^{1,a)}, Manuel F. Rañada

^{1,b)}and Mariano Santander

^{2,c)}

### Abstract

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, (κ > 0) and (κ < 0), to the standard *spherical waves* in *E* ^{3}. The curvature κ is considered as a parameter and for any κ we show how the radial Schrödinger equation can be transformed into a κ-dependent Gauss hypergeometric equation that can be considered as a κ-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 κ-dependent and κ-independent), and are explicitly obtained.

The authors are indebted to the referee for some interesting remarks which have improved the presentation of this paper. J.F.C. and M.F.R. acknowledge support from research projects MTM–2009–11154 (MCI, Madrid) and DGA–E24/1 (DGA, Spain); and M.S. from research projects MTM–2009–10751 (MCI, Madrid).

I. INTRODUCTION

II. GEODESIC MOTION, κ-DEPENDENT FORMALISM, AND QUANTIZATION

A. Lagrangian formalism, Noether symmetries, and Noether momenta

B. κ-dependent Hamiltonian and quantization

III. κ-DEPENDENT SCHRÖDINGER EQUATION

IV. SPHERICAL κ > 0 CASE

A. Solutions of type I

B. Solutions of type II

C. Final solution for the sphere and its Euclidean limit

V. HYPERBOLIC κ < 0 CASE

VI. FINAL COMMENTS AND OUTLOOK

### Key Topics

- Polynomials
- 40.0
- Lagrangian mechanics
- 13.0
- Numerical solutions
- 9.0
- Differential equations
- 8.0
- Vector fields
- 8.0

## Figures

Several polynomials with *n* = 4 and values of *L* = 0, 1, 2, 3, 4, 5, 6, 8. The polynomial with *L* = 0 is in dark grey, thick, and increasing *L* corresponds to lighter grey. The variable *r* in abscissas is the geodesic length on a sphere with curvature κ = 1, so *r* = 0 corresponds to the North Pole and *r* = π corresponds to the South pole. Notice all these polynomials have precisely 4 zeros and all are even around the equator, *r* = π/2.

Several polynomials with *n* = 4 and values of *L* = 0, 1, 2, 3, 4, 5, 6, 8. The polynomial with *L* = 0 is in dark grey, thick, and increasing *L* corresponds to lighter grey. The variable *r* in abscissas is the geodesic length on a sphere with curvature κ = 1, so *r* = 0 corresponds to the North Pole and *r* = π corresponds to the South pole. Notice all these polynomials have precisely 4 zeros and all are even around the equator, *r* = π/2.

Several polynomials with *n* = 5 and values of *L* = 0, 1, 2, 3, 4, 5, 6, 8. The polynomial with *L* = 0 is in dark grey, thick. The variable *r* in abscissas is the geodesic length on a sphere with curvature κ = 1, so *r* = 0 corresponds to the North Pole and *r* = π corresponds to the South pole. Notice all these polynomials have precisely 5 zeros and all are odd around the equator, *r* = π/2.

Several polynomials with *n* = 5 and values of *L* = 0, 1, 2, 3, 4, 5, 6, 8. The polynomial with *L* = 0 is in dark grey, thick. The variable *r* in abscissas is the geodesic length on a sphere with curvature κ = 1, so *r* = 0 corresponds to the North Pole and *r* = π corresponds to the South pole. Notice all these polynomials have precisely 5 zeros and all are odd around the equator, *r* = π/2.

Several polynomials with *n* = 12 and values of *L* = 0, 1, 2, 3, 4, 5, 6, 8. Same conventions as in preceding figures. Notice all these polynomials have precisely 12 zeros and all are even around the equator, *r* = π/2; the amplitude of oscillations at the equatorial band decreases when *L* grows, and to allow this to be seen clearly the range displayed has been reduced.

Several polynomials with *n* = 12 and values of *L* = 0, 1, 2, 3, 4, 5, 6, 8. Same conventions as in preceding figures. Notice all these polynomials have precisely 12 zeros and all are even around the equator, *r* = π/2; the amplitude of oscillations at the equatorial band decreases when *L* grows, and to allow this to be seen clearly the range displayed has been reduced.

Several polynomials with *n* = 13 and values of *L* = 0, 1, 2, 3, 4, 5, 6, 8. Notice all these polynomials have precisely 13 zeros and all are odd around the equator, *r* = π/2.

Several polynomials with *n* = 13 and values of *L* = 0, 1, 2, 3, 4, 5, 6, 8. Notice all these polynomials have precisely 13 zeros and all are odd around the equator, *r* = π/2.

Approaching the Euclidean radial function for *L* = 0 (the spherical Bessel function *j* _{0}(*kr*) for *k* = 10 with a sequence of the solutions for the sphere, corresponding to the values *n* = 20, 24, 32, 40 and *L* = 0. In the limit κ → 0, the upper hemisphere goes to the full Euclidean plane and the sphere equator goes to Euclidean infinity, so here only the left-hand side of the graphics in Figures 1–4 (which corresponds to the upper hemisphere) is pertinent.

Approaching the Euclidean radial function for *L* = 0 (the spherical Bessel function *j* _{0}(*kr*) for *k* = 10 with a sequence of the solutions for the sphere, corresponding to the values *n* = 20, 24, 32, 40 and *L* = 0. In the limit κ → 0, the upper hemisphere goes to the full Euclidean plane and the sphere equator goes to Euclidean infinity, so here only the left-hand side of the graphics in Figures 1–4 (which corresponds to the upper hemisphere) is pertinent.

Same as Figure 5 but for *L* = 3, approaching the spherical Bessel function *j* _{3}(*kr*).

Same as Figure 5 but for *L* = 3, approaching the spherical Bessel function *j* _{3}(*kr*).

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