Hyperspherical Harmonics

where d 3 3)2 ( L x - -- i3x j3x j i ij 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.

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.