Structural Origin of Square-Root Mass Scaling inQuantum Tunnelling

Structural Origin of Square-Root Mass Scaling inQuantum Tunnelling | 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 Structural Origin of Square-Root Mass Scaling inQuantum Tunnelling Krishna Kingkar Pathak This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8813754/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Quantum tunnelling phenomena across physics and chemistry are governed by an exponential sensitivity to the effective mass associated with the tunnelling coordinate. Within semiclassical WKB and instanton formalisms, this dependence takes a characteristic form, $\ln \mathcal{R} \propto -\sqrt{M_{\mathrm{eff}}}$, arising from the factorisation of the Euclidean action into mass-dependent and geometric contributions. Although this scaling is well established at the level of theory, its status as a unifying structural principle across distinct physical regimes has not been systematically clarified. In this work, we examine the extent to which the square-root effective-mass dependence functions as a universal semiclassical descriptor of tunnelling across radically different physical platforms. By reanalysing representative experimental datasets from molecular isotopologues, collective tunnelling in ultracold atomic systems, and macroscopic quantum escape in Josephson circuits, we demonstrate that the leading semiclassical trend persists over more than twenty orders of magnitude in tunnelling rate. When expressed in terms of $\ln \mathcal{R}$ versus $\sqrt{M_{\mathrm{eff}}}$, all platforms exhibit near-linear behaviour, indicating that the dominant contribution to the tunnelling exponent is controlled by a common Euclidean-action structure rather than system-specific microscopic details. We further identify the physical mechanisms—finite temperature, dissipation, many-body effects, and mass-dependent barrier reshaping—that delimit the regime of validity of this leading-order scaling. Taken together, the results support the view that the $\sqrt{M_{\mathrm{eff}}}$ dependence reflects a form of mechanical similarity in imaginary time, providing an ontologically neutral and cross-domain organising principle for quantum tunnelling phenomena from molecular systems to engineered macroscopic quantum devices. quantum Tunnelling instanton theory cold atoms Josephson junctions nonlinear scaling Full Text Additional Declarations No competing interests reported. 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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-8813754","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":588021352,"identity":"45762a8a-62d1-4f3b-9c63-4a8bf2bc5cb3","order_by":0,"name":"Krishna Kingkar Pathak","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA2ElEQVRIiWNgGAWjYBACCR7GBgaGAwwM/CBeQgEpWiQbQFoMiNICIoFaDA6AGMRokew53Pbwyxm7fOPzqxM/PDBgkOcXO4BfizRvY7uxzI1ky2033m6WADrMcObsBPxa5PgZ26QlPjAbmN04uwGkJcHgNnFa6g2MZ5zd/IMoLUCHtUl+uHHYwIC/dxtxtkj2HGyTZjhz3EDiBu82iwQDCcJ+kTiT/kzyx7FqA/7+s5tv/qiwkeeXJqAFBJjBcSMBVilBWDkIMP4AkfwHiFM9CkbBKBgFIw8AABq/RV+gR3AbAAAAAElFTkSuQmCC","orcid":"","institution":"Arya Vidyapeeth College","correspondingAuthor":true,"prefix":"","firstName":"Krishna","middleName":"Kingkar","lastName":"Pathak","suffix":""}],"badges":[],"createdAt":"2026-02-07 08:40:06","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8813754/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8813754/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":103832295,"identity":"28759fe4-7e80-4f0c-be1e-61024a59bf20","added_by":"auto","created_at":"2026-03-03 12:56:26","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":277862,"visible":true,"origin":"","legend":"","description":"","filename":"FOPmain.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8813754/v1_covered_63312ec4-697c-4384-8283-298d43818b24.pdf"},{"id":102362439,"identity":"52b10580-96db-4261-b23c-176ceeb58ce6","added_by":"auto","created_at":"2026-02-11 00:42:00","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":212249,"visible":true,"origin":"","legend":"","description":"","filename":"SIFOP.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8813754/v1/43ac340a0f16fca6bdd44b4b.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Structural Origin of Square-Root Mass Scaling inQuantum Tunnelling","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"quantum Tunnelling, instanton theory, cold atoms, Josephson junctions, nonlinear scaling","lastPublishedDoi":"10.21203/rs.3.rs-8813754/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8813754/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Quantum tunnelling phenomena across physics and chemistry are governed by an exponential sensitivity to the effective mass associated with the tunnelling coordinate. Within semiclassical WKB and instanton formalisms, this dependence takes a characteristic form, $\\ln \\mathcal{R} \\propto -\\sqrt{M_{\\mathrm{eff}}}$, arising from the factorisation of the Euclidean action into mass-dependent and geometric contributions. Although this scaling is well established at the level of theory, its status as a unifying structural principle across distinct physical regimes has not been systematically clarified.\nIn this work, we examine the extent to which the square-root effective-mass dependence functions as a universal semiclassical descriptor of tunnelling across radically different physical platforms. By reanalysing representative experimental datasets from molecular isotopologues, collective tunnelling in ultracold atomic systems, and macroscopic quantum escape in Josephson circuits, we demonstrate that the leading semiclassical trend persists over more than twenty orders of magnitude in tunnelling rate. When expressed in terms of $\\ln \\mathcal{R}$ versus $\\sqrt{M_{\\mathrm{eff}}}$, all platforms exhibit near-linear behaviour, indicating that the dominant contribution to the tunnelling exponent is controlled by a common Euclidean-action structure rather than system-specific microscopic details.\nWe further identify the physical mechanisms—finite temperature, dissipation, many-body effects, and mass-dependent barrier reshaping—that delimit the regime of validity of this leading-order scaling. 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