{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,4]],"date-time":"2025-12-04T18:50:09Z","timestamp":1764874209168,"version":"build-2065373602"},"reference-count":56,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,15]],"date-time":"2025-01-15T00:00:00Z","timestamp":1736899200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper explores the implications of modifying the canonical Heisenberg commutation relations over two simple systems, such as the free particle and the tunnel effect generated by a step-like potential. The modified commutation relations include position-dependent and momentum-dependent terms analyzed separately. For the position deformation case, the corresponding free wave functions are sinusoidal functions with a variable wave vector k(x). In the momentum deformation case, the wave function has the usual sinusoidal behavior, but the energy spectrum becomes non-symmetric in terms of momentum. Tunneling probabilities depend on the deformation strength for both cases. Also, surprisingly, the quantum mechanical model generated by these modified commutation relations is related to the Black\u2013Scholes model in finance. In fact, by taking a particular form of a linear position deformation, one can derive a Black\u2013Scholes equation for the wave function when an external electromagnetic potential is acting on the particle. In this way, the Scholes model can be interpreted as a quantum-deformed model. Furthermore, by identifying the position coordinate x in quantum mechanics with the underlying asset S, which in finance satisfies stochastic dynamics, this analogy implies that the Black\u2013Scholes equation becomes a quantum mechanical system defined over a random spatial geometry. If the spatial coordinate oscillates randomly about its mean value, the quantum particle\u2019s mass would correspond to the inverse of the variance of this stochastic coordinate. Further, because this random geometry is nothing more than gravity at the microscopic level, the Black\u2013Scholes equation becomes a possible simple model for understanding quantum gravity.<\/jats:p>","DOI":"10.3390\/axioms14010060","type":"journal-article","created":{"date-parts":[[2025,1,15]],"date-time":"2025-01-15T09:01:02Z","timestamp":1736931662000},"page":"60","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Modified Heisenberg Commutation Relations, Free Schr\u00f6dinger Equations, Tunnel Effect and Its Connections with the Black\u2013Scholes Equation"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6132-8032","authenticated-orcid":false,"given":"Mauricio","family":"Contreras Gonz\u00e1lez","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica, Facultad de Ciencias B\u00e1sicas, Universidad Metropolitana de Ciencias de la Educaci\u00f3n (UMCE), Santiago 7760197, Chile"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1940-3443","authenticated-orcid":false,"given":"Roberto","family":"Ortiz Herrera","sequence":"additional","affiliation":[{"name":"Facultad de Ingenier\u00eda, Universidad Diego Portales, Santiago 8370191, Chile"},{"name":"Facultad de Ciencias Econ\u00f3micas y Administrativas FACEA, Universidad Cat\u00f3lica de la Sant\u00edsima Concepci\u00f3n, Concepci\u00f3n 4070129, Chile"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2413-0681","authenticated-orcid":false,"given":"Jos\u00e9","family":"Gonz\u00e1lez Su\u00e1rez","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica y Astronom\u00eda, Universidad Andres Bello, Sazi\u00e9 2212, Chile"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"126001","DOI":"10.1088\/0034-4885\/78\/12\/126001","article-title":"A review of the generalized uncertainty principle","volume":"78","author":"Tawfik","year":"2015","journal-title":"Rep. 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