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

7. Conclusion

As a summary account of this paper that investigates a macroscopic context for Schroedinger equation at the scale of the Solar System, let us emphasize the following new results:

1. There is a `dynamical degeneracy' of any stationary quantum state . Indeed, to each such state defined by the energy eigenvalue E, one can associate an infinity of dynamical states of the system that are defined through Eqs. (22) and (29-32) by a particular value of the Wronskian A of the Schroedinger equation.

2. These dynamical states, the existence of which is guaranteed by i) the Planck-DeBroglie postulate stating that the phase of the wavefunction (3) is Hamilton's characteristic (or principal) function S of the system, and ii) the canonical Hamilton-Jacobi equation (31), must be understood in the classical (causal) sense.

3. The quantity (averaged over the polar angles) is the local classical statistical probability distribution (36) of the particle position in terms of the corresponding velocity flow derived from the radial momentum field (29-32).

4. There exists for any energy level E of the system a particular state defined by Eqs. (40-43). It describes a classical-like dynamical behaviour of the system within the classically allowed region in accordance with Eq. (39). We suggest to call this quantum state the `classical-like quantum state', and the corresponding steady-state radial-velocity matter flow the `classical-like static soliton mode'.

© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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