Hyperspherical Harmonics Expansion Techniques PDF Download
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Author: Tapan Kumar Das
Publisher: Springer
ISBN: 8132223616
Category : Science
Languages : en
Pages : 170
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Book Description
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.
Author: Tapan Kumar Das
Publisher: Springer
ISBN: 8132223616
Category : Science
Languages : en
Pages : 170
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Book Description
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.
Author: John S. Avery
Publisher: Springer Science & Business Media
ISBN: 0306469448
Category : Science
Languages : en
Pages : 202
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Book Description
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.
Author: John S. Avery
Publisher: Springer Science & Business Media
ISBN: 9400923236
Category : Science
Languages : en
Pages : 265
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Book Description
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.
Author: James Emil Avery
Publisher: World Scientific
ISBN: 9813229314
Category : Science
Languages : en
Pages : 300
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Book Description
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.
Author: Jan Mohring
Publisher:
ISBN:
Category :
Languages : en
Pages : 141
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Book Description
Author: Rajmund Krivec
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
Author: N. A. Orr
Publisher: Springer Nature
ISBN: 3030323579
Category : Science
Languages : en
Pages : 968
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Book Description
Few-body physics covers a rich and wide variety of phenomena, ranging from the very lowest energy scales of atomic and molecular physics to high-energy particle physics. The papers contained in the present volume provide an apercu of recent progress in the field from both the theoretical and experimental perspectives and are based on work presented at the “22nd International Conference on Few-Body Problems in Physics”. This book is geared towards academics and graduate students involved in the study of systems which present few-body characteristics and those interested in the related mathematical and computational techniques.
Author: Kendall Atkinson
Publisher: Springer Science & Business Media
ISBN: 3642259839
Category : Mathematics
Languages : en
Pages : 253
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Book Description
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.
Author: Hongen Liao
Publisher: Springer Science & Business Media
ISBN: 3642156983
Category : Computers
Languages : en
Pages : 590
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Book Description
This book constitutes the refereed proceedings of the 5th International Workshop on Medical Imaging and Augmented Reality, MIAR 2010, held in Beijing, China, in September 2010. The 60 revised full papers presented were carefully reviewed and selected from 139 submissions. The papers are organized in topical sections on image segmentation, image registration, shape modeling and morphometry, image analysis, diffusion tensor image, computer assisted intervention, medical image computing, visualization and application, segmentation and classification, medical image understanding, image-guided surgery, and augmented reality.
Author: C.A. Weatherford
Publisher: Springer Science & Business Media
ISBN: 9400979215
Category : Science
Languages : en
Pages : 188
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Book Description
The First International Conference on ETO Multicenter Molecular Integrals was held August 3-6, 1981, on the Florida A&M university campus in Tallahassee, Florida, USA. Thirty four scientists from eight countries assembled in Tallahassee under the sponsorship of the Institute for Molecular Computations and the Physics Department at Florida A&M. Financial support is gratefully acknowledged from the National Science Foundation, U.S. Army Research Office (Durham), Office of Naval Research, the National Aeronautics and Space Admini stration (NASA), and Florida A&M University. In particular, the editors would like to thank Dr. Joe Majowicz and Dr. David Squire of the U.S. Army, and Dr. Aaron Temkin of NASA for their support and encouragement. We would also like to acknowledge the Atlanta University Resource Center for Science and Engineering for financial support in the pre paration of the manuscript. Also, of course, we sincerely appreciate the participation of the attendees and especially the contributors to this work. As a result of their presentations, the conference was a very intense and fertile forum for the exchange of ideas on a very important and historic problem of quantum chemistry. Finally, we want to thank Ms. Sonja Richardson for the enthusiastic, diligent and competent preparation of a very difficult manuscript. Charles A. Weatherford Herbert W. Jones vii C. A. Weatherford and H. W. Jones (eds.), ETO Multicenter Molecular Inteffrals, vii.
