Classical Representation for Quantum States of a Particle in $\lambda z^{2m}$ potential

TL;DR

This paper develops a classical ensemble representation for quantum eigenstates in polynomial potentials, linking classical trajectories with quantum energies and analyzing the classical form of the Schrödinger equation.

Contribution

It introduces a novel classical representation for quantum states in polynomial potentials, including negative energy distributions and singularities, bridging classical and quantum descriptions.

Findings

Classical ensembles can represent quantum eigenstates in polynomial potentials.

Mean energies of ensembles match quantum eigenenergies.

The classical Schrödinger equation form is analyzed for these systems.

Abstract

A classical representation for quantum eigenstates of a particle bound in $\lambda z^{2m}$ $(\lambda >0, m=1,2,...)$ potentials is developed. It is represented by ensembles of classical trajectories with energy distributions that can take on negative values, for $m>1$ have integrable singularities at zero energy and whose mean energies coincide with quantum eigenenergies. The corresponding Schr\"odinger equation in classical representation is analyzed.