2. In two dimensions the relation is sharp and saturated by Gaussians, identifying the dimension as the origin of the non-Gaussian optimizer in one dimension. Finally, we discuss other complementary pairs and the relation of the width to interferometric predictability and visibility.
ER -