{"product_id":"9789401075442","title":"Hyperspherical Harmonics: Applications in Quantum Theory","description":"\u003ch1\u003eHyperspherical Harmonics: Applications in Quantum Theory\u003c\/h1\u003e \u003ch2\u003eAvery, John S.\u003c\/h2\u003e \u003cp\u003ewhere d 3 3)2 ( L x - -- i3x j3x j i i\u0026gt;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.\u003c\/p\u003e \u003ch3\u003eDetails\u003c\/h3\u003e \u003cp\u003ePublished by: Springer\u003c\/p\u003e \u003cp\u003ePublication Date: 2012-01-21\u003c\/p\u003e \u003cp\u003eFormat: Paperback\u003c\/p\u003e \u003cp\u003eISBN-13: 9789401075442\u003c\/p\u003e \u003cp\u003eDOI: 10.1007\/978-94-009-2323-2\u003c\/p\u003e \u003cp\u003eDimensions: 235.0cm x155.0cm\u003c\/p\u003e \u003cp\u003ePages: 256.0\u003c\/p\u003e ","brand":"Springer Netherlands","offers":[{"title":"Default Title","offer_id":44422477447308,"sku":"9789401075442","price":152.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0710\/9545\/1788\/files\/9789401075442.jpg?v=1756266592","url":"https:\/\/fh90cf-fv.myshopify.com\/products\/9789401075442","provider":"Late Knight Books and Services, LLC","version":"1.0","type":"link"}