Abstract
A nonlinear model representing the quantum harmonic oscillator on the three-dimensional spherical and hyperbolic spaces, \(S_{\kappa }^{3}\) (κ>0) and \(H_{k}^{3}\) (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 dμκ and expressing the Hamiltonian as a function of the Noether momenta.
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