Curvature-dependent formalism, Schrödinger equation and energy levels for the harmonic oscillator on three-dimensional spherical and hyperbolic spaces

Abstract

A nonlinear model representing the quantum harmonic oscillator on the threedimensional spherical and hyperbolic spaces, S 3 κ (κ > 0) and H 3 k (κ < 0), is studied. The curvature κ is considered as a parameter and then the radial Schrödinger equation becomes a κ-dependent Gauss hypergeometric equation that can be considered as a κ-deformation of the confluent hypergeometric equation that appears in the Euclidean case. The energy spectrum and the wavefunctions are exactly obtained in both the three-dimensional sphere S 3 κ (κ > 0) and the hyperbolic space H 3 k (κ < 0). A comparative study between the spherical and the hyperbolic quantum results is presented.Keywords: Quantization. Quantum mechanics on spaces of constant curvature. Orthogonal polynomials.Running title: The quantum harmonic oscillator on spaces with curvature.

Position dependent mass Scarf Hamiltonians generated via the Riccati equation

“…that has the same form as (15) for a = − 1 4 . This means that the Hamiltonian H can be factorized in the form (72) with…”

“…where and are 2 real-valued functions that do not dependent on the hierarchy label . In this way, the set (14) and (15) transforms into the set…”

“…Finally, we may consider either the Riccati equation (14) or (15) in order to obtain the potential V . After some calculations, we obtain a family of PDM trigonometric Scarf potentials given by…”

“…Those include shape‐invariant and supersymmetric potentials . Of particular interest is the generation of exactly solvable Hamiltonians endowed with position dependent mass (PDM) due to the fact that they have multiple applications in many areas of physics such as the description of optical and electronic properties of semiconductors and quantum dots, the construction of mechanical models in spaces with curvature, and the study of galactic mass loss among others. The use of the factorization method in considering PDM Hamiltonians has deserved growing attention in the last decades .…”

On Hamiltonians with position-dependent mass from Kaluza–Klein compactifications

“…thereby the exactly solvable nonlinear oscillator Hamiltonian introduced in [19,21] arising. Also, the corresponding quantum model can exactly be solved [22,23,28,31]. In this context, if the size of the extra dimension is assumed to be given by b(r) (which is the inverse of the conformal factor) 18) this means that as long as r grows, the extra dimension becomes larger and the effective mass of the particle becomes small.…”