The Spherical Harmonics Expansion Method

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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.

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.

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.

Tables Of Integrals For The Spherical Harmonic Expansion Of The Hydromagnetic Equations

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Author : Keith L. McDonald
language : en
Publisher:
Release Date : 1969

Tables Of Integrals For The Spherical Harmonic Expansion Of The Hydromagnetic Equations written by Keith L. McDonald and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1969 with Geomagnetism categories.

The method of spherical harmonic expansion of the hydromagnetic equations for incompressible media involves two well-known integral forms that are evaluated by standard numerical integration methods. The results are tabulated to 10 significant figure accuracy for all sets of principled integers inclusive of an 8th-degree harmonic analysis. Extension to compressible media introduces two further surface integral forms that are each simply evaluated in therms of a finite sum of products of the above tabulated integrals. A general development is presented of the known inter-relationships between these four basis integrals, and their selection rules are discussed. Their occurrence in the coupling elements is made explicit in the appendix.

On The Equilvalence Of The Spherical Harmonics Method And The Discrete Ordinate Method Using Gauss Quadrature For The Boltzman Equation

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Author : Robert Gast
language : en
Publisher:
Release Date : 1958

On The Equilvalence Of The Spherical Harmonics Method And The Discrete Ordinate Method Using Gauss Quadrature For The Boltzman Equation written by Robert Gast and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1958 with Harmonic analysis categories.

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.

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.

Spherical Harmonic Reduction Of The Fokker Planck Equation

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Author : Thomas A. Oliphant
language : en
Publisher:
Release Date : 1965

Spherical Harmonic Reduction Of The Fokker Planck Equation written by Thomas A. Oliphant and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1965 with Fokker-Planck equation categories.

The Fokker-Planck equation is reduced to aform that is useful from the viewpoint of doing practical calculations of problems involving configuration space as well as velocity space. The basic technique is a spherical harmonic decomposition in velocity space that reduces the number of independent variables by two. As an example, we show how to apply this method to a problem with theta-pinch geometry.

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.

A Modification Of The Spherical Harmonics Method In Neutron Transport Theory

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Author : A. Sauer
language : en
Publisher:
Release Date : 1958

A Modification Of The Spherical Harmonics Method In Neutron Transport Theory written by A. Sauer and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1958 with Neutron transport theory categories.