Effective Geometry and Position-Dependent Mass in Dual-$q$ Quantum Mechanics
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Quantum Physics arXiv:2606.12444 (quant-ph) [Submitted on 27 May 2026] Title:Effective Geometry and Position-Dependent Mass in Dual-$q$ Quantum Mechanics Authors:A. Boumali, A. Makhlouf View a PDF of the paper titled Effective Geometry and Position-Dependent Mass in Dual-$q$ Quantum Mechanics, by A. Boumali and A. Makhlouf View PDF HTML (experimental) Abstract:This work investigates the deformed-derivative formalism introduced by Borges, with emphasis on the relation between the linear operator $D_{(q)}$ and its nonlinear dual counterpart $D^{(q)}$. Directly inserting the dual derivative into the kinetic term leads to a nonlinear Schrödinger equation and obscures the usual interpretation of superposition and probability. We show that this nonlinearity can be removed by a simultaneous transformation of the coordinate and of the wave function. The transformed problem is an ordinary linear Schrödinger equation in a deformed coordinate, and its representation in the physical coordinate is equivalent to a Hermitian position-dependent-mass (PDM) Hamiltonian. In this formulation, the deformation parameter $q$ determines both the effective mass profile and the associated metric. The formalism is applied to the free particle, the infinite square well, the rectangular barrier, and the harmonic oscillator in the weak-deformation regime. Comparison with the nonadditive-translation approach of Costa Filho \emph{et al.} shows that the Borges dual-$q$ framework provides an alternative route to the same effective geometric structure. For $q1$, the effective length is increased, which lowers the spectrum and suppresses tunneling relative to the undeformed limit $q=1$. Subjects: Quantum Physics (quant-ph); High Energy Physics - Theory (hep-th) Cite as: arXiv:2606.12444 [quant-ph] (or arXiv:2606.12444v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2606.12444 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Abdelmalek Boumali [view email] [v1] Wed, 27 May 2026 11:29:44 UTC (449 KB) Full-text links: Access Paper: View a PDF of the paper titled Effective Geometry and Position-Dependent Mass in Dual-$q$ Quantum Mechanics, by A. Boumali and A. MakhloufView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-06 Change to browse by: hep-th 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?) 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?)