An update on the classical and quantum harmonic oscillators on the sphere and the hyperbolic plane in polar coordinates

Abstract

A simple derivation of the classical solutions of a nonlinear model describing a harmonic oscillator on the sphere and the hyperbolic plane is presented in polar coordinates. These solutions are then related to those in cartesian coordinates, whose form was previously guessed. In addition, the nature of the classical orthogonal polynomials entering the bound-state radial wavefunctions of the corresponding quantum model is identified.

arXiv comment13 pages, 2 figures, published version

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Cited by in corpus (9)

• Quantum oscillator and Kepler-Coulomb problems in curved spaces: deformed shape invariance, point canonical transformations, and rational extensions
• On Hamiltonians with position-dependent mass from Kaluza-Klein compactifications
• Families of quasi-exactly solvable extensions of the quantum oscillator in curved spaces
• On the classical and quantum dynamics of a class of nonpolynomial oscillators
• Deformed shape invariance symmetry and potentials in curved space with two known eigenstates
• Quasi-exactly solvable extensions of the Kepler-Coulomb potential on the sphere
• Novel exactly solvable Schrödinger equations with a position-dependent mass in multidimensional spaces obtained from duality
• Addendum to An update on the classical and quantum harmonic oscillators on the sphere and the hyperbolic plane in polar coordinates
• Algebraic derivation of the Energy Eigenvalues for the quantum oscillator defined on the Sphere and the Hyperbolic plane