The Dirac–Coulomb problem: a mathematical revisit

Alhaidari, A. D.; Bahlouli, H.; Ismail, Mourad E. H.

doi:10.1088/1751-8113/45/36/365204

Cited by 5 publications

(3 citation statements)

References 19 publications

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“…(1) We believe that this formulation could easily be extended to relativistic quantum mechanics. Our recent work on the Dirac-Coulomb problem suggests that this is feasible [23].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Quantum mechanics without potential function

Alhaidari

Ismail

2015

Journal of Mathematical Physics

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In the standard formulation of quantum mechanics, one starts by proposing a potential function that models the physical system. The potential is then inserted into the Schr\"odinger equation, which is solved for the wave function, bound states energy spectrum and/or scattering phase shift. In this work, however, we propose an alternative formulation in which the potential function does not appear. The aim is to obtain a set of analytically realizable systems, which is larger than in the standard formulation and may or may not be associated with any given or previously known potential functions. We start with the wavefunction, which is written as a bounded infinite sum of elements of a complete basis with polynomial coefficients that are orthogonal on an appropriate domain in the energy space. Using the asymptotic properties of these polynomials, we obtain the scattering phase shift, bound states and resonances. This formulation enables one to handle not only the well-known quantum systems but also previously untreated ones. Illustrative examples are given for two- and there-parameter systems.Comment: 25 pages, 1 table, and 3 figure

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“…(1) We believe that this formulation could easily be extended to relativistic quantum mechanics. Our recent work on the Dirac-Coulomb problem suggests that this is feasible [23].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Quantum mechanics without potential function

Alhaidari

Ismail

2015

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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“…All physical 28 information about the system, both structural and dynamical, are contained in these expansion 29 coefficients. The "Tridiagonal Representation Approach (TRA)" is an algebraic method for solving the 30 wave equation (e.g., the Schrödinger or Dirac equation) [1][2][3][4]…”

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confidence: 99%

Special Orthogonal Polynomials in Quantum Mechanics

Ad¹

2019

Preprint

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Using an algebraic method for solving the wave equation in quantum mechanics, we 8 encountered a new class of orthogonal polynomials on the real line. One of these is a four-parameter 9 polynomial with a discrete spectrum. Another that appeared while solving a Heun-type equation 10 has a mix of continuous and discrete spectra. Based on these results and on our recent study of the 11 solution space of an ordinary differential equation of the second kind with four singular points, we 12 introduce a modification of the hypergeometric polynomials in the Askey scheme. Up to now, all of 13 these polynomials are defined only by their three-term recursion relations and initial values. 14 However, their other properties like the weight function, generating function, orthogonality, 15 Rodrigues-type formula, etc. are yet to be derived analytically. This is an open problem in 16 orthogonal polynomials.

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The Dirac–Coulomb problem: a mathematical revisit

Cited by 5 publications

References 19 publications

Quantum mechanics without potential function

Quantum mechanics without potential function

Open Problem in Orthogonal Polynomials

Special Orthogonal Polynomials in Quantum Mechanics

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