Trajectory Interpretation of Correspondence Principle: Solution of Nodal Issue

TL;DR

This paper introduces a novel approach using complex quantum trajectories to resolve the nodal issue in the correspondence principle, aligning quantum and classical distributions without nodes.

Contribution

It proposes a method employing complex stochastic trajectories and optimal guidance to address the nodal issue in quantum-classical correspondence.

Findings

Point set B avoids nodes and matches classical distributions at high quantum numbers.

Complex trajectories provide a quantum distribution compatible with the harmonic oscillator.

Distribution of point set B is validated by Fokker-Planck equation solutions.

Abstract

The 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 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 under the optimal guidance law. It turns out that point set A which is collected by the intersections of complex random trajectories and the real axis can present 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 distribution without appearance of nodes. Moreover, point set B can represent 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.