Hyperspherical Harmonics - Applications in Quantum Theory (Paperback, Softcover reprint of the original 1st ed. 1989)

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.

General

Imprint:	Springer
Country of origin:	Netherlands
Series:	Reidel Texts in the Mathematical Sciences, 5
Release date:	2012
First published:	1989
Authors:	John S. Avery
Dimensions:	235 x 155 x 14mm (L x W x T)
Format:	Paperback
Pages:	256
Edition:	Softcover reprint of the original 1st ed. 1989
ISBN-13:	978-9401075442
Categories:	Books > Science & Mathematics > Physics > Atomic & molecular physics Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Functional analysis Books > Science & Mathematics > Physics > Quantum physics (quantum mechanics) > General Books > Science & Mathematics > Chemistry > Physical chemistry > General
LSN:	9401075441
Barcode:	9789401075442

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