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 by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two world...

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Bibliographic Details
Published in:Journal of physics. Conference series 2014-01, Vol.504 (1), p.12005-11
Main Author: Schuch, Dieter
Format: Article
Language:English
Subjects:
Cosmology
Formalism
Linearity
Mathematical analysis
Nonlinear evolution equations
Nonlinearity
Physics
Quadratic programming
Quantum mechanics
Quantum physics
Riccati equation
Schrodinger equation
Schroedinger equation
Statistical thermodynamics
Theoretical physics
Time dependence
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Summary: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 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. I claim that linearity in quantum mechanics is not as essential as it apparently seems since quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown where complex Riccati equations appear in time-dependent quantum mechanics and how they can be treated and compared with similar space-dependent Riccati equations in supersymmetric quantum mechanics. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation. Finally, it will be shown that (real and complex) Riccati equations also appear in many other fields of physics, like statistical thermodynamics and cosmology.
ISSN:1742-6588
1742-6596
DOI:10.1088/1742-6596/504/1/012005