On the nodes of wave function and the quantum Hamilton-Jacobi solution

TL;DR

This paper introduces a straightforward method to analytically solve the stationary quantum Hamilton-Jacobi equation, demonstrating its effectiveness with central potentials and connecting quantum action-angle variables to wave function nodes.

Contribution

It develops a simple, exact solution method for the quantum Hamilton-Jacobi equation and links quantum action-angle variables to wave function nodes, validated on key potentials.

Findings

Exact solutions for harmonic oscillator and Coulomb potentials

Quantum action-angle variables linked to wave function nodes

Method validated through bound-state solutions

Abstract

We present the analytic solution for the stationary quantum HamiltonJacobi equation. Knowing the strong relation between the Riccati and quantum Hamilton-Jacobi equations, we develop a simple method to obtain the exact solution. Then, in order to prove the validity of the proposed method, we use two central potentials: the three-dimensional harmonic oscillator and Coulomb potential, both with bound-states. Finally, we compute the action-angle variables in a entirely quantum version for to achieve connect with the nodes of the wave function.