TY - JOUR

T1 - Trajectory Interpretation of Correspondence Principle

T2 - Solution of Nodal Issue

AU - Yang, Ciann Dong

AU - Han, Shiang Yi

N1 - Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2020/9/1

Y1 - 2020/9/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85088557482&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85088557482&partnerID=8YFLogxK

U2 - 10.1007/s10701-020-00363-3

DO - 10.1007/s10701-020-00363-3

M3 - Article

AN - SCOPUS:85088557482

VL - 50

SP - 960

EP - 976

JO - Foundations of Physics

JF - Foundations of Physics

SN - 0015-9018

IS - 9

ER -