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

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.