Hyperspherical Harmonics Expansion Techniques
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Hyperspherical Harmonics Expansion Techniques
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Author : Tapan Kumar Das
language : en
Publisher: Springer
Release Date : 2015-11-26
Hyperspherical Harmonics Expansion Techniques written by Tapan Kumar Das and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-11-26 with Science categories.
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.
Hyperspherical Harmonics And Generalized Sturmians
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Author : John S. Avery
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-04-11
Hyperspherical Harmonics And Generalized Sturmians written by John S. Avery and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-04-11 with Science categories.
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
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Author : John S. Avery
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Hyperspherical Harmonics written by John S. Avery and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Science categories.
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
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Author : Avery James Emil
language : en
Publisher: World Scientific
Release Date : 2017-11-27
Hyperspherical Harmonics And Their Physical Applications written by Avery James Emil and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-11-27 with Science categories.
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
The Spherical Harmonics Expansion Method
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Author : Jan Mohring
language : en
Publisher:
Release Date : 1996
The Spherical Harmonics Expansion Method written by Jan Mohring and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996 with categories.
Spherical Harmonics And Approximations On The Unit Sphere An Introduction
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Author : Kendall Atkinson
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-02-17
Spherical Harmonics And Approximations On The Unit Sphere An Introduction written by Kendall Atkinson and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-02-17 with Mathematics categories.
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.
Correlation Function Hyperspherical Harmonics Method Cfhhm Program Package
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Author : Rajmund Krivec
language : en
Publisher:
Release Date : 2003
Correlation Function Hyperspherical Harmonics Method Cfhhm Program Package written by Rajmund Krivec and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003 with categories.
Fast Convergent Hyperspherical Expansion And Its Application To Precise Nonvariational Atomic Calculations
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Author :
language : en
Publisher:
Release Date : 1985
Fast Convergent Hyperspherical Expansion And Its Application To Precise Nonvariational Atomic Calculations written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1985 with categories.
An efficient method of solving the three-body Schroedinger equation is presented. The wave function is decomposed into the product of a correlation factor describing the singularity and clustering structure, and a smooth factor expanded in hyperspherical harmonics. The application to the Helium atom yields a ground state energy of 2.9037244 (2.9033052) au for infinite (finite) nuclear mass. The convergence pattern shows that the accuracy of these values is better than a few parts in 10 to the 8th power.
Variational Boundary Conditions For The Spherical Harmonics Approximation To The Neutron Transport Equation
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Author : Gerald C. Pomraning
language : en
Publisher:
Release Date : 1963
Variational Boundary Conditions For The Spherical Harmonics Approximation To The Neutron Transport Equation written by Gerald C. Pomraning and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1963 with Neutron transport theory categories.
Computation Of Spherical Harmonics And Approximation By Spherical Harmonic Expansions
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Author : Willi Freeden
language : en
Publisher:
Release Date : 1985
Computation Of Spherical Harmonics And Approximation By Spherical Harmonic Expansions written by Willi Freeden and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1985 with Gravity categories.
A technique is developed for generating spherical harmonics by exact computation (in integer mode) thereby circumventing any source of rounding errors. Essential results of the theory of spherical harmonics are recapitulated by intrinsic properties of the space of homogeneous harmonic polynomials. Exact computation of (maximal) linearly independent and orthonormal systems of spherical harmonics is explained using exclusively integer operations. The numerical efficiency is discussed. The development of exterior gravitational potential in a series of outer (spherical) harmonics is investigated. Some numerical examples are given for solving exterior Dirichlet's boundary-value problems by use of outer (spherical) harmonic expansions for not-necessarily spherical boundaries. Keywords: Homogeneous harmonic polynomials; Spherical harmonics; Exact computation in integer mode; Series expansion into spherical harmonics; Exterior dirichlet's problem.