The Harmonic Oscillator on Three-Dimensional Spherical and Hyperbolic Spaces: Curvature Dependent Formalism and Quantization

Abstract

A nonlinear model representing the quantum harmonic oscillator on the threedimensional spherical and hyperbolic spaces, S 3 κ (κ > 0) and H 3 k (κ < 0), is studied using geodesic spherical coordinates (r, θ, φ). The curvature κ is considered as a parameter and the results are formulated in explicit dependence of κ. The first part of the paper is concerned with the existence of Killing vectors, the existence of Noether symmetries and the properties of the Noether momenta. The second part is devoted to the transition from classical to quantum mechanics. The classical system is quantized by obtaining a κ-dependent invariant measure dμ κ and expressing the Hamiltonian as a function of the Noether momenta.

Superintegrable deformations of superintegrable systems: Quadratic superintegrability and higher-order superintegrability

“…(i) The Hamiltonian H 1 (κ), that has been studied in Refs. [48,50,51,52] (although in some cases with a trigonometric-hyperbolic notation), represents the harmonic oscillator in a space of constant curvature κ (sphere S 2 κ with κ > 0, and Hiperbolic plane H 2 κ with κ < 0). We note that the kinetic term includes not only the factor (1 − κ r 2 ) but also a contribution of the angular momentum J with the curvature κ as coefficient.…”

Superintegrability on the three-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the sphere S 3 and on the hyperbolic space H 3

“…The hyperbolic potential V κ , κ < 0, is a well with finite depth since lim r→∞ V κ = 1/|κ|. A quantization of this system was studied in [56] and the Schrödinger equation determined by this Hamiltonian was solved in [57] (the radial Schrödinger equation becomes a κ-dependent Gauss hypergeometric equation that can be considered as a κ-deformation of the confluent hypergeometric equation that appears in the Euclidean case) but making use of another system of coordinates (see Appendix II). We also mention that this oscillator was also studied in [62] using as an approach first a stereographic projection and then both Poincaré and Beltrami coordinates.…”

Curvature-dependent formalism, Schrödinger equation and energy levels for the harmonic oscillator on three-dimensional spherical and hyperbolic spaces

“…The measure dµ κ , that was obtained as the unique measure (up to a multiplicative constant) invariant under the Killing vectors [46], coincides with the corresponding Riemann volume in a space with curvature κ.…”

“…• The differential element of distance ds κ , in the family M 3 κ = (S 3 κ , lE 3 , H 3 κ ) of three-dimensional spaces with constant curvature κ, can be written is some different but equivalent ways (this question is discussed in [46]- [47]; see also [48]). For example, if we make use of the following κ-dependent trigonometric (either circular, parabolic or hyperbolic) functions…”

“…This paper is concerned with the study of the quantum harmonic oscillator on threedimensional spherical and hyperbolic spaces making use of a set of coordinates (r, θ, φ) obtained by introducing a small change in the radial part of the geodesic spherical coordinates. It can be considered as a new paper in a series devoted to the study of classical [38][39][40][41][42] and quantum [43][44][45][46][47] systems on Riemannian configuration spaces with constant curvature κ = 0. We follow an approach that can be summarized in the following two points.…”