Comment on “Fourier transform of hydrogen-type atomic orbitals”

Abstract

Podolsky and Pauling (Phys. Rev. 34, 109 (1929) doi:10.1103/PhysRev.34.109) were the first ones to derive an explicit expression for the Fourier transform of a bound-state hydrogen eigenfunction. Yükçü and Yükçü (Can. ), who were apparently unaware of the work of Podolsky and Pauling or of the numerous other earlier references on this Fourier transform, proceeded differently. They expressed a generalized Laguerre polynomial as a finite sum of powers, or equivalently, they expressed a bound-state hydrogen eigenfunction as a finite sum of Slater-type functions. This approach looks very simple, but it leads to comparatively complicated expressions that cannot match the simplicity of the classic result obtained by Podolsky and Pauling. It is, however, possible to reproduce not only Podolsky and Pauling's formula for the bound-state hydrogen eigenfunction, but to obtain results of similar quality also for the Fourier transforms of other, closely related, functions, such as Sturmians, Lambda functions, or Guseinov's functions, by expanding generalized Laguerre polynomials in terms of so-called reduced Bessel functions. Résumé: Podolsky et Pauling (Phys. Rev. 34, 109 (1929) doi:10.1103/PhysRev.34.109) ont été les premiers à développer une expression explicite pour la transformée de Fourier de la fonction propre d'un état lié de l'hydrogène. Yükçü et Yükçü (Can. ) ont procédé différemment, apparemment sans connaître le travail de Podolsky et Pauling, ni les nombreuses références antérieures sur la transformée de Fourier. Ils ont exprimé un polynôme généralisé de Laguerre comme une somme finie de puissances ou, de façon équivalente, ils ont exprimé la fonction propre d'un état lié de l'hydrogène comme une somme finie de fonctions de type Slater. Cette approche semble très simple, mais elle mène à des expressions comparativement compliquées qui ne peuvent pas égaler la simplicité du résultat classique de Podolsky et Pauling. Il reste néanmoins possible de reproduire, non seulement la formule de Podolsky et Pauling, mais d'obtenir des résultats de qualité similaires pour les transformées de Fourier d'autres fonctions étroitement reliées, comme les Sturmiens, les fonctions Lambda ou de Guseinov, en faisant l'expansion des polynômes généralisés de Laguerre en termes de fonctions appelées fonctions réduites (reduced) de Bessel. [Traduit par la Rédaction]Mots-clés : fonctions propres d'états liés de l'hydrogène, transformation de Fourier, polynômes généralisés de Laguerre, série hypergéométrique, fonctions de type Slater, fonctions réduites de Bessel.

Wave functions of the Hydrogen atom in the momentum representation

“…However, contrary to the case of the Fourier transforms of the radial wave functions considered in [1,4,5], in the present case we cannot avoid this problem by invoking the generating functions of the Laguerre and Gegenbauer polynomials. The reason for this is that, following this path, the hypergeometric function which would appear instead of 2 F 1 in equation ( 20) in [5], cannot be transformed to the binomial series (as 1 F 0 , in equation ( 21) in [5] does).…”

“…Notice that the use of Slater-type functions in (30) above, can cause numerical instabilities of the solutions (due to the alternating sign of the (−1) t factor) when the summation extends to very high (N − ℓ + 1) values, a critical remark already expressed in [5]. However, contrary to the case of the Fourier transforms of the radial wave functions considered in [1,4,5], in the present case we cannot avoid this problem by invoking the generating functions of the Laguerre and Gegenbauer polynomials. The reason for this is that, following this path, the hypergeometric function which would appear instead of 2 F 1 in equation ( 20) in [5], cannot be transformed to the binomial series (as 1 F 0 , in equation ( 21) in [5] does).…”

“…In order to find the transformation H, we follow the heuristic idea exposed after (5), and start by finding the (formal) eigenfunctions of pr , that is, for each p r ∈ ]0, ∞[ , we want the solutions to pr ψ pr (r) = ±p r ψ pr (r). As this equation is separable, we readily get…”

“…The aim of this paper is to study the construction of the wave functions of the Hydrogen atom in the momentum representation, a topic which has been considered in [1][2][3][4][5][6][7], among others. To understand why this is a non-trivial exercise in basic quantum mechanics, we need to recall some physical and mathematical ideas that will be important in the following sections.…”