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

This paper is the second part of a study of the quantum free particle on spherical and hyperbolic spaces by making use of a curvature-dependent formalism. Here we study the analogues, on the three-dimensional spherical and hyperbolic spaces, $S_\k^3$ ($κ>0$) and $H_\k^3$ ($κ<0$), to the standard {\itshape spherical waves} in $E^3$. The curvature $\k$ is considered as a parameter and for any $\k$ we show how the radial Schrödinger equation can be transformed into a $\k$-dependent Gauss hypergeometric equation that can be considered as a $\k$-deformation of the (spherical) Bessel equation. The specific properties of the spherical waves in the spherical case are studied with great detail. These have a discrete spectrum and their wave functions, which are related with families of orthogonal polynomials (both $\k$-dependent and $\k$-independent), and are explicitly obtained.