Phase space representation for the inversely quadratic Hellmann-Kratzer potential

Phase space representation for the inversely quadratic Hellmann-Kratzer potential | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Phase space representation for the inversely quadratic Hellmann-Kratzer potential Othmane Cherroud, Sid-Ahmed Yahiaoui This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9007658/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 8 You are reading this latest preprint version Abstract In this work, we develop a phase space formulation of quantum mechanics to investigate the inversely quadratic Hellmann-Kratzer (IQHK) potential, with the aim of simultaneously constructing the Wigner distribution and the corresponding characteristic functions. By employing Weyl transformations, we establish two independent computational approaches, each yielding explicit and generalized analytical expressions for higher-order moments and momentum. In addition, we derive a generalized analytical formulation of the momentum operator and establish a corresponding generalized expression of Heisenberg’s uncertainty principle, explicitly dependent on the quantum numbers n and L. This constitutes a novel and complementary contribution to the study of the IQHK potential. Moreover, our approach enables the identification of a dimensionless principal parameter β in phase-space transformations, which further refines the generalized analytical expressions for the energy levels En,L. These energy levels are numerically evaluated and exhibit excellent agreement with previously published results for all special cases of the IQHK potential across various diatomic molecular systems. Furthermore, we verify that our results are consistent with the canonical Heisenberg-Born-Jordan-Dirac commutation relations and Heisenberg’s uncertainty principle, thereby confirming the robustness and reliability of the proposed methodology within this framework. Phase space representation Hellmann-Kratzer potential Higher-order moments Expectation values Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 07 Apr, 2026 Reviewers agreed at journal 10 Mar, 2026 Reviews received at journal 04 Mar, 2026 Reviewers agreed at journal 03 Mar, 2026 Reviewers invited by journal 03 Mar, 2026 Editor assigned by journal 03 Mar, 2026 Submission checks completed at journal 03 Mar, 2026 First submitted to journal 02 Mar, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9007658","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":601440912,"identity":"1f81c191-f6f4-47f5-aebb-716602204867","order_by":0,"name":"Othmane Cherroud","email":"data:image/png;base64,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","orcid":"","institution":"University Yahia Fares of Medea","correspondingAuthor":true,"prefix":"","firstName":"Othmane","middleName":"","lastName":"Cherroud","suffix":""},{"id":601440913,"identity":"945b9d00-e5ad-4feb-ba6e-da7fea9343ab","order_by":1,"name":"Sid-Ahmed Yahiaoui","email":"","orcid":"","institution":"Blida 1 University","correspondingAuthor":false,"prefix":"","firstName":"Sid-Ahmed","middleName":"","lastName":"Yahiaoui","suffix":""}],"badges":[],"createdAt":"2026-03-02 08:38:59","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9007658/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9007658/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104403223,"identity":"d9be303f-18df-42fe-a0f2-7c653012aa25","added_by":"auto","created_at":"2026-03-11 12:17:46","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":968520,"visible":true,"origin":"","legend":"","description":"","filename":"FewBodySystemsHellmannKratzerpotential.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9007658/v1_covered_67b8c757-896f-46d0-9317-13fe44f75b16.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Phase space representation for the inversely quadratic Hellmann-Kratzer potential","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"few-body-systems","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"fbsy","sideBox":"Learn more about [Few-Body Systems](http://link.springer.com/journal/601)","snPcode":"601","submissionUrl":"https://submission.nature.com/new-submission/601/3","title":"Few-Body Systems","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Phase space representation, Hellmann-Kratzer potential, Higher-order moments, Expectation values","lastPublishedDoi":"10.21203/rs.3.rs-9007658/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9007658/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"In this work, we develop a phase space formulation of quantum mechanics to investigate the inversely quadratic Hellmann-Kratzer (IQHK) potential, with the aim of simultaneously constructing the Wigner distribution and the corresponding characteristic functions. 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