## 3. Madelung's central-field problemFor 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 ## 3.1. Micro-versus macroscopic quantizationChaotic planetary systems and atoms are 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 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 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
## 3.2. The Ermakov-Milne-Pinney (EMP) nonlinear differential equationConsider a stationary quantum state described by its complex-valued wavefunction: in agreement with Eq. (2) (where the amplitude
and the phase Now assume that is a confining central-field
potential where where This equation defines the two following constants of integration: while Eqs. (4) and (6) yield: The treatment of the angular part of Eq. (10) is fairly standard (White, 1931, 1934; Messiah, 1962; Rae, 1992). Let (with ) be the associated Legendre polynomial defined by: and normalized according to the convention: Now assume the following particular separation of the variables
Then Eqs. (8 - 11) yield: Now because of the nonlinear term in
Eq. (14) the wave equation does not where is any (say, positive) constant that has the dimension of a momentum flux (momentum times surface), and Then defining: 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 |