The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach

Abstract

The quantum free particle on the sphere $S_{\kappa }^2$ ($\kappa >0$) and on the hyperbolic plane $H_{\kappa }^2$ ($\kappa <0$) is studied using a formalism that considers the curvature $\k$ as a parameter. The first part is mainly concerned with the analysis of some geometric formalisms appropriate for the description of the dynamics on the spaces ($S_{\kappa }^2$, $\IR^2$, $H_{\kappa }^2$) and with the the transition from the classical $\kappa$-dependent system to the quantum one using the quantization of the Noether momenta. The Schrödinger separability and the quantum superintegrability are also discussed. The second part is devoted to the resolution of the $\kappa$-dependent Schrödinger equation. First the characterization of the $\kappa$-dependent `curved' plane waves is analyzed and then the specific properties of the spherical case are studied with great detail. It is proved that if $\kappa >0$ then a discrete spectrum is obtained. The wavefunctions, that are related with a $\kappa$-dependent family of orthogonal polynomials, are explicitly obtained.

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• Quantization of a Particle on a Two-Dimensional Manifold of Constant Curvature
• Free particle and isotropic harmonic oscillator on a spheroidal surface: the Higgs-like approach
• Elementary atoms in spaces of constant curvature by the Nikiforov-Uvarov method
• Motion on Constant Curvature Spaces and Quantization Using Noether Symmetries
• Higgs algebraic symmetry of screened system in a spherical geometry