Hyperspherical Harmonics: Applications in Quantum Theory - Couverture rigide

Avery, John S.

Couverture rigide

ISBN 10 : 079230165X ISBN 13 : 9780792301653

Editeur : Kluwer Academic Publishers, 1989

Synopsis

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.

Détails de l'édition

• Éditeur: Kluwer Academic Publishers
• Date d'édition: 1989
• Langue: anglais
• ISBN 10: 079230165X
• ISBN 13: 9780792301653
• Reliure: Relié
• Nombre de pages: 272
• Coordonnées du fabricant: Springer Nature Customer Service Center GmbH; ProductSafety@springernature.com; ; Europaplatz 3; Heidelberg; 69115; Allemagne