Hyperspherical Harmonics - John S. Avery

Hyperspherical Harmonics

Applications in Quantum Theory

(Autor)

Buch | Softcover
256 Seiten
2012 | Softcover reprint of the original 1st ed. 1989
Springer (Verlag)
978-94-010-7544-2 (ISBN)
171,19 inkl. MwSt
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.

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.

Reihe/Serie Reidel Texts in the Mathematical Sciences ; 5
Zusatzinfo XVI, 256 p.
Verlagsort Dordrecht
Sprache englisch
Maße 155 x 235 mm
Themenwelt Mathematik / Informatik Mathematik Analysis
Naturwissenschaften Chemie Physikalische Chemie
Naturwissenschaften Physik / Astronomie Atom- / Kern- / Molekularphysik
Naturwissenschaften Physik / Astronomie Quantenphysik
Naturwissenschaften Physik / Astronomie Theoretische Physik
ISBN-10 94-010-7544-1 / 9401075441
ISBN-13 978-94-010-7544-2 / 9789401075442
Zustand Neuware
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