Supersymmetry approaches to the radial bound states of the hydrogen‐like atoms

Chenaghlou, A.; Fakhri, H.

doi:10.1002/qua.20276

Cited by 15 publications

(8 citation statements)

References 25 publications

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“…We note in passing, though, that Jafarizadeh and Fakhri [6] discuss the associated equation and quote an alternative form of the Rodrigues formula (28); also, special cases of (28) are presented in Chenaghlou and Fakhri [3,4]. Finally, Gonçalves treatment of the class of integrals represented by relations (30) is very specific and a somewhat different problem that that treated here and by Areal et al [1] is considered [5].…”

Section: Discussionmentioning

confidence: 99%

On the Rodrigues formula solution of the hypergeometric-type differential equation

Robin

2013

imf

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In this paper, we present a new systematic approach to the solution of the hypergeometric-like differential equation and its associated equation. The method produces, tout court, the general solution of these equations in the form of a combination of a standard Rodrigues formula and a 'generalized' Rodrigues formula, of a type due originally to Gonçalves [5] and recently considered, again, by Area et al [1]. In addition, a novel analysis of a class of integrals determining the generalized Rodrigues formulae is given, which complements an original analysis of Area et al [1]. Finally, the relation between the hypergeometric-type differential equation and the hypergeometric equation is elucidated further, following the work of Koepf and Masjed-Jamel [7]. Mathematics Subject Classification: 33C25; 33C45

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Section: Discussionmentioning

confidence: 99%

On the Rodrigues formula solution of the hypergeometric-type differential equation

Robin

2013

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“…The square integrable associate Bessel functions can be applied to obtain bound states corresponding to some one-dimensional supersymmetric potentials and also some two-dimensional quantum mechanical models having a Lie algebra symmetry [23,38]. Therefore, exponential generating functions corresponding to the formal power series of associate Bessel functions B…”

Section: Exponential Generating Functions For the Associated Bessel Fmentioning

confidence: 99%

“…Generating function corresponding to a given set of special functions is not unique. This manuscript has been devoted to introducing new generating functions for the solutions of the differential equation of associated Bessel functions which can be applied to obtain bound states of some solvable models in the framework of supersymmetric quantum mechanics, such as radial bound states of the hydrogen-like atoms [23]. Then, let us remember that Krall and Frink have first studied the Bessel polynomials in the formalism of hypergeometric functions [24].…”

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confidence: 99%

Exponential generating functions for the associated Bessel functions

Fakhri¹,

Mojaveri²,

Nobary³

2008

J. Phys. A: Math. Theor.

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Similar to the associated Legendre functions, the differential equation for the associated Bessel functions B l,m (x) is introduced so that its form remains invariant under the transformation l → −l − 1. A Rodrigues formula for the associated Bessel functions as squared integrable solutions in both regions l < 0 and l ≥ 0 is presented. The functions with the same m but with different positive and negative values of l are not independent of each other, while the functions with the same l + m (l − m) but with different values of l and m are independent of each other. So, all the functions B l,m (x) may be taken into account as the union of the increasing (decreasing) infinite sequences with respect to l. It is shown that two new different types of exponential generating functions are attributed to the associated Bessel functions corresponding to these rearranged sequences.[1, 2, 3], random graphs and complex networks [4], polymerization kinetics [5], counting problems in combinatorics [6], are some applications of the theory of generating functions. The generating functions are used to obtain expected values (averages), variances, moments and cumulants of distributions, and also, to establish relationships between distributions [7].The exponential generating functions and their numerous generalizations have been alternatively introduced and studied by various methods for the orthogonal polynomials and special functions (see Refs. [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]). The generating functions as continuous functions generally describe the convergence of an infinite summation of given infinite sequences of functions. In this sense, the special polynomials and functions of a given sequence in one variable x can be defined as the coefficients in the expansion of their generating functions. The generating functions include various useful properties and all information that is needed to generate the solutions corresponding to a differential equation or a set of recursion relations between those solutions. Therefore, generating functions are very useful to analyze problems involving summations on the infinite sequences of functions such as coherent states. The application of generating functions for known orthonormal special functions, allows one to derive a compact formula for the coherent states. Generating function corresponding to a given set of special functions is not unique. This manuscript has been devoted to introducing new generating functions for the solutions of the differential equation of associated Bessel functions which can be applied to obtain bound states of some solvable models in the framework of supersymmetric quantum mechanics, such as radial bound states of the hydrogen-like atoms [23]. Then, let us remember that Krall and Frink have first studied the Bessel polynomials in the formalism of hypergeometric functions [24]. Also, some authors have introduced some generating functions for Bessel polynomials [25,26,27]. Moreover, the generating functions associated with the group-theoretic techni...

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“…The Calogero-Sutherland model, half-oscillator, radial part of 3D harmonic oscillator, Morse potential, and radial part of a hydrogen-like atom are the five examples [20][21][22][23][24][25]. After a short presentation of the associated Laguerre functions (polynomials), this article follows the joint way of the first three models, realizing su(1, 1) Lie algebra symmetry, through the mode-independent second-order differential operators with the degree of the polynomial as mode number.…”

Section: Introductionmentioning

confidence: 99%

Approach of the associated Laguerre functions to the su(1,1) coherent states for some quantum solvable models

Fakhri

Dehghani

Mojaveri

2009

Int J of Quantum Chemistry

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Using second-order differential operators as a realization of the su(1, 1) Lie algebra by the associated Laguerre functions, it is shown that the quantum states of the Calogero-Sutherland, half-oscillator and radial part of a 3D harmonic oscillator constitute the unitary representations for the same algebra. This su(1, 1) Lie algebra symmetry leads to derivation of the Barut-Girardello and Klauder-Perelomov coherent states for those models. The explicit compact forms of these coherent states are calculated. Also, to realize the resolution of the identity, their corresponding positive definite measures on the complex plane are obtained in terms of the known functions.

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Supersymmetry approaches to the radial bound states of the hydrogen‐like atoms

Cited by 15 publications

References 25 publications

On the Rodrigues formula solution of the hypergeometric-type differential equation

On the Rodrigues formula solution of the hypergeometric-type differential equation

Exponential generating functions for the associated Bessel functions

Approach of the associated Laguerre functions to the su(1,1) coherent states for some quantum solvable models

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