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

TL;DR

The paper addresses the classical and quantum dynamics of a particle confined in a three-dimensional paraboloidal cavity formed by two confocal paraboloids. It develops an integrable classical framework via Hamilton–Jacobi separability, deriving closed-form actions $J_\\sigma$ and $J_\\tau$, caustics, winding numbers, and a complete $(s,t,\\ell)$ classification of periodic orbits; concurrently, it solves the quantum problem by separating the Schrödinger equation in parabolic coordinates, obtaining eigenmodes $\\psi_{l,n,m}$ described by Whittaker functions and a discretized spectrum with geometry-induced degeneracies. A direct classical–quantum correspondence is established by matching the invariants $\\alpha$ and $\\beta$ to quantum numbers, and by comparing classical caustics to quantum probability densities, which exhibit penetration into classically forbidden regions quantified by the ratio $\\Pi$. These results provide exact solvable benchmarks for 3D parabolic confinement and have potential relevance for nanoscale parabolic dots and resonator designs where geometric anisotropy controls wave patterns and spectra.

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 Schrödinger 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.

Figures (8)

• Figure 1: (a) Paraboloidal cavity defined by paraboloids $\sigma_0$ and $\tau_0$. (b) Parabolic coordinates $\mathbf{r} = (\sigma,\tau,\phi)$ defined in Eqs. \ref{['pcs']}. Surfaces of constant $\sigma$ and $\tau$ correspond to confocal paraboloids opening in the $z<0$ and $z>0$ directions, respectively.
• Figure 2: Classification of trajectories in the paraboloidal cavity. The blue-gray surface is the $\sigma$ caustic ($\sigma=\sigma_c$) and the apricot surface is the $\tau$ caustic ($\tau=\tau_c$). (a) Generic 3D motion with $\alpha\approx 0$ and $\beta>0$, bounded by both caustics. (b) Lower wall biased motion as $\alpha\to -\sigma_0^{2}$, with the $\sigma$–caustic approaching the boundary. (c) Meridional planar motion ($\beta=0$) that recovers the planar parabolic billiard.
• Figure 3: Poincaré phase maps $\alpha(\sigma,p_\sigma)$ and $\beta(\sigma,p_\sigma)$ for representative values $\alpha\in\{-5,0,2\}$ and $\beta\in\{0.0,0.1,0.2\}$ in a cavity with walls $\sigma_0=3$ and $\tau_0=2$.
• Figure 4: Periodic trajectories $(s,t,\ell)$ in the parabolic cavity with $\sigma_0=3$ and $\tau_0=2$. Panels (a)–(d) show representative examples with $\ell=0,1,2,3$, respectively. Here $s$ counts bounces on the $\sigma_0$ wall, $t$ counts bounces on the $\tau_0$ wall, and $\ell$ counts full revolutions about the $z$-axis.
• Figure 5: (a) Contour lines $S(\sigma_{0})=0$ and $T(\tau_{0})=0$ in the $(a,k)$ plane for $m= \{0,1,2\}$. Eigenpairs $(a,k)$ are given by the intersections of the two families.