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Potential function of the Wilson-Racah quantum system. Appendix: The Wilson and Racah orthogonal polynomials. (English) Zbl 1431.81017

Summary: We show how to establish a correspondence between the recently introduced formulation of quantum mechanics based on orthogonal polynomials in the energy and physical parameters (without reference to a potential function) and the convention formulation. In this work, we obtain the potential function for the newly introduced quantum system associated with the Wilson and Racah polynomials [A. D. Alhaidari and the 2nd author, ibid. 58, No. 2, 022101, 8 p. (2017; Zbl 1357.81156)] using any of the proposed procedures in the work of A. D. Alhaidari [Commun. Theor. Phys. 68, No. 6, 711–728 (2017; Zbl 1382.81006)]. To achieve that, we use the matrix elements of the potential function and the basis in which these were calculated.
©2020 American Institute of Physics

MSC:

81P05 General and philosophical questions in quantum theory
12E10 Special polynomials in general fields
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
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References:

[1] Alhaidari, A. D.; Ismail, M. E. H., Quantum mechanics without potential function, J. Math. Phys., 56, 072107 (2015) · Zbl 1330.81105 · doi:10.1063/1.4927262
[2] Alhaidari, A. D., Formulation of quantum mechanics without potential function, Quantum Phys. Lett., 4, 51 (2015)
[3] Alhaidari, A. D.; Taiwo, T. J., Wilson-Racah quantum system, J. Math. Phys., 58, 022101 (2017) · Zbl 1357.81156 · doi:10.1063/1.4975138
[4] Alhaidari, A. D., Reconstructing the potential function in a formulation of quantum mechanics Basedon orthogonal polynomials, Commun. Theor. Phys., 68, 711-728 (2017) · Zbl 1382.81006 · doi:10.1088/0253-6102/68/6/711
[5] Alhaidari, A. D.; Li, Y.-T., Quantum systems associated with the Hahn and continuous Hahn polynomials, Rep. Math. Phys., 82, 285 (2018) · Zbl 1441.81132 · doi:10.1016/s0034-4877(19)30002-3
[6] Alhaidari, A. D., Representation of the quantum mechanical wavefunction by orthogonal polynomials in the energy and physical parameters, Commun. Theor. Phys., 72, 015104 (2020) · Zbl 1452.81094
[7] Koekoek, R.; Swarttouw, R., The Askey Scheme of Hypergeometric Orthogonal Polynomials and its Q-Analogues (1998), Delft University of Technology: Delft University of Technology, Delft
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