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Hyperspherical Harmonics: Applications in Quantum Theory by John S. Avery

Hyperspherical Harmonics: Applications in Quantum Theory

by John S. Avery
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Book cover type: Paperback
  • ISBN13: 9789401075442
  • Binding: Paperback
  • Subject: N/A
  • Publisher: Springer
  • Publisher Imprint: Springer
  • Publication Date:
  • Pages: 256
  • Original Price: EUR 149.99
  • Language: English
  • Edition: Softcover Repri
  • Item Weight: 421 grams
  • BISAC Subject(s): Physics / Quantum Theory, Chemistry / Physical & Theoretical, and Physics / Atomic & Molecular

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.

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