Classical and quantum dynamics of a particle confined in a paraboloidal cavity
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Quantum Physics arXiv:2512.08021 (quant-ph) [Submitted on 8 Dec 2025] Title:Classical and quantum dynamics of a particle confined in a paraboloidal cavity Authors:Ángel E. Reyna-Cruz, Julio C. Gutiérrez-Vega View a PDF of the paper titled Classical and quantum dynamics of a particle confined in a paraboloidal cavity, by \'Angel E. Reyna-Cruz and 1 other authors View PDF HTML (experimental) Abstract:We present a classical and quantum analysis of a particle confined in a three-dimensional paraboloidal cavity formed by two confocal paraboloids. Classically, the system is integrable and presents three independent constants of motion, namely, the energy, the $z$-component of the angular momentum, and a third dynamical constant associated with the paraboloidal geometry, which can be derived from the separability of the Hamilton--Jacobi equation. We derive closed-form analytical expressions for the actions, which allow us to determine the two conditions to get periodic closed trajectories. We classify these trajectories through the indices $(s,t,\ell)$. The caustic paraboloids that bound the motion provide a complete geometric characterization of admissible trajectories. Quantum mechanically, separability of the Schrödinger equation in parabolic coordinates yields eigenmodes described by Whittaker functions. We determine the energy spectrum and identify degeneracies arising not only from azimuthal symmetry but also from specific cavity deformations. A direct correspondence between classical trajectories and quantum eigenstates reveals that probability densities concentrate in the classically allowed region with controlled penetration into forbidden zones. Comments: Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Classical Physics (physics.class-ph) Cite as: arXiv:2512.08021 [quant-ph] (or arXiv:2512.08021v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2512.08021 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Ángel Emiliano Reyna Cruz [view email] [v1] Mon, 8 Dec 2025 20:31:33 UTC (11,183 KB) Full-text links: Access Paper: View a PDF of the paper titled Classical and quantum dynamics of a particle confined in a paraboloidal cavity, by \'Angel E. Reyna-Cruz and 1 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2025-12 Change to browse by: math math-ph math.MP physics physics.class-ph References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) Links to Code Toggle Papers with Code (What is Papers with Code?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)