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The quantum harmonic oscillator on a circle -- fragmentation of the algebraic method

arXiv Quantum Physics
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⚡ Quantum Brief
Physicists Daniel Burgarth and Paolo Facchi challenge a foundational quantum assumption by examining a particle confined to a circular path under a quadratic potential, revealing non-harmonic energy spectra despite algebraic similarities to the standard harmonic oscillator. The study identifies a critical flaw in the traditional algebraic method, which predicts integer energy gaps but fails here due to topological constraints imposed by the circular geometry, breaking conventional quantum mechanical expectations. Their analysis uncovers unexpected physical phenomena in this deceptively simple model, suggesting broader implications for quantum systems with non-trivial boundary conditions or curved spaces. The findings bridge algebraic quantum mechanics and geometric effects, offering insights into how spatial constraints can fundamentally alter energy quantization, even in systems resembling well-understood oscillators. Published in March 2026, this work highlights how fundamental quantum principles may require revision when applied to non-Euclidean or constrained geometries, opening new avenues for theoretical exploration.
The quantum harmonic oscillator on a circle -- fragmentation of the algebraic method

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Quantum Physics arXiv:2603.23774 (quant-ph) [Submitted on 24 Mar 2026] Title:The quantum harmonic oscillator on a circle -- fragmentation of the algebraic method Authors:Daniel Burgarth, Paolo Facchi View a PDF of the paper titled The quantum harmonic oscillator on a circle -- fragmentation of the algebraic method, by Daniel Burgarth and 1 other authors View PDF HTML (experimental) Abstract:A quantum particle on a circle in a quadratic potential exhibits a spectrum that is not harmonic, despite having all algebraic properties of the quantum harmonic oscillator. This raises the question where the usual algebraic argument -- implying integer gaps -- fails. The answer is illuminating and covers a surprisingly rich range of physical phenomena for such a simple model. Comments: Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2603.23774 [quant-ph] (or arXiv:2603.23774v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2603.23774 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Daniel Burgarth [view email] [v1] Tue, 24 Mar 2026 23:13:17 UTC (845 KB) Full-text links: Access Paper: View a PDF of the paper titled The quantum harmonic oscillator on a circle -- fragmentation of the algebraic method, by Daniel Burgarth and 1 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-03 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?)

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Source: arXiv Quantum Physics