scholarly journals Fourier transform of hydrogen-type atomic orbitals

2018 ◽  
Vol 96 (7) ◽  
pp. 724-726 ◽  
Author(s):  
N. Yükçü ◽  
S.A. Yükçü
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Transform Method ◽  
Atomic Orbitals ◽  
Mathematical Properties ◽  
The Fourier Transform ◽  
Exponential Type Orbital ◽  
Analytical Expressions

Hydrogen-type atomic orbitals (HTOs) are an important type of exponential-type orbital. These orbitals have some mathematical properties and they are used usually in the theoretical atomic and molecular investigations as special functions to figure out analytical expressions. The Fourier transform method is a great way to convert basis functions into the momentum space, because their Fourier transforms are easier to use in mathematical calculations. In this paper, we obtain new and useful mathematical representations for the Fourier transform of HTOs related with Gegenbauer polynomials and hypergeometric functions, by using recurrence relations of Laguerre polynomials, Rayleigh expansion and some properties of normalized HTOs.

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

2019 ◽  
Vol 97 (12) ◽  
pp. 1349-1360 ◽  
Author(s):  
Ernst Joachim Weniger
Keyword(s):  
Fourier Transform ◽  
Bessel Functions ◽  
Fourier Transforms ◽  
Laguerre Polynomials ◽  
Bound State ◽  
Atomic Orbitals ◽  
Generalized Laguerre Polynomials ◽  
Classic Result ◽  
The Fourier Transform ◽  
Finite Sum

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. J. Phys. 96, 724 (2018) doi: 10.1139/cjp-2017-0728 ), 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.

Multicomponent Analysis Using Fourier Transform Infrared and UV Spectra

1988 ◽  
Vol 42 (2) ◽  
pp. 353-359 ◽  
Author(s):  
Steven M. Donahue ◽  
Chris W. Brown ◽  
Robert J. Obremski
Keyword(s):  
Factor Analysis ◽  
Fourier Transform ◽  
Fourier Transforms ◽  
Uv Spectra ◽  
Data Sets ◽  
Transform Method ◽  
Multicomponent Analysis ◽  
P Matrix ◽  
The Fourier Transform ◽  
Selection Of

Two- and three-component mixtures of methylated benzenes were analyzed with the use of both infrared and UV spectra. The spectra of known mixtures were Fourier transformed and coefficients from the transforms selected to form coordinates of vectors. The resulting vectors were subjected to factor analysis to obtain representations for multicomponent analysis. A total of eight data sets were analyzed by factor analysis after preprocessing by taking the Fourier transforms of the spectra. The eight data sets were also analyzed by the P-matrix method (inverse Beer's law) in the spectral domain after preprocessing of the data to allow selection of the optimum analytical wavenumbers. This spectral method was compared to the Fourier transform method using cross-validation, in which one sample at a time was left out of the standards and treated as an unknown. The Standard Error of Prediction (SEP) was calculated for the two methods for all possible numbers of vectors and numbers of wavenumbers, starting with the number equal to the number of components and increasing up to a total number of standards or some reasonable cut-off value. Processing in the Fourier domain clearly produced the best results for seven of the data sets and equal results for the other set.

Linear and nonlinear generalized Fourier transforms

2006 ◽  
Vol 364 (1849) ◽  
pp. 3231-3249 ◽  
Author(s):  
Beatrice Pelloni
Keyword(s):  
Fourier Transform ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fourier Transforms ◽  
Nonlinear Partial Differential Equations ◽  
Transform Method ◽  
Fourier Transform Method ◽  
Linear And Nonlinear ◽  
Mathematical Techniques ◽  
The Fourier Transform

This article presents an overview of a transform method for solving linear and integrable nonlinear partial differential equations. This new transform method, proposed by Fokas, yields a generalization and unification of various fundamental mathematical techniques and, in particular, it yields an extension of the Fourier transform method.

Wavelet packets associated with linear canonical transform on spectrum

2021 ◽  
pp. 2150030
Author(s):  
M. Younus Bhat ◽  
Aamir H. Dar
Keyword(s):  
Fourier Transform ◽  
Multiresolution Analysis ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Integral Transforms ◽  
Wavelet Packets ◽  
Linear Canonical Transform ◽  
Fresnel Transform ◽  
The Fourier Transform ◽  
Vector Valued

The linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we, in this paper, constructed associated wavelet packets. First, we construct wavelet packets corresponding to nonuniform Multiresolution analysis (MRA) associated with LCT and then those corresponding to vector-valued nonuniform MRA associated with LCT. We investigate their various properties by means of LCT.

scholarly journals Interaction between line cracks in an orthotropic layer

2002 ◽  
Vol 29 (1) ◽  
pp. 31-42 ◽  
Author(s):  
Subir Das
Keyword(s):  
Fourier Transform ◽  
Stress Intensity ◽  
Finite Thickness ◽  
Intensity Factors ◽  
Orthotropic Layer ◽  
Finite Hilbert Transform ◽  
Crack Tips ◽  
The Fourier Transform ◽  
Analytical Expressions ◽  
Elastostatic Problem

We deal with the interaction between three coplanar Griffith cracks located symmetrically in the mid plane of an orthotropic layer of finite thickness2h. The Fourier transform technique is used to reduce the elastostatic problem to the solution of a set of integral equations which have been solved by using the finite Hilbert transform technique and Cooke's result. The analytical expressions for the stress intensity factors at the crack tips are obtained for largeh. Numerical values of the interaction effect have been computed for and results show that interaction effects are either shielding or amplification depending on the location of each crack with respect to each other and crack tip spacing as well as the thickness of the layer.

An analysis of the spectrum of a gravity anomaly over an inclined dike

Geophysics ◽  
1985 ◽  
Vol 50 (9) ◽  
pp. 1500-1501
Author(s):  
B. N. P. Agarwal ◽  
D. Sita Ramaiah
Keyword(s):  
Fourier Transform ◽  
Gravity Anomaly ◽  
Fourier Spectrum ◽  
Fourier Transforms ◽  
Weighted Average ◽  
Window Function ◽  
Average Spectrum ◽  
Amplitude Spectra ◽  
Distance Data ◽  
The Fourier Transform

Bhimasankaram et al. (1977) used Fourier spectrum analysis for a direct approach to the interpretation of gravity anomaly over a finite inclined dike. They derived several equations from the real and imaginary components and from the amplitude and phase spectra to relate various parameters of the dike. Because the width 2b of the dike (Figure 1) appears only in sin (ωb) term—ω being the angular frequency—they determined its value from the minima/zeroes of the amplitude spectra. The theoretical Fourier spectrum uses gravity field data over an infinite distance (length), whereas field observations are available only for a limited distance. Thus, a set of observational data is viewed as a product of infinite‐distance data with an appropriate window function. Usually, a rectangular window of appropriate distance (width) and of unit magnitude is chosen for this purpose. The Fourier transform of the finite‐distance and discrete data is thus represented by convolution operations between Fourier transforms of the infinite‐distance data, the window function, and the comb function. The combined effect gives a smooth, weighted average spectrum. Thus, the Fourier transform of actual observed data may differ substantially from theoretic data. The differences are apparent for low‐ and high‐frequency ranges. As a result, the minima of the amplitude spectra may change considerably, thereby rendering the estimate of the width of the dike unreliable from the roots of the equation sin (ωb) = 0.

scholarly journals Apparent Variation of v sin i and Rotational Modulation in Be Stars

2004 ◽  
Vol 215 ◽  
pp. 93-94
Author(s):  
C. Neiner ◽  
S. Jankov ◽  
M. Floquet ◽  
A. M. Hubert
Keyword(s):  
Fourier Transform ◽  
Magnetic Dipole ◽  
Be Stars ◽  
Line Profiles ◽  
Transform Method ◽  
Rotational Modulation ◽  
Be Star ◽  
Apparent Variation ◽  
The Fourier Transform ◽  
First Time

v sin i was determined by applying the Fourier transform method to the line profiles of two classical Be Stars. A variation is observed in the apparent v sin i which corresponds to the main frequencies associated to nrp modes. Rotational modulation is observed in wind sensitive UV lines of the Be star ω Ori and is associated with an oblique magnetic dipole which is discovered for the first time in a classical Be star.

Bleustein-Gulyaev Waves in an Inhomogeneous Layered Piezoelectric Half-Space

2006 ◽  
Vol 306-308 ◽  
pp. 1223-1228
Author(s):  
Fei Peng ◽  
Hua Rui Liu
Keyword(s):  
Fourier Transform ◽  
Phase Velocity ◽  
Half Space ◽  
Electric Potential ◽  
Piezoelectric Layer ◽  
Field Equations ◽  
Transform Method ◽  
Coupled Field ◽  
Coupled Field Equations ◽  
The Fourier Transform

The propagation of Bleustein-Gulyaev (BG) waves in an inhomogeneous layered piezoelectric half-space is investigated in this paper. Application of the Fourier transform method and by solving the electromechanically coupled field equations, solutions to the mechanical displacement and electric potential are obtained for the piezoelectric layer and substrate, respectively. The phase velocity equations for BG waves are obtained for the surface electrically shorted case. When the layer and the substrate are homogenous, the dispersion equations are in agreement with the corresponding results. Numerical calculations are performed for the case that the layer and the substrate are identical LiNbO3 except that they are polarized in opposite directions. Effects of the inhomogeneities induced by either the layer or substrate are discussed in detail.

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