3. Madelung's central-field problem
For the sake of simplicity, we shall continue to make use of standard QM notations (hence the reduced quantum of action ). The translation of the results in terms of macroscopic quantization will simply be made in Sect. 7 by changing into (cf. Eq. (62)). Here µ is the (reduced) mass of the particle and is the Bohr radius of the system. The quantity has the dimension of a velocity. It is the major concept that is introduced by Nottale in his Scale-Relativistic treatment of the CPSS (Nottale 1993, 1996a,b, 1997). Recent observations of the Tifft effect suggest that it has an universal value quite close to (Guthrie & Napier, 1996; Nottale, 1996a).
3.1. Micro-versus macroscopic quantization
Chaotic planetary systems and atoms are not identical copies of the same thing on different scales, although the present paper aims to show that both may derive their respective physical properties from the same Schroedinger equation. In the case of the atom, the quantum of action, as an universal constant, does of course not depend on the mass, but the Bohr radius (where defines the the Keplerian central potential ) obviously does. In the case of the CPSS, the situation is reverse: the `macro quantum of action' is now the mass-dependent quantity while the `macro Bohr radius' yields as emphasized in Sect. 1, regardless of the planet that is under consideration. This simply means that the quantization will be made in terms of a velocity field describing the chaotic matter flow in the CPSS instead of a momentum field (cf Eqs. (63-64) and (69-72)). And this is indeed expected since the parallel we draw between the atom and the solar system has an obvious limit: while the orbiting particles in the atom (namely the electrons) have all the same mass, such is of course not the case for the planets of the solar system.
There is another huge difference that concerns the build-up of the CPSS with respect to standard atom theory. In first approximation, the way the atoms are built is simply sequentially filling up all the available quantum `boxes' (which are characterized by the three quantum numbers N, l and m) by couples of electrons, because of the spin (hence the periodic table). Once the lower-energy `quantum-boxes' are filled out, this very situation immediately affects the energy balance for the next available outer-electron couples which are to fill out the remaining boxes (obviously of higher energy) because these outer electrons do feel less Coulomb attraction from the nucleus, due to the partial electrostatic screening opposed by the already existing inner electrons.
Such is clearly not the case for the solar system where the inner orbiting planets do not seriously affect the gravitational attraction force between any outer planet and the sun. Moreover, although the planet spin and the Pauli exclusion principle have explicitely been considered in early investigations with the amazingly correct prediction that Venus' revolution should be retrograde (Barnothy, 1946), the interpretation of such spin quantization phenomenon in the planetary solar system remains problematic. Therefore, one should not be tempted to look for a sequential filling up of a sort of planet periodic table in the CPSS. The macroscopic Schroedinger theory should be indeed, as clearly stated by Nottale in his Scale-Relativity theory, a global theory of chaos: what will be quantized does not really concern individual (particle-like) physical properties such as energy, action and spin, as in (micro) QM; rather, it will mostly concern extended field properties such as the CPSS velocity field of the matter flow .
We now want to show that the independent planet histories occurring while they are being formed as stationary patterns in the chaotic stage of the proto solar system can be related to the phenomenon of static solitons . Indeed, solitons (and in the present case, one-dimensional radial-velocity solitons) are nonlinear fields which, due to their integrability by use of the inverse-scattering-transform, preserve their shape and, more generally, their identity during all their dynamical history, whatever the number of collisions, interactions and/or weak perturbations affecting them (Bullough & Caudrey 1980; Dodd et al., 1982; Reinisch, 1992). This dynamical stability is actually due to the one-to-one relationship between the set of soliton parameters and the eigenvalues of a corresponding (linear) ODE. In the following sub-sections, we now want to exhibit that relationship which is the hall-mark of the soliton phenomenon.
3.2. The Ermakov-Milne-Pinney (EMP) nonlinear differential equation
Consider a stationary quantum state described by its complex-valued wavefunction:
in agreement with Eq. (2) (where the amplitude and the phase S are now real-valued functions of the space variable only). Note that this ansatz agrees with Scale Relativity where in which is complex-valued (). The stationary Schroedinger equation corresponding to the applied potential and the energy eigenvalue E yields the following Madelung system of two coupled PDE's:
Now assume that is a confining central-field potential where r is the value of the radius. Using the spherical polar coordinates and taking into account both the single-valuedness of the wave function at and the spherical symmetry of the system (all meridian planes through the -axis are equally probable), we have:
where m is any integer. Then Eq. (5) becomes:
This equation defines the two following constants of integration:
while Eqs. (4) and (6) yield:
and normalized according to the convention:
Now assume the following particular separation of the variables r and :
Then Eqs. (8 - 11) yield:
Now because of the nonlinear term in Eq. (14) the wave equation does not a priori separate. The unique transformation which restitutes this separation is defined by:
where is any (say, positive) constant that has the dimension of a momentum flux (momentum times surface), and
Eqs. (14-16) become:
Adopting the appropriate dimensionless radial variable:
(since and for ), Eq. (19) becomes:
where , and
© European Southern Observatory (ESO) 1998
Online publication: August 6, 1998