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arXiv:2512.08021

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

Ángel E. Reyna-Cruz, Julio C. Gutiérrez-Vega

quant-phmath-phphysics.class-phSubmitted Dec 8, 2025• Announced Dec 10, 2025• 0 citations
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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.

Comments
24 pages, 8 figures

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Abstract In the first part of this work, using the quantum potential approach, we show that a solution to the time-independent Schrödinger equation determines a subset of classical solutions, only if the region corresponding to the zeroes of the quantum potential is tangent to the caustic region determined by the classical trajectories. Thus, the solutions of the time-independent Schrödinger equation, according to their caustic and the zeros of the quantum potential, can be classified in three different cases given by the following conditions: the two regions coincide, they are tangent at certain subset of points, and the two regions are not tangent at any point. In the second part, as examples of the first type of wave functions, we present the solutions of the Schrödinger equation for the 2D isotropic harmonic oscillator, which are eigenfunctions of both the Hamiltonian and the angular momentum operators. That is, we show that for this family of solutions, the zeroes of the quantum potential coincide with the caustic region. Furthermore, we find that the classical trajectories, determined from the quantum ones and the zeroes of the quantum potential, conform to a family of elliptical curves for a particle with energy, (2n + l + 1)ℏ ω, and orbital angular momentum l ℏ.
The quantization method based on the quantum Hamiltonian Jacobi equation, is extended to two-dimensional non-separable but integrable Hamiltonians. It is shown that each wave function for those systems corresponds to a well-defined family of classical trajectories, enveloped by a caustic. The energy eigenvalues and the values of the wave functions on the caustic are obtained by solving the 1-dim quantum Hamilton Jacobi equation of the caustic' arcs. Results are in good agreement with those obtained by usual methods.
It is shown that it is possible to construct the quantum wave functions for non-separable but integrable two-dimensional Hamiltonian systems, by solving suitable Dirichlet boundary values problems inside and outside the regions spanned by particular families of classical trajectories, in one-to-one correspondence with the quantum state. The method is applied both to the Schrodinger equation, and to the quantum Hamilton-Jacobi equation. The boundary values are obtained by integrating the one-dim equations on the caustics arcs enveloping the classical trajectories. This approach gives the same results as the usual methods, and furthermore clarifies the links between quantum and classical mechanics.
We develop a dynamical theory, based on a system of ordinary differential equations describing the motion of particles which reproduces the results of quantum mechanics. The system generalizes the Hamilton equations of classical mechanics to the quantum domain, and turns into them in the classical limit ℏ→0\hbar\rightarrow 0ℏ→0. The particles' motions are completely determined by the initial conditions. In this theory, the wave function ψψψ of quantum mechanics is equal to the exponent of an action function, obtained by integrating some Lagrangian function along particle trajectories, described by equations of motion. Consequently, the equation for the logarithm of a wave function is related to the equations of motion in the same way as the Hamilton-Jacobi equation is related to the Hamilton equations in classical mechanics. We demonstrate that the probability density of particles, moving according to these equations, should be given by a standard quantum-mechanical relation, ρ=∣ψ∣2ρ=|ψ|^2ρ=∣ψ∣2. The theory of quantum measurements is presented, and the mechanism of nonlocal correlations between results of distant measurements with entangled particles is revealed. In the last section, we extend the theory to particles with nonzero spin.
Abstract The main contribution of the present work is to use the probability density of an Airy beam to identify its maxima with the family of caustics associated with the wavefronts determined by the level curves of a one-parameter family of solutions to the Hamilton–Jacobi equation with a given potential . To this end, we give a classical mechanics characterization of a solution of the one-dimensional Schrödinger equation in free space determined by a complete integral of the Hamilton–Jacobi and Laplace equations in free space. That is, with this type of solution, we associate a two-parameter family of wavefronts in the spacetime, which are the level curves of a one-parameter family of solutions to the Hamilton–Jacobi equation with a determined potential, and a one-parameter family of caustics. The general results are applied to an Airy beam to show that the maxima of its probability density provide a discrete set of: caustics, wavefronts and potentials. The results presented here are a natural generalization of those obtained by Berry and Balazs in 1979 for an Airy beam. Finally, we remark that, in a natural manner, each maxima of the probability density of an Airy beam determines a Hamiltonian system.
General analytical solutions of the Quantum Hamilton Jacobi Equation for conservative one-dimensional or reducible motion are presented and discussed. The quantum Hamilton's characteristic function and its derivative, i.e. the quantum momentum function, are obtained in general, and it is shown that any one-dimensional wave function can be exactly represented in a WKB-like form. The formalism is applied to the harmonic oscillator and to the electron's motion in the hydrogen atom, and the above mentioned functions are computed and discussed for different quantum numbers. It is analyzed how the quantum quantities investigated tend to the corresponding classical ones, in the limit ℏ→0\hbar\to 0ℏ→0. These results demonstrate that the Quantum Hamilton Jacobi Equation is not only completely equivalent to the Schrödinger Equation, but allows also to fully investigate the transition from quantum to classical mechanics.