Relations between nonlinear Riccati equations and other equations in fundamental physics

16th Symmetries in Science

Schuch, Dieter^1

Institut für Theoretische Physik, J.W. Goethe-Universität Frankfurt Am Main, Max-von-Laue-Str. 1, Frankfurt am Main

D-60438, Germany^1

关键词: Bose-Einstein condensates;    Cosmological models;    Dissipative effects;    Fundamental physics;    Fundamental properties;    Nonlinear evolution equation;    Nonlinear formulation;    Supersymmetric quantum mechanics;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/538/1/012019/pdf
DOI  :  10.1088/1742-6596/538/1/012019

来源: IOP

【 摘 要 】

Many phenomena in the observable macroscopic world obey nonlinear evolution equations while the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. Linearizing nonlinear dynamics would destroy the fundamental property of this theory, however, it can be shown that quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown that the information about the dynamics of quantum systems with analytical solutions can not only be obtainable from the time-dependent Schrödinger equation but equally-well from a complex Riccati equation. Comparison with supersymmetric quantum mechanics shows that even additional information can be obtained from the nonlinear formulation. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation for any potential. Extension of the Riccati formulation to include irreversible dissipative effects is straightforward. Via (real and complex) Riccati equations, other fields of physics can also be treated within the same formalism, e.g., statistical thermodynamics, nonlinear dynamical systems like those obeying a logistic equation as well as wave equations in classical optics, Bose- Einstein condensates and cosmological models. Finally, the link to abstract "quantizations" such as the Pythagorean triples and Riccati equations connected with trigonometric and hyperbolic functions will be shown.