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An Algebraic Approach to the Harmonic Oscillator Plus an Inverse Square Potential in Three Dimensions

Shi-Hai Dong and M. Lozada-Cassou

Abstract

The eigenfunctions and eigenvalues of the three-dimensional Schrödinger equation with a harmonic oscillator plus an inverse square interaction are obtained. A realization of the ladder operators for the wave functions is studied. It is found that these operators satisfy the commutation relations of an SU(1,1) group. The closed analytical expressions for the matrix elements of different functions ρ and ρd/dρ with ρ = r 2 are evaluated. Another hidden symmetry explores the relations between the eigenvalues and eigenfunctions for substituting r→ ir. PACS number(s): 03. 65. Fd, 03. 65.Ge and 02. 20.Qs.

American Journal of Applied Sciences
Volume 2 No. 1, 2005, 376-382

DOI: https://doi.org/10.3844/ajassp.2005.376.382

Submitted On: 28 June 2005 Published On: 31 January 2005

How to Cite: Dong, S. & Lozada-Cassou, M. (2005). An Algebraic Approach to the Harmonic Oscillator Plus an Inverse Square Potential in Three Dimensions. American Journal of Applied Sciences, 2(1), 376-382. https://doi.org/10.3844/ajassp.2005.376.382

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Keywords

  • Inverse Square Interaction
  • Ladder Operators
  • SU (1, 1) Group
  • Matrix Elements