Modeling quantum harmonic oscillator in complex domain

In this paper, we show the equivalence between a classical Hamilton–Jacobi equation defined in complex domain and the Schrödinger equation. This equivalence allows us to model quantum behavior of harmonic oscillator, which was commonly described probabilistically by Schrödinger equation, in terms of deterministic complex dynamics. The consequence of this correspondence is the possibility to realize abstract quantum eigen-state in terms of deterministic eigen-trajectory in complex space. Using this correspondence in the visualization of spin motion, we find that the eigen-trajectories for the spin motion in ground state form a vortex-like circulating flow. With the aid of residue theorem, the contour integration over the closed eigen-trajectories in each eigen-state indicates that various quantization phenomena such as, quantization of action, quantization of period of oscillation, and quantization of quantum potential, can be completely exhibited via deterministic complex dynamics. When harmonic oscillator is modeled deterministically by complex dynamics, the discussion of how quantum motion transits to classical motion becomes straightforward and unambiguous, since classical motion happens in real space so that we can say that the classical limit is achieved as the complex motion of a particle falls entirely on a special subset of complex space – the real space. Thus, transition from quantum world to classical world can be performed continuously by tracing a complex trajectory connecting the two worlds without any conceptual discontinuity, and the conventional confliction between the probabilistic view and the deterministic view across the transition boundary no longer exists. This conclusion is consistent with that advanced by G. Ord and M.S. Elnaschie [Elnaschie MS. Chaos, Solitons & Fractals 2006;27:39, Elnaschie MS. Chaos, Solitons & Fractals, 2006;29:23, Weibel P, Ord G, Rõsslev O. Space time physics and fractality. Festschrift in honour of Mohamed Elnaschie, New York: Springer-Vienna; 2005].