Astron. Astrophys. 337, 299-310 (1998)

Macroscopic Schroedinger quantization of the early chaotic solar system

Gilbert Reinisch

Observatoire de la Côte d'Azur, BP 4229, Département Fresnel, F-06304 Nice Cedex 04, France

Received 12 August 1997 / Accepted 20 May 1998

Abstract

Classical mechanics is compatible with Schroedinger's equation without any approximation of the standard WKB semiclassical quantum type. In order to illustrate this property, we propose an exact Schroedinger theory for Nottale's macro-quantization of the solar system in which neither semiclassical approximations nor the probabilistic Born postulate are assumed. We point out the existence and properties of radial-velocity proto-planetary solitons that describe the swarm of chaotic orbits supposed to dominate at the later stage of the solar system formation, in accordance with Nottale's scale-relativity theory.

Key words: relativity gravitation chaos quantization

This article contains no SIMBAD objects.

Contents

• 1. Introduction
• 2. Quantum probability and complex wavefunctions
  • 2.1. Continuity equation
  • 2.2. Divergence of for stationary states
  • 2.3. Schroedinger equation's irregular eigenmode
  • 2.4. Schroedinger equation's Hamilton-Jacobi dynamics
  • 2.5. Schroedinger equation's new parameter
• 3. Madelung's central-field problem
  • 3.1. Micro-versus macroscopic quantization
  • 3.2. The Ermakov-Milne-Pinney (EMP) nonlinear differential equation
• 4. Classical interpretation of Schroedinger equation
  • 4.1. Milne's nonlinear superposition formula
  • 4.2. Hamilton-Jacobi definition of the radial matter flow
  • 4.3. Schroedinger equation and classical mechanics
  • 4.4. Choice of initial conditions
  • 4.5. Classical quantization
• 5. The Kepler-Madelung system
  • 5.1. The nonlinear radial EMP equation
  • 5.2. The pseudo-circular case
• 6. Schroedinger equation on the scale of the solar system
  • 6.1. Radial-velocity solitions
  • 6.2. Quantized patterns
  • 6.3. Conjecture about planetary orbit quantization
• 7. Conclusion
• Acknowledgements
• References

© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
helpdesk.link@springer.de