Hyperspherical Harmonics and Generalized Sturmians

Hyperspherical Harmonics and Generalized Sturmians PDF

Author: John S. Avery

Publisher: Springer Science & Business Media

Published: 2006-04-11

Total Pages: 202

ISBN-13: 0306469448

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This text explores the connections between the theory of hyperspherical harmonics, momentum-space quantum theory and generalized Sturmian basis functions. It also introduces methods which may be used to solve many-particle problems directly, without the use of the self-consistent-field approximation.; The method of many-electron Sturmians offers an interesting alternative to the usual SCF-CI methods for calculating atomic and molecular structure. When many-electron Sturmians are used, and when the basis potential is chosen to be the attractive potential of the nuclei in the system, the following advantages are offered: the matrix representation of the nuclear attraction potential is diagonal; the kinetic energy term vanishes from the secular equation; the Slater exponents of the atomic orbitals are automatically optimized; convergence is rapid; a correlated solution to the many-electron problem can be obtained directly, without the use of the SCF approximation; and excited states can be obtained with good accuracy.; The text should be of interest to advanced students and research workers in theoretical chemistry, physics and mathematics.

Hyperspherical Harmonics

Hyperspherical Harmonics PDF

Author: John S. Avery

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 265

ISBN-13: 9400923236

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where d 3 3)2 ( L x - -- i3x j3x j i i>j Thus the Gegenbauer polynomials play a role in the theory of hyper spherical harmonics which is analogous to the role played by Legendre polynomials in the familiar theory of 3-dimensional spherical harmonics; and when d = 3, the Gegenbauer polynomials reduce to Legendre polynomials. The familiar sum rule, in 'lrlhich a sum of spherical harmonics is expressed as a Legendre polynomial, also has a d-dimensional generalization, in which a sum of hyper spherical harmonics is expressed as a Gegenbauer polynomial (equation (3-27»: The hyper spherical harmonics which appear in this sum rule are eigenfunctions of the generalized angular monentum 2 operator A , chosen in such a way as to fulfil the orthonormality relation: VIe are all familiar with the fact that a plane wave can be expanded in terms of spherical Bessel functions and either Legendre polynomials or spherical harmonics in a 3-dimensional space. Similarly, one finds that a d-dimensional plane wave can be expanded in terms of HYPERSPHERICAL HARMONICS xii "hyperspherical Bessel functions" and either Gegenbauer polynomials or else hyperspherical harmonics (equations ( 4 - 27) and ( 4 - 30) ) : 00 ik·x e = (d-4)!!A~oiA(d+2A-2)j~(kr)C~(~k'~) 00 (d-2)!!I(0) 2: iAj~(kr) 2:Y~ (["2k)Y (["2) A A=O ). l). l)J where I(O) is the total solid angle. This expansion of a d-dimensional plane wave is useful when we wish to calculate Fourier transforms in a d-dimensional space.

Hyperspherical Harmonics And Their Physical Applications

Hyperspherical Harmonics And Their Physical Applications PDF

Author: Avery James Emil

Publisher: World Scientific

Published: 2017-11-27

Total Pages: 300

ISBN-13: 9813229314

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Hyperspherical harmonics are extremely useful in nuclear physics and reactive scattering theory. However, their use has been confined to specialists with very strong backgrounds in mathematics. This book aims to change the theory of hyperspherical harmonics from an esoteric field, mastered by specialists, into an easily-used tool with a place in the working kit of all theoretical physicists, theoretical chemists and mathematicians. The theory presented here is accessible without the knowledge of Lie-groups and representation theory, and can be understood with an ordinary knowledge of calculus. The book is accompanied by programs and exercises designed for teaching and practical use. Contents: PrefaceHarmonic FunctionsGeneralized Angular MomentumGegenbauer PolynomialsFourier Transforms in d DimensionsFock's Treatment of Hydrogenlike Atoms and Its GeneralizationD-Dimensional Hydrogenlike Orbitals in Direct SpaceGeneralized SturmiansChoosing Appropriate Hyperspherical RepresentationsMolecular Integrals from Hyperspherical HarmonicsLagrangians for Particles and FieldsCoordinate Transformations for N BodiesSome Illustrative ExamplesAppendices: The D-Dimensional Harmonic OscillatorMolecular Integrals for Slatertype OrbitalsBibliographyIndex Readership: Scientists and researchers in theoretical physics, theoretical chemistry, and mathematics. Keywords: Harmonic Functions;Reactive Scattering Theory; Nuclear Physics;Gegenbauer Polynomials;Generalized Sturmians;Slatertype OrbitalsReview: Key Features: Exercises are included at the end of each chapterThe e-version of the exercises and solutions can be found in the supplementary tab

Hyperspherical Harmonics Expansion Techniques

Hyperspherical Harmonics Expansion Techniques PDF

Author: Tapan Kumar Das

Publisher: Springer

Published: 2015-11-26

Total Pages: 170

ISBN-13: 8132223616

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The book provides a generalized theoretical technique for solving the fewbody Schrödinger equation. Straight forward approaches to solve it in terms of position vectors of constituent particles and using standard mathematical techniques become too cumbersome and inconvenient when the system contains more than two particles. The introduction of Jacobi vectors, hyperspherical variables and hyperspherical harmonics as an expansion basis is an elegant way to tackle systematically the problem of an increasing number of interacting particles. Analytic expressions for hyperspherical harmonics, appropriate symmetrisation of the wave function under exchange of identical particles and calculation of matrix elements of the interaction have been presented. Applications of this technique to various problems of physics have been discussed. In spite of straight forward generalization of the mathematical tools for increasing number of particles, the method becomes computationally difficult for more than a few particles. Hence various approximation methods have also been discussed. Chapters on the potential harmonics and its application to Bose-Einstein condensates (BEC) have been included to tackle dilute system of a large number of particles. A chapter on special numerical algorithms has also been provided. This monograph is a reference material for theoretical research in the few-body problems for research workers starting from advanced graduate level students to senior scientists.

Spherical Harmonics and Approximations on the Unit Sphere: An Introduction

Spherical Harmonics and Approximations on the Unit Sphere: An Introduction PDF

Author: Kendall Atkinson

Publisher: Springer Science & Business Media

Published: 2012-02-17

Total Pages: 253

ISBN-13: 3642259820

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These notes provide an introduction to the theory of spherical harmonics in an arbitrary dimension as well as an overview of classical and recent results on some aspects of the approximation of functions by spherical polynomials and numerical integration over the unit sphere. The notes are intended for graduate students in the mathematical sciences and researchers who are interested in solving problems involving partial differential and integral equations on the unit sphere, especially on the unit sphere in three-dimensional Euclidean space. Some related work for approximation on the unit disk in the plane is also briefly discussed, with results being generalizable to the unit ball in more dimensions.

Spherical Harmonics In P Dimensions

Spherical Harmonics In P Dimensions PDF

Author: Costas Efthimiou

Publisher: World Scientific

Published: 2014-03-07

Total Pages: 156

ISBN-13: 981459671X

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The current book makes several useful topics from the theory of special functions, in particular the theory of spherical harmonics and Legendre polynomials in arbitrary dimensions, available to undergraduates studying physics or mathematics. With this audience in mind, nearly all details of the calculations and proofs are written out, and extensive background material is covered before exploring the main subject matter.

Geometric Applications of Fourier Series and Spherical Harmonics

Geometric Applications of Fourier Series and Spherical Harmonics PDF

Author: H. Groemer

Publisher: Cambridge University Press

Published: 1996-09-13

Total Pages: 343

ISBN-13: 0521473187

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This book provides a comprehensive presentation of geometric results, primarily from the theory of convex sets, that have been proved by the use of Fourier series or spherical harmonics. An important feature of the book is that all necessary tools from the classical theory of spherical harmonics are presented with full proofs. These tools are used to prove geometric inequalities, stability results, uniqueness results for projections and intersections by hyperplanes or half-spaces and characterisations of rotors in convex polytopes. Again, full proofs are given. To make the treatment as self-contained as possible the book begins with background material in analysis and the geometry of convex sets. This treatise will be welcomed both as an introduction to the subject and as a reference book for pure and applied mathematics.

Approximation Theory and Harmonic Analysis on Spheres and Balls

Approximation Theory and Harmonic Analysis on Spheres and Balls PDF

Author: Feng Dai

Publisher: Springer Science & Business Media

Published: 2013-04-17

Total Pages: 447

ISBN-13: 1461466601

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This monograph records progress in approximation theory and harmonic analysis on balls and spheres, and presents contemporary material that will be useful to analysts in this area. While the first part of the book contains mainstream material on the subject, the second and the third parts deal with more specialized topics, such as analysis in weight spaces with reflection invariant weight functions, and analysis on balls and simplexes. The last part of the book features several applications, including cubature formulas, distribution of points on the sphere, and the reconstruction algorithm in computerized tomography. This book is directed at researchers and advanced graduate students in analysis. Mathematicians who are familiar with Fourier analysis and harmonic analysis will understand many of the concepts that appear in this manuscript: spherical harmonics, the Hardy-Littlewood maximal function, the Marcinkiewicz multiplier theorem, the Riesz transform, and doubling weights are all familiar tools to researchers in this area.

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