*Astron. Astrophys. 337, 299-310 (1998)*
## 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:
where
It is well-known that there exists a regular (normalized)
eigenstate for Eq. (23) provided that the
energy *E* takes any of the following negative discrete
eigenvalues (White, 1931, 1934; Rae, 1992):
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:
Therefore, defining:
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
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