Astronomy and Astrophysics – Astronomy
Scientific paper
Sep 2008
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2008cemda.102..219c&link_type=abstract
Celestial Mechanics and Dynamical Astronomy, Volume 102, Issue 1-3, pp. 219-239
Astronomy and Astrophysics
Astronomy
4
Orbits, Quantum Mechanics, Bohmian Approach, Bohmian Trajectories
Scientific paper
We discuss the issue of ordered and chaotic trajectories in the Bohmian approach of Quantum Mechanics from points of view relevant to the methods of Celestial Mechanics. The Bohmian approach gives the same results as the orthodox (Copenhagen) approach, but it considers also underlying trajectories guided by the wave. The Bohmian trajectories are rather different from the corresponding classical trajectories. We give examples of a classical chaotic system that is ordered quantum-mechanically and of a classically ordered system that is mostly chaotic quantum mechanically. Then we consider quantum periodic orbits and ordered orbits, that can be represented by formal series of the “third integral” type, and we study their asymptotic properties leading to estimates of exponential stability. Such orbits do not approach the “nodal points” where the wavefunction ψ vanishes. On the other hand, when an orbit comes close to a nodal point, chaos is generated in the neighborhood of a hyperbolic point (called X-point). The generation of chaos is maximum when the X-point is close to the nodal point. Finally we remark that high order periodic orbits may behave as “effectively ordered” or “effectively chaotic” for long times before reaching the period.
Contopoulos George
Efthymiopoulos Christos
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