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arXiv · 1304.3739

An exactly-solvable three-dimensional nonlinear quantum oscillator

Axel Schulze-Halberg·John R. Morris
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Abstract

Exact analytical, closed-form solutions, expressed in terms of special functions, are presented for the case of a three-dimensional nonlinear quantum oscillator with a position dependent mass. This system is the generalization of the corresponding one-dimensional system, which has been the focus of recent attention. In contrast to other approaches, we are able to obtain solutions in terms of special functions, without a reliance upon a Rodrigues-type of formula. The wave functions of the quantum oscillator have the familiar spherical harmonic solutions for the angular part. For the s-states of the system, the radial equation accepts solutions that have been recently found for the one-dimensional nonlinear quantum oscillator, given in terms of associated Legendre functions, along with a constant shift in the energy eigenvalues. Radial solutions are obtained for all angular momentum states, along with the complete energy spectrum of the bound states.

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Axel Schulze-Halberg, John R. Morris. 2013-04-12. An exactly-solvable three-dimensional nonlinear quantum oscillator. https://doi.org/10.1063/1.4829669

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