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Hyperspherical Harmonics
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Book Synopsis Hyperspherical Harmonics by : John S. Avery
Download or read book Hyperspherical Harmonics written by John S. Avery and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 265 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Hyperspherical Harmonics and Generalized Sturmians by : John S. Avery
Download or read book Hyperspherical Harmonics and Generalized Sturmians written by John S. Avery and published by Springer Science & Business Media. This book was released on 2006-04-11 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Hyperspherical Harmonics And Their Physical Applications by : James Emil Avery
Download or read book Hyperspherical Harmonics And Their Physical Applications written by James Emil Avery and published by World Scientific. This book was released on 2017-11-27 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Hyperspherical Harmonics Expansion Techniques by : Tapan Kumar Das
Download or read book Hyperspherical Harmonics Expansion Techniques written by Tapan Kumar Das and published by Springer. This book was released on 2015-11-26 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Spherical Harmonics and Approximations on the Unit Sphere: An Introduction by : Kendall Atkinson
Download or read book Spherical Harmonics and Approximations on the Unit Sphere: An Introduction written by Kendall Atkinson and published by Springer Science & Business Media. This book was released on 2012-02-17 with total page 253 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Spherical Harmonics by : Claus Müller
Download or read book Spherical Harmonics written by Claus Müller and published by Springer. This book was released on 2006-11-14 with total page 50 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Spherical Harmonics In P Dimensions by : Costas Efthimiou
Download or read book Spherical Harmonics In P Dimensions written by Costas Efthimiou and published by World Scientific. This book was released on 2014-03-07 with total page 156 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Approximation Theory and Harmonic Analysis on Spheres and Balls by : Feng Dai
Download or read book Approximation Theory and Harmonic Analysis on Spheres and Balls written by Feng Dai and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 447 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Geometric Applications of Fourier Series and Spherical Harmonics by : H. Groemer
Download or read book Geometric Applications of Fourier Series and Spherical Harmonics written by H. Groemer and published by Cambridge University Press. This book was released on 1996-09-13 with total page 343 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Hyperspherical Harmonics by : John S Avery
Download or read book Hyperspherical Harmonics written by John S Avery and published by . This book was released on 1989-04-30 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Spherical Harmonic by : Catherine Asaro
Download or read book Spherical Harmonic written by Catherine Asaro and published by Macmillan. This book was released on 2001-12-14 with total page 10 pages. Available in PDF, EPUB and Kindle. Book excerpt: Catherine Asaro is a popular SF writer, combining her diverse talents to blend hard science fiction and heartrending romance into a sweeping epic known as the Saga of the Skolian Empire. This is her trademark series. Ever since Primary Inversion, her very first novel, this series has continued to grow, building a significant readership and receiving widespread praise. All of Asaro's considerable talent is on display in Spherical Harmonic, the direct sequel to The Radiant Seas. Separated for decades by circumstance and political machinations, the Ruby Dynasty, hereditary rulers of the Skolian Empire, struggle to bring together the tattered remnants of their family in the shadow of a disastrous interstellar war. Too many have died, others are presumed lost, yet they must move quickly if they are reassume their rightful place as rulers of Skolia.
Book Synopsis The Theory of Potential and Spherical Harmonics by : Wolfgang Sternberg
Download or read book The Theory of Potential and Spherical Harmonics written by Wolfgang Sternberg and published by . This book was released on 1964 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Theory of Potential and Spherical Harmonics by : Wolfgang J. Sternberg
Download or read book The Theory of Potential and Spherical Harmonics written by Wolfgang J. Sternberg and published by . This book was released on 1952 with total page 332 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author :David Michael P. Mingos Publisher :Springer Science & Business Media ISBN 13 :3642273777 Total Pages :247 pages Book Rating :4.6/5 (422 download)
Book Synopsis Molecular Electronic Structures of Transition Metal Complexes II by : David Michael P. Mingos
Download or read book Molecular Electronic Structures of Transition Metal Complexes II written by David Michael P. Mingos and published by Springer Science & Business Media. This book was released on 2012-01-11 with total page 247 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book reviews current and future trends in modern chemical research, focusing on chemical structure and bonding. Covers development of electronic structure theories for transition metal complexes, orbital models and electronic structure theory and more.
Book Synopsis Symmetry-Adapted Basis Sets by : John Avery
Download or read book Symmetry-Adapted Basis Sets written by John Avery and published by World Scientific. This book was released on 2012 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: In theoretical physics, theoretical chemistry and engineering, one often wishes to solve partial differential equations subject to a set of boundary conditions. This gives rise to eigenvalue problems of which some solutions may be very difficult to find. For example, the problem of finding eigenfunctions and eigenvalues for the Hamiltonian of a many-particle system is usually so difficult that it requires approximate methods, the most common of which is expansion of the eigenfunctions in terms of basis functions that obey the boundary conditions of the problem. The computational effort needed in such problems can be much reduced by making use of symmetry-adapted basis functions. The conventional method for generating symmetry-adapted basis sets is through the application of group theory, but this can be difficult. This book describes an easier method for generating symmetry-adapted basis sets automatically with computer techniques. The method has a wide range of applicability, and can be used to solve difficult eigenvalue problems in a number of fields. The book is of special interest to quantum theorists, computer scientists, computational chemists and applied mathematicians.
Book Synopsis Introduction To Modern Methods Of Quantum Many-body Theory And Their Applications by : Adelchi Fabrocini
Download or read book Introduction To Modern Methods Of Quantum Many-body Theory And Their Applications written by Adelchi Fabrocini and published by World Scientific. This book was released on 2002-08-19 with total page 427 pages. Available in PDF, EPUB and Kindle. Book excerpt: This invaluable book contains pedagogical articles on the dominant nonstochastic methods of microscopic many-body theories — the methods of density functional theory, coupled cluster theory, and correlated basis functions — in their widest sense. Other articles introduce students to applications of these methods in front-line research, such as Bose-Einstein condensates, the nuclear many-body problem, and the dynamics of quantum liquids. These keynote articles are supplemented by experimental reviews on intimately connected topics that are of current relevance. The book addresses the striking lack of pedagogical reference literature in the field that allows researchers to acquire the requisite physical insight and technical skills. It should, therefore, provide useful reference material for a broad range of theoretical physicists in condensed-matter and nuclear theory.
Book Synopsis Hyperspherical Harmonics and Generalized Sturmians by : John S. Avery
Download or read book Hyperspherical Harmonics and Generalized Sturmians written by John S. Avery and published by Springer. This book was released on 2014-03-14 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt:
