Hyperspherical Harmonics

Name: Hyperspherical Harmonics
Brand: Springer Netherlands
SKU: 9789401075442
Price: 829.14 RON
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Applications in Quantum Theory

Autor: John S. Avery

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Editura: Springer Netherlands

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where d 3 3)2 ( L x - -- i3x j3x j i ij Thus the Gegenbauer polynomials play a role in the theory of...

Informații despre carte

Autor

John S. Avery

Limbă

engleză

Legare

Carte - Carte broșată

Publicat

2012

Pagini

256

EAN

9789401075442

ISBN

9401075441

Enbook ID

02177902

Editura

Springer Netherlands

Greutate

427

Dimensiuni

156 x 236 x 14

Descriere completă

where d 3 3)2 ( L x - -- i3x j3x j i ij 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.

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Hyperspherical Harmonics

Applications in Quantum Theory

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