Synthesizing quantum probability by a single chaotic complex-valued trajectory

The current trajectory interpretation of quantum mechanics is based on an ensemble viewpoint that the evolution of an ensemble of Bohmian trajectories guided by the same wavefunction converges asymptotically to the quantum probability vertical bar psi vertical bar(2). Instead of the Bohm's ensemble-trajectory interpretation, the present paper gives a single-trajectory interpretation of quantum mechanics by showing that the distribution of a single chaotic complex-valued trajectory is enough to synthesize the quantum probability. A chaotic complex-valued trajectory manifests both space-filling (ergodic) and ensemble features. The space-filling feature endows a chaotic trajectory with an invariant statistical distribution, while the ensemble feature enables a complex-valued trajectory to envelop the motion of an ensemble of real trajectories. The comparison between complex-valued and real-valued Bohmian trajectories shows that without the participation of its imaginary part, a single real-valued trajectory loses the ensemble information contained in the wavefunction , and this explains the reason why we have to employ an ensemble of real-valued Bohmian trajectories to recover the quantum probability vertical bar Psi vertical bar(2). (c) 2015 Wiley Periodicals, Inc.