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
Get Book Here
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: 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: 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: Yasuyuki Suzuki
Publisher: Springer Science & Business Media
ISBN: 354049541X
Category : Science
Languages : en
Pages : 314
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Book Description
The quantum-mechanical few-body problem is of fundamental importance for all branches of microphysics and it has substantially broadened with the advent of modern computers. This book gives a simple, unified recipe to obtain precise solutions to virtually any few-body bound-state problem and presents its application to various problems in atomic, molecular, nuclear, subnuclear and solid state physics. The main ingredients of the methodology are a wave-function expansion in terms of correlated Gaussians and an optimization of the variational trial function by stochastic sampling. The book is written for physicists and, especially, for graduate students interested in quantum few-body physics.
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: Debashis Mukherjee
Publisher: Springer Science & Business Media
ISBN: 3642613306
Category : Science
Languages : en
Pages : 551
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Book Description
This volume features invited lectures presented in the workshop-cum-symposium on aspects of many-body effects in molecules and extended systems, Calcutta, February 1 - 10, 1988. The organizers invited leading experts to present recent developments of many-body methods as applied to molecules and condensed systems. The panorama portrayed is quite broad, but by no means exhaustive. The emphasis is undoubtedly on a "molecular point of view".
Author: Adelchi Fabrocini
Publisher: World Scientific
ISBN: 9814488135
Category : Science
Languages : en
Pages : 427
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Book Description
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.
Author: Hisashi Horiuchi
Publisher: World Scientific
ISBN: 9814544728
Category :
Languages : en
Pages : 554
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Book Description
The recent rapid innovations in supercomputer technology are changing the concepts of numerical calculations employed in solving a wide variety of nuclear many-body problems. The purpose of the XVII RCNP International Symposium on Innovative Computational Methods in Nuclear Many-Body Problems (INNOCOM97) was to discuss the frontiers of various computational methods and to exchange ideas in wide fields of nuclear physics. The subjects discussed at the symposium covered almost all the areas of nuclear physics.
Author: Antonino Zichichi
Publisher: World Scientific
ISBN: 9814489719
Category : Science
Languages : en
Pages : 690
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Book Description
In August/September 2000, a group of 80 physicists from 53 laboratories in 15 countries met in Erice, Italy, to participate in the 38th Course of the International School of Subnuclear Physics. This book constitutes the proceedings of that meeting. It focuses on the theoretical investigation of several basic unity issues, including: (1) the understanding of gauge theories in both their continuum and lattice versions; (2) the possible existence and relevance of large extra dimensions together with the resultant lowering of the Planck/string scale to the TeV range; (3) the origin and structure of flavour mixing in the quark and lepton (neutrino) sectors.
