# Comment on "Fourier transform of hydrogen-type atomic orbitals'', Can. J. Phys. Vol.\ 96, 724 - 726 (2018) by N. Y\"{u}k\c{c}\"{u} and S. A. Y\"{u}k\c{c}\"{u}

## TL;DR

This paper critiques a recent derivation of the Fourier transform of hydrogenic orbitals, highlighting its complexity compared to classical results and discussing alternative expansion methods for related functions.

## Contribution

It clarifies the differences between a recent approach and classical results, and explores alternative expansions for Fourier transforms of hydrogen-like functions.

## Key findings

- Classical Fourier transform results by Podolsky and Pauling are simpler and more elegant.
- Recent methods express hydrogen eigenfunctions as finite sums of Slater-type functions, leading to complex expressions.
- Alternative expansions using reduced Bessel functions can reproduce classical results and extend to related functions.

## Abstract

Podolsky and Pauling (Phys. Rev. \textbf{34}, 109 - 116 (1929)) were the first ones to derive an explicit expression for the Fourier transform of a bound-state hydrogen eigenfunction. Y\"{u}k\c{c}\"{u} and Y\"{u}k\c{c}\"{u}, 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 the Podolsky and Pauling 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.