Complex Trajectories Interpretation of Correspondence Principle

Shiang-Yi Han^1*, Ciann-Dong Yang²

¹Department of Applied Physics, National University of Kaohsiung, Kaohsiung, Taiwan
²Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan

* Presenter:Shiang-Yi Han, email:karate0221@nuk.edu.tw

Correspondence Principle states that the quantum system will approach to the classical system in high quantum numbers. Indeed, the average of the probability density distribution reflects a classical-like distribution; however, the likelihood of finding a particle at nodes of the wave function is zero. Quantum mechanics cannot surmount the problem of zero probability at nodes. In this study, we attempt to tackle this issue by means of complex quantum random trajectories which are obtained by solving the stochastic differential equation under the optimal guidance law. It turns out that point set {A} collected by the crossovers made by complex random trajectories across the real axis presents quantum mechanical compatible distribution in both low and high quantum number of the quantum harmonic oscillator system. Point set {B} formed by the complex quantum random trajectories projected onto the real axis presents non-existence of nodes in low quantum number; while the probability distribution coheres with the classical result in high quantum number.

Keywords: Correspondence Principle, Quantum Random Trajectory, Quantum Harmonic Oscillator