Trajectory Interpretation of Correspondence Principle: Solution of Nodal Issue
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Trajectory Interpretation of Correspondence Principle: Solution of Nodal Issue Ciann‑Dong Yang1 · Shiang‑Yi Han2 Received: 30 November 2019 / Accepted: 20 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The correspondence principle states that the quantum system will approach the classical system in high quantum numbers. Indeed, the average of the quantum probability density distribution reflects a classical-like distribution. However, the probability of finding a particle at the node of the wave function is zero. This condition is recognized as the nodal issue. In this paper, we propose a solution for this issue by means of complex quantum random trajectories, which are obtained by solving the stochastic differential equation derived from the optimal guidance law. It turns out that point set A, which is formed by the intersections of complex random trajectories with the real axis, can represent the quantum mechanical compatible distribution of the quantum harmonic oscillator system. Meanwhile, the projections of complex quantum random trajectories on the real axis form point set B that gives a spatial distribution without the appearance of nodes, and approaches the classical compatible distribution in high quantum numbers. Furthermore, the statistical distribution of point set B is verified by the solution of the Fokker–Planck equation. Keywords Correspondence principle · Complex quantum random trajectory · Complex Fokker–Planck equation
1 Introduction The correspondence principle not only plays an important role in quantum mechanics but also holds a very crucial key to connecting the microscopic and macroscopic worlds. It states that quantum mechanics will reduce to classical mechanics within the limits of the quantum number approaching infinity, and has been widely discussed even in the infancy of the quantum era [1–5]. The quantum probability * Shiang‑Yi Han [email protected] 1
Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan, ROC
2
Department of Applied Physics, National University of Kaohsiung, Kaohsiung, Taiwan, ROC
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Vol.:(0123456789)
Foundations of Physics
density is in good agreement with the classical probability, on average, with the principal discrepancy being the rapid oscillations in |Ψ|2 . These oscillations generate the same number of nodes as the quantum number in Fig. 1 shows. This nodal issue remains unsolved despite the fact that many semiclassical or classical-like interpretations of quantum mechanics have been proposed. A proper interpretation of the correspondence principle is needed as technology starts to transcend the limits of both quantum and classical boundaries. The nodal issue and classical compatible probability distribution are two main challenges encountered by the new interpretation of the correspondence principle. The correspondence principle has been discussed from different perspectives by different interpretations [6–10]. The agreement between the classical and quantum pro
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