| Hyperspherical Harmonics: Applications in Quantum Theory 1989 Edition Contributor(s): Avery, John S. (Author) |
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ISBN: 079230165X ISBN-13: 9780792301653 Publisher: Springer OUR PRICE: $161.49 Product Type: Hardcover Published: April 1989 |
| Additional Information |
| BISAC Categories: - Science | Physics - Quantum Theory - Science | Chemistry - Physical & Theoretical - Mathematics | Mathematical Analysis |
| Dewey: 515.5 |
| LCCN: 89-31039 |
| Series: Reidel Texts in the Mathematical Sciences |
| Physical Information: 0.69" H x 6.14" W x 9.21" (1.24 lbs) 256 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 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. |