Wilson-Racah quantum system

Alhaidari, A. D.; Taiwo, T. J.

doi:10.1063/1.4975138

Cited by 27 publications

(48 citation statements)

References 7 publications

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“…That is, either becomes the discrete version of the Wilson polynomial whose analytic properties are well known (see, for example, the Appendices in Ref. [30] and citations therein). Table 1: The orthogonal polynomial(s) representing the physical system as a function of the type of its energy spectrum with examples.…”

Section: Appendix B: Tra Solution For a New Class Of Potential Functionsmentioning

confidence: 99%

Tridiagonal representation approach in quantum mechanics

Alhaidari¹,

Bahlouli²

2019

Phys. Scr.

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We present an algebraic approach for finding exact solutions of the wave equation. The approach, which is referred to as the Tridiagonal Representation Approach (TRA), is inspired by the J-matrix method and based on the theory of orthogonal polynomials. The class of exactly solvable problems in this approach is larger than the conventional class. All properties of the physical system (energy spectrum of the bound states, phase shift of the scattering states, energy density of states, etc.) are obtained in this approach directly and simply from the properties of the associated orthogonal polynomials.

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Section: Appendix B: Tra Solution For a New Class Of Potential Functionsmentioning

confidence: 99%

Tridiagonal representation approach in quantum mechanics

Alhaidari¹,

Bahlouli²

2019

Phys. Scr.

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“…For instance, the potential functions might be non-analytic, Nonlocal, energy dependent, or the corresponding differential wave equation is higher second order and so on. As a result of this formulation, new quantum systems that do not belong to the conventional solvable class in quantum mechanics were obtained [1][2][3] However; in [3], we obtained a new quantum system-Wilson-Racah Quantum Systemwithits discrete energy spectrum, scattering phase shift, and discrete wavefunction (Racah quantum system). But for this physical system, we are yet to get the potential function (either analytically or numerically).…”

Section: Introductionmentioning

confidence: 99%

Potential function of the Wilson-Racah quantum system

Taiwo

2020

Journal of Mathematical Physics

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“…All physical information about the system, both structural and dynamical, are contained in these expansion coefficients. The "Tridiagonal Representation Approach (TRA)" is an algebraic method for solving the wave equation (e.g., the Schrödinger or Dirac equation) [1][2][3][4]. In the TRA, the basis elements are chosen such that the matrix representation of the wave operator is tridiagonal.…”

Section: Introductionmentioning

confidence: 99%

Open Problem in Orthogonal Polynomials

Alhaidari

2019

Reports on Mathematical Physics

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Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line and two of its discrete versions; one with a finite spectrum and another with countably infinite spectrum. A second class of these new orthogonal polynomials appeared recently while solving a Heun-type equation. Based on these results and on our recent study of the solution space of an ordinary differential equation of the second kind with four singular points, we introduce a modification of the Askey scheme of hypergeometric orthogonal polynomials. Up to now, these polynomials are defined by their three-term recursion relations and initial values. However, their other properties like the weight functions, generating functions, orthogonality, Rodrigues-type formulas, etc. are yet to be derived analytically. Due to the prime significance of these polynomials in physics and mathematics, we call upon experts in the field of orthogonal polynomials to study them, derive their properties and write them in closed form (e.g., in terms of hypergeometric functions).

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Wilson-Racah quantum system

Cited by 27 publications

References 7 publications

Tridiagonal representation approach in quantum mechanics

Tridiagonal representation approach in quantum mechanics

Potential function of the Wilson-Racah quantum system

Open Problem in Orthogonal Polynomials

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