Modified Heisenberg Commutation Relations, Free Schrödinger Equations, Tunnel Effect and Its Connections with the Black–Scholes Equation
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 ana...
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2025-01-01
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| author | Mauricio Contreras González Roberto Ortiz Herrera José González Suárez |
| author_facet | Mauricio Contreras González Roberto Ortiz Herrera José González Suárez |
| author_sort | Mauricio Contreras González |
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| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. 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–Scholes model in finance. In fact, by taking a particular form of a linear position deformation, one can derive a Black–Scholes 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 <i>x</i> in quantum mechanics with the underlying asset <i>S</i>, which in finance satisfies stochastic dynamics, this analogy implies that the Black–Scholes 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’s 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–Scholes equation becomes a possible simple model for understanding quantum gravity. |
| format | Article |
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| institution | Kabale University |
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| language | English |
| publishDate | 2025-01-01 |
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| series | Axioms |
| spelling | doaj-art-4b4a698990b9435ca392a11794e014902025-01-24T13:22:18ZengMDPI AGAxioms2075-16802025-01-011416010.3390/axioms14010060Modified Heisenberg Commutation Relations, Free Schrödinger Equations, Tunnel Effect and Its Connections with the Black–Scholes EquationMauricio Contreras González0Roberto Ortiz Herrera1José González Suárez2Departamento de Física, Facultad de Ciencias Básicas, Universidad Metropolitana de Ciencias de la Educación (UMCE), Santiago 7760197, ChileFacultad de Ingeniería, Universidad Diego Portales, Santiago 8370191, ChileDepartamento de Física y Astronomía, Universidad Andres Bello, Sazié 2212, ChileThis 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. 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–Scholes model in finance. In fact, by taking a particular form of a linear position deformation, one can derive a Black–Scholes 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 <i>x</i> in quantum mechanics with the underlying asset <i>S</i>, which in finance satisfies stochastic dynamics, this analogy implies that the Black–Scholes 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’s 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–Scholes equation becomes a possible simple model for understanding quantum gravity.https://www.mdpi.com/2075-1680/14/1/60modified Heisenberg commutation relationsquantum mechanicstunnel effectBlack–Scholes equationeconophysics |
| spellingShingle | Mauricio Contreras González Roberto Ortiz Herrera José González Suárez Modified Heisenberg Commutation Relations, Free Schrödinger Equations, Tunnel Effect and Its Connections with the Black–Scholes Equation Axioms modified Heisenberg commutation relations quantum mechanics tunnel effect Black–Scholes equation econophysics |
| title | Modified Heisenberg Commutation Relations, Free Schrödinger Equations, Tunnel Effect and Its Connections with the Black–Scholes Equation |
| title_full | Modified Heisenberg Commutation Relations, Free Schrödinger Equations, Tunnel Effect and Its Connections with the Black–Scholes Equation |
| title_fullStr | Modified Heisenberg Commutation Relations, Free Schrödinger Equations, Tunnel Effect and Its Connections with the Black–Scholes Equation |
| title_full_unstemmed | Modified Heisenberg Commutation Relations, Free Schrödinger Equations, Tunnel Effect and Its Connections with the Black–Scholes Equation |
| title_short | Modified Heisenberg Commutation Relations, Free Schrödinger Equations, Tunnel Effect and Its Connections with the Black–Scholes Equation |
| title_sort | modified heisenberg commutation relations free schrodinger equations tunnel effect and its connections with the black scholes equation |
| topic | modified Heisenberg commutation relations quantum mechanics tunnel effect Black–Scholes equation econophysics |
| url | https://www.mdpi.com/2075-1680/14/1/60 |
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