{"product_id":"hyperspherical-harmonics-applications-in-quantum-theory-9789401075442","title":"Hyperspherical Harmonics: Applications in Quantum Theory","description":"\u003cp\u003e • Author(s): John S. Avery\u003cbr\u003e • Publisher: Springer\u003cbr\u003e • Publisher Imprint: Springer\u003cbr\u003e • BISAC: Physics - Quantum Theory\u003c\/p\u003e\u003cp\u003e\u003c\/p\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","brand":"Springer","offers":[{"title":"Paperback","offer_id":45282042675351,"sku":"9789401075442","price":10900.0,"currency_code":"INR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0666\/3471\/1191\/files\/9789401075442.webp?v=1769300977","url":"https:\/\/atlanticbooks.com\/products\/hyperspherical-harmonics-applications-in-quantum-theory-9789401075442","provider":"Atlantic Books","version":"1.0","type":"link"}