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Riccati Equations as a Scale-Relativistic Gateway to Quantum Mechanics

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Abstract

Applying the resolution–scale relativity principle to develop a mechanics of non-differentiable dynamical paths, we find that, in one dimension, stationary motion corresponds to an Itô process driven by the solutions of a Riccati equation. We verify that the corresponding Fokker–Planck equation is solved for a probability density corresponding to the squared modulus of the solution of the Schrödinger equation for the same problem. Inspired by the treatment of the one-dimensional case, we identify a generalization to time dependent problems in any number of dimensions. The Itô process is then driven by a function which is identified as establishing the link between non-differentiable dynamics and standard quantum mechanics. This is the basis for the scale relativistic interpretation of standard quantum mechanics and, in the case of applications to chaotic systems, it leads us to identify quantum-like states as characterizing the entire system rather than the motion of its individual constituents.

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References

  1. 1.

    Al Rashid, S.N.T.: Some applications of the scale relativity theory in quantum physics. Ph.D thesis, Al-Mustansiriyah University, Baghdad, Iraq (2006)

  2. 2.

    Al Rashid, S.N.T.: Numerical simulations of particle in a double oscillators. J. Al-Anbar Univ. Pure Sci. 1(3), 86–95 (2007)

  3. 3.

    Al Rashid, S.N.T., Habeeb, M.A.Z., Ahmed, K.A.: Application of scale relativity (ScR) theory to the problem of a particle in a finite one-dimensional square well (FODSW) potential. J. Quantum Inf. Sci. 1, 7–1, (2011). https://doi.org/10.4236/jqis.2011.11002

  4. 4.

    Bohm, D.: A suggested interpretation of the quantum theory in terms of hidden variables. Phys. Rev. 85, 166–179 (1952)

  5. 5.

    Bonilla, M., Rosas-Ortiz, O.: The harmonic oscillator in the framework of scale relativity. IOP Conf. Ser.: J. Phys.: Conf. Ser. 839, 012009 (2017)

  6. 6.

    Haley, S.B.: An underrated entanglement: Riccati and Schrödinger equations. AJP 65, 237 (1997)

  7. 7.

    Hermann, R.P.: Numerical simulation of a quantum patrical in a box. J. Phys. A 30, 3967–3975 (1997)

  8. 8.

    Hermann, R., Schumacher, G., Guyard, R.: Scale relativity and quantization of the solar system. Orbit quantization of the planet’s satellites. Astron. Astrophys. 335, 281:286 (1998)

  9. 9.

    LeBohec, S.: Scale relativistic signature in the Brownian motion of micro-spheres in optical traps. Int. J. Mod. Phys. A 32(26), 1750156 (2017)

  10. 10.

    Madelung, E.: Quantentheorie in hydrodynamischer form. Z. Phys. 40, 322 (1927) (See http://www.neo-classical-physics.info/uploads/3/0/6/5/3065888/madelung_-_hydrodynamical_interp..pdf for a translation by D.H. Delphenich)

  11. 11.

    McClendon, M., Rabitz, H.: Numerical simulations in stochastic mechanics. Phys. Rev. A 37, 3479 (1988)

  12. 12.

    Teh, M.-H., Nottale, L., LeBohec, S.: Resolution-scale relativistic formulation of non-differentiable mechanics. Eur. Phys. J. Plus 134, 438 (2019)

  13. 13.

    Nelson, E.: Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev. 150, 1079 (1966)

  14. 14.

    Nottale, L.: Fractal Space-Time and Microphysics. World Scientific Publishing Company, Singapore (1993). ISBN: 978-981-02-0878-3

  15. 15.

    Nottale, L., Schumacher, G., Gay, J.: Scale relativity and quantization of the solar system. Astron. Astrophys. 322, 1018–1025 (1997)

  16. 16.

    Nottale, L.: Scale-relativity and quantization of exoplanet orbital semi-major axes. Astron. Astrophys. 361, 379–387 (2000)

  17. 17.

    Nottale, L., Célérier, M.-N.: Derivation of the postulates of quantum mechanics from the first principles of scale relativity. J. Phys. A 40, 14471–14498 (2007)

  18. 18.

    Nottale, L.: Generalized quantum potentials. J. Phys A: Math 42, 275306 (2009)

  19. 19.

    Nottale, L.: Scale Relativity And Fractal Space-Time: A New Approach to Unifying Relativity and Quantum Mechanics. World Scientific Publishing Company, Singapore (2011). ISBN: 978-1-84816-650-9

  20. 20.

    Nowakowski, M., Rosu, H.C.: Newton’s laws of motion in form of Riccati equation. Phys.Rev. 202, E65 (2002) 047602, arXiv:physics/0110066

  21. 21.

    Rogers, G.W.: Riccati equations and perturbation expansions in quantum mechanics. J. Math. Phys. 26(14), 567–575 (1985)

  22. 22.

    Reid, W.T.: Riccati Differential Equations. Academic Press, New York (1972). ISBN 978-0124110861

  23. 23.

    Schuch, D.: Nonlinear Riccati equations as unifying link between linear quantum mechanics and other fields of physics. J. Phys.: Conf. Ser. (2014). https://doi.org/10.1088/1742-6596/504/1/012005

  24. 24.

    Wheeler, N.: Quantum applications of the Riccati equation. Notes, Reed College Physics Department, (June 2004)

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Acknowledgements

S. LeBohec is grateful to Yong-Shi Wu for his helpful conversation and to Dirk Pützfeld for his attentive reading and corrections of the manuscript.

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Correspondence to Tugdual S. LeBohec.

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Al-Rashid, S.N.T., Habeeb, M.A.Z. & LeBohec, T.S. Riccati Equations as a Scale-Relativistic Gateway to Quantum Mechanics. Found Phys (2020). https://doi.org/10.1007/s10701-020-00324-w

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