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Unified treatment of the Coulomb and harmonic oscillator potentials in $D$ dimensions

G. Lévai·B. Kónya·Z. Papp
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Abstract

Quantum mechanical models and practical calculations often rely on some exactly solvable models like the Coulomb and the harmonic oscillator potentials. The $D$ dimensional generalized Coulomb potential contains these potentials as limiting cases, thus it establishes a continuous link between the Coulomb and harmonic oscillator potentials in various dimensions. We present results which are necessary for the utilization of this potential as a model and practical reference problem for quantum mechanical calculations. We define a Hilbert space basis, the generalized Coulomb-Sturmian basis, and calculate the Green's operator on this basis and also present an SU(1,1) algebra associated with it. We formulate the problem for the one-dimensional case too, and point out that the complications arising due to the singularity of the one-dimensional Coulomb problem can be avoided with the use of the generalized Coulomb potential.

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G. Lévai, B. Kónya, Z. Papp. 1998-02-03. Unified treatment of the Coulomb and harmonic oscillator potentials in $D$ dimensions. https://doi.org/10.1063/1.532595

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