Riccati Equations as a Scale-Relativistic Gateway to Quantum Mechanics

  title={Riccati Equations as a Scale-Relativistic Gateway to Quantum Mechanics},
  author={Saeed N. T. Al-Rashid and Mohammed A. Z. Habeeb and Tugdual LeBohec},
  journal={Foundations of Physics},
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… Expand


Resolution-scale relativistic formulation of non-differentiable mechanics
Abstract.This article motivates and presents the scale relativistic approach to non-differentiability in mechanics and its relation to quantum mechanics. It stems from the scale relativity proposalExpand
Generalized quantum potentials
We first recall that the system of fluid mechanics equations (Euler and continuity) that describes a fluid in irrotational motion subjected to a generalized quantum potential (depending on a constantExpand
Nonlinear Riccati equations as a unifying link between linear quantum mechanics and other fields of physics
Theoretical physics seems to be in a kind of schizophrenic state. Many phenomena in the observable macroscopic world obey nonlinear evolution equations, whereas the microscopic world is governed byExpand
Scale Relativistic signature in the Brownian motion of micro-spheres in optical traps
The development of mechanics of nondifferentiable paths36 suggested by Scale Relativity31,32 results in a foundation of Quantum Mechanics30,37 including Schrodinger’s equation and all the otherExpand
Scale Relativity and Fractal Space-Time: A New Approach to Unifying Relativity and Quantum Mechanics
This book provides a comprehensive survey of the development of the theory of scale relativity and fractal space-time. It suggests an original solution to the disunified nature of theExpand
Derivation of the Schrodinger equation from Newtonian mechanics
We examine the hypothesis that every particle of mass $m$ is subject to a Brownian motion with diffusion coefficient $\frac{\ensuremath{\hbar}}{2m}$ and no friction. The influence of an externalExpand
Application of Scale Relativity (ScR) Theory to the Problem of a Particle in a Finite One-Dimensional Square Well (FODSW) Potential
This work presents a new example where scale relativity theory, based on a fractal space-time concept, can accurately reproduce quantum mechanical results without invoking the Schrodinger equation. Expand
Riccati equations and perturbation expansions in quantum mechanics
General perturbation expansions, which allow corrections to any order to be written in quadrature, are presented for Riccati and other nonlinear first‐order equations. These results are valid forExpand
Newton's laws of motion in the form of a Riccati equation.
  • M. Nowakowski, H. Rosu
  • Mathematics, Medicine
  • Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2002
An analogy to barotropic Friedmann-Robertson-Lemaitre cosmology is shown where the expansion of the universe can be also shown to obey a Riccati equation. Expand
Derivation of the postulates of quantum mechanics from the first principles of scale relativity
Quantum mechanics is based on a series of postulates which lead to a very good description of the microphysical realm but which have, up to now, not been derived from first principles. In the presentExpand