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" Hyperspherical Harmonics : "
by John Avery.
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BL
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| Record Number
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772843
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b592837
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| Main Entry
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by John Avery.
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| Title & Author
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Hyperspherical Harmonics : : Applications in Quantum Theory\ by John Avery.
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| Publication Statement
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Dordrecht : Springer Netherlands, 1989
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| Series Statement
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Reidel Texts in the Mathematical Sciences, A Graduate-Level Book Series, 5.
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(XVI, 256 pages).
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| ISBN
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9400923236
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: 9789400923232
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| Contents
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Harmonic polynomials --; Generalized angular momentum --; Gegenbauer polynomials --; Fourier transforms in d dimensions --; Fock's treatment of hydrogenlike atoms and its generalization --; Many-dimensional hydrogenlike wave functions in direct space --; Solutions to the reciprocal-space Schrödinger equation for the many-center Coulomb problem --; Matrix representations of many-particle Hamiltonians in hyper spherical coordinates --; Iteration of integral forms of the Schrödinger equation --; Symmetry-adapted hyperspherical harmonics --; The adiabatic approximation --; Appendix A: Angular integrals in a 6-dimensional space --; Appendix B: Matrix elements of the total orbital angular momentum operator --; Appendix C: Evaluation of the transformation matrix U --; Appendix D: Expansion of a function about another center --; References.
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| Abstract
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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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Functions, Special.
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Physical organic chemistry.
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Physics.
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| LC Classification
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QC174.26.W28B956 1989
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| Added Entry
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John Avery
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| Parallel Title
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Reidel Texts in the Mathematical Sciences, vol. 5
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