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Hyperspherical Harmonics - Applications in Quantum Theory (Hardcover, 1989 ed.)
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Hyperspherical Harmonics - Applications in Quantum Theory (Hardcover, 1989 ed.)
Series: Reidel Texts in the Mathematical Sciences, 5
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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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