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arXiv · 1210.5055

Curvature-dependent formalism, Schr\"odinger equation and energy levels for the harmonic oscillator on three-dimensional spherical and hyperbolic spaces

José F. Cariñena·Manuel F. Rañada·Mariano Santander
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Abstract

A nonlinear model representing the quantum harmonic oscillator on the three-dimensional spherical and hyperbolic spaces, $S_\k^3$ ($\kappa>0$) and $H_k^3$ ($\kappa<0$), is studied. The curvature $\k$ is considered as a parameter and then the radial Schr\"odinger equation becomes a $\k$-dependent Gauss hypergeometric equation that can be considered as a $\k$-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 $S_\k^3$ ($\kappa>0$) and the hyperbolic space $H_k^3$ ($\kappa<0$). A comparative study between the spherical and the hyperbolic quantum results is presented.

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José F. Cariñena, Manuel F. Rañada, Mariano Santander. 2012-10-18. Curvature-dependent formalism, Schr\"odinger equation and energy levels for the harmonic oscillator on three-dimensional spherical and hyperbolic spaces. https://doi.org/10.1088/1751-8113%2F45%2F26%2F265303

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