Modified Heisenberg Commutation Relations, Free Schrödinger Equations, Tunnel Effect and Its Connections with the Black–Scholes Equation

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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–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 x in quantum mechanics with the underlying asset S, 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.