{"product_id":"hyperspherical-harmonics-applications-in-quantum-theory-von-john-s-avery","title":"Hyperspherical Harmonics","description":"where 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.\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9789401075442\"\u003e\u003ch3\u003eApplications in Quantum Theory\u003c\/h3\u003e\u003c\/div\u003e\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9780792301653\"\u003e\u003ch3\u003eApplications in Quantum Theory\u003c\/h3\u003e\u003c\/div\u003e","brand":"Libri","offers":[{"title":"Softcover - 9789401075442","offer_id":39415575478365,"sku":"9789401075442","price":160.49,"currency_code":"EUR","in_stock":true},{"title":"Hardcover - 9780792301653","offer_id":51579467462,"sku":"9780792301653","price":160.49,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/f36c93ed-f210-4278-8fb0-a06d7fb90df0.jpg?v=1774587862","url":"https:\/\/shop.autorenwelt.de\/products\/hyperspherical-harmonics-applications-in-quantum-theory-von-john-s-avery","provider":"Autorenwelt Shop","version":"1.0","type":"link"}