2008, Vol.11, No.2, pp.107-120

We study the manifestation of quantum chaos in the Bohmian approach to quantum mechanics. The Bohmian orbits correspond to the streamlines of the Madelung probability flow in the orthodox approach to quantum mechanics. These orbits can be very different from the classical orbits in the same Hamiltonian system. We identify cases were the orbits are chaotic classically and ordered quantum mechanically, or vice versa. We study in detail the orbits of two quantum models, namely the particle in a rectangular box and in a double harmonic oscillator potential. In both cases we consider the orbits under a pilot wave (wave function) given as the superposition of three stationary states. In these systems we identify a) periodic, b) regular, and c) chaotic orbits. The periodic orbits are regular, but if the period is extremely long they exhibit a transient behavior for long time intervals (shorter than the period) that allows us to characterize them as "effectively ordered" or "effectively chaotic". The chaotic, or effectively chaotic orbits have close approaches to "nodal points" in which the equations of motion are singular. The regular orbits are always far from the nodal points and they can be represented by an analytical theory yielding the orbits in terms of a series in a small parameter. We discuss the consequences of Bohmian quantum chaos in the Bohm - Vigier theory, namely in the asymptotic approach of the probability p (of an initially arbitrary distribution of particles) to the square of the wavefunction , when chaotic mixing of the orbits takes place in the configuration space.
Key words: quantum chaos, particle in a rectangular box, double harmonic oscillator

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