{
  "abstract": "We introduce the average confidence width $Δ_{a} x=\\int_0^1Δ_{c} x (θ)\\,\\mathrm{d}θ$: 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 $Δ_{a} x\\,Δ_{a} p\\ge c^\\ast\\hbar$ for pure and mixed states. A mean--entropy argument with the Bialynicki-Birula--Mycielski relation gives $c^\\ast\\geπ/e$, while the ground state $ψ_0$ of the Fourier-invariant operator $|x|+|p|$, with $E_0=1.1040744$, gives $c^\\ast\\le E_0^2\\approx1.2190<4/π$: 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 $Δ_{a} x \\cdot 2 \\langle |p-p_0| \\rangle \\ge E_0^2\\hbar$ and $2\\langle|x-x_0| \\rangle \\cdotΔ_{a} p\\ge E_0^2\\hbar$ hold for all states, so $Δ_{a} x\\,Δ_{a} p\\ge E_0^2\\hbar$ 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 $E_0^2\\hbar$ and consistently returns $ψ_0$, supporting the conjecture $c^\\ast=E_0^2$. 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.31443",
  "author_total": 1,
  "authors": [
    {
      "name": "Shengjun Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4t4hm1gx9k6d4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.31443",
  "fulltext_url": null,
  "id": "4t4hm1gx9k6d4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.31443",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "3107|Quantum Mechanics and Applications"
    ]
  ],
  "openalex": "W7166886356",
  "pdf_url": "https://arxiv.org/pdf/2606.31443",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4t4hm1gx9k6d4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:39Z",
      "native_id": "2606.31443",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.31443"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Position--momentum uncertainty relation for the average confidence width",
  "title_text": "Position--momentum uncertainty relation for the average confidence width",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/4t4hm1gx9k6d4",
  "venue": "arXiv",
  "versions": []
}