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

José F. Cariñena, Manuel F. Rañada, Mariano Santander
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 Schr\"odinger
separability and the quantum superintegrability are also discussed. The second
part is devoted to the resolution of the $\kappa$-dependent Schr\"odinger
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.
View original: http://arxiv.org/abs/1201.5589