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Hyperspherical Harmonics: Applications in Quantum Theory by John S. Avery (Hardcover, 1989)

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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.

Product Identifiers

Publisher
Springer
ISBN-13
9780792301653
eBay Product ID (ePID)
96781436

Product Key Features

Book Title
Hyperspherical Harmonics: Applications in Quantum Theory
Author
John S. Avery
Format
Hardcover
Language
English
Topic
Chemistry, Mathematics, Physics
Publication Year
1989
Type
Textbook
Number of Pages
256 Pages

Dimensions

Item Height
235mm
Item Width
155mm
Volume
5
Item Weight
1250 g

Additional Product Features

Title_Author
John S. Avery
Series Title
Reidel Texts in the Mathematical Sciences
Country/Region of Manufacture
Netherlands

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Hyperspherical Harmonics: Applications in Quantum Theory by John S. Avery (Hardcover, 1989)

About this product

Product Information

Product Identifiers

Product Key Features

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Additional Product Features

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