5. The Kepler-Madelung system
Let us illustrate the above theory by considering the Kepler problem:
5.1. The nonlinear radial EMP equation
In terms of the reduced variable defined by Eqs. (20), we have:
The radial quantum number that defines the quantum condition (45) is equal to (Messiah, 1962):
where the azimuthal quantum number l related to the orbital angular momentum is defined by:
When the energy parameter E is tuned in accordance with Eq. (49), Eq. (21) becomes:
Equivalently, one can measure the radius r in units of the Bohr radius:
Eq. (52) becomes:
while the discrete energy levels (49) now read:
5.2. The pseudo-circular case
For , there are no radial nodes ( according to Eq. (50)) and, hence, a single maximum for the regular eigenfunction that is located at:
(cf. Eq. (54)). Indeed the abscissa that defines the initial conditions Eqs. (40-43) is then given by Eq. (57).
Moreover, the radial action (45) has the remarkable property to be independent of both the Wronskian A and the level N. Therefore it appears as a fundamental invariant of the pseudo-circular Kepler system since it is equal to one single quantum of action h:
The case where the azimuthal quantum number l has its largest possible value is of importance here for it semiclassically corresponds to the quasi-circular orbits which one would like to associate with the planetary orbits of the solar system (Nottale, 1993, 1996a,b, 1997). However this correspondence only plays for large quantum numbers N. For such low values of N as those being considered here (for instance for the `inner solar system'), it is irrelevant to make use of it. This point is clearly illustrated by White (1934) who displays the four models used by different investigators in order to accommdate the newly discovered Schroedinger equation with classical orbit descriptions. Their `effective azimuthal quantum number' k, which is basically the angular momentum of the system, ranges from to , which means for an uncertainty of about for, say, the energy level. A crude (i.e. of the semiclassical type) application to QM of the CM formula for the orbit eccentricity e, where are the appropriate action variables (Goldstein, 1980), yields . Accordingly the square eccentricity would range from to for . Nottale proposes a third expresssion, namely , that also yields zero for (Nottale, 1993, 1996b, 1997). These few examples illustrate the ambiguity of the very concept of orbit eccentricity and, hence, of its value at low quantum numbers.
It would at least be more approriate to use the well-known concept of quantum expectation value and define the expectation value of the radial distance and its corresponding mean square deviation , namely (White, 1934; Messiah, 1962):
Note that the relative mean square deviation may reach or more for values of N about 3 or less. In the present nonlinear QM theory, this mean square deviation is basically the width of the static radial-velocity soliton.
© European Southern Observatory (ESO) 1998
Online publication: August 6, 1998