Classical and quantum dynamics of a particle confined in a paraboloidal cavity

TL;DR

This paper provides a comprehensive classical and quantum analysis of a particle in a paraboloidal cavity, deriving analytical solutions, classifying trajectories, and exploring quantum eigenstates and energy spectra.

Contribution

It introduces a detailed analytical framework for the classical and quantum behavior of particles in paraboloidal cavities, including trajectory classification and eigenmode characterization.

Findings

Classical system is integrable with explicit constants of motion.

Quantum eigenstates are described by Whittaker functions with specific degeneracies.

Classical trajectories correlate with quantum probability densities.

Abstract

We present a classical and quantum analysis of a particle confined in a three-dimensional paraboloidal cavity formed by two confocal paraboloids. Classically, the system is integrable and presents three independent constants of motion, namely, the energy, the $z$-component of the angular momentum, and a third dynamical constant associated with the paraboloidal geometry, which can be derived from the separability of the Hamilton--Jacobi equation. We derive closed-form analytical expressions for the actions, which allow us to determine the two conditions to get periodic closed trajectories. We classify these trajectories through the indices $(s,t,\ell)$. The caustic paraboloids that bound the motion provide a complete geometric characterization of admissible trajectories. Quantum mechanically, separability of the Schr\"odinger equation in parabolic coordinates yields eigenmodes described by Whittaker functions. We determine the energy spectrum and identify degeneracies arising not only from azimuthal symmetry but also from specific cavity deformations. A direct correspondence between classical trajectories and quantum eigenstates reveals that probability densities concentrate in the classically allowed region with controlled penetration into forbidden zones.