Three-body problem, hyperspherical

The full three-body problem in the physical three-dimensional space required development of hyperspherical harmonic expansions [39]. Crucial for further progress was the introduction of discrete analogues for the latter [40-43], based on hyperangular momentum theory [44,45] and leading to the efficient hyperquantization algorithm [46 19]. For other hyperspherical approaches to reaction dynamics, see [50-63],... 

V. Aquilanti and S. Tonzani, Three-body problem in quantum mechanics Hyperspherical elliptic coordinates and harmonic basis sets. J. Chem. Phys., 120(9) 4073, 2004. 

V. Aquilanti, G. Grossi, A. Lagana, E. Pelikan, and H. Klar, A decoupling scheme for a three-body problem treated by expansions into hyperspherical harmonics the hydrogen molecular ion. Lett. Nuovo Cim., 41 541-544, 1984. 

For a prototypical three body problem, the Helium-like atom, the procedure is well known since the early days of quantum mechanics. More recently, Fano, Macek and Klar [18-23] identified a near separable variable p = (rf -I- rD, where rj and T2 are the two Jacobi vectors of the system, named hyperradius , corresponding to the radius of a six-dimensional hypersphere pareuneterized by five hyperangles . However, note that the hyperradius is independent of the numbering of particles and is therefore very useful for rearrangement problems. 

To further illustrate the tree-method, consider the six-dimensional hypersphere which parameterizes the components of Jacobi vectors for the three-body problem the symmetric tree, see fig. 5, corresponds to the hyperspherical harmonics... 

Considering the literature it has to be realized that the hyperspherical coordinates (sometimes called Delves coordinates or mass weighted polar coordinates, too) have guite a long standing tradition in describing three body problems in a variety of physical fields. Originally they seem to have appeared in studies of the helium atom (1932) [1] and since then a continuous stream of publications indicate their application to the treatment of two electron atoms [2] and the molecule [3]. 

The tail of the wave function as is, however, not too well reproduced, since the exponential vanishing of u r) is replaced by a zero of finite order for w(jc) at jc = 1. The main advantage of this method, for our purpose, is its immediate generation to coupled equations. This will be very useful for Faddeev, hyperspherical, or Born-Oppenheimer approaches to the three-body problem. 

Hyperspherical coordinates were introduced by Delves [52] and the formalism of hyperspherical expansion was further developed by many authors [40,53,54] for three-body or more complicated bound states. The usefulness of this method for baryon spectroscopy was shown by several groups [55]. The basic idea is rather simple the two relative coordinates are merged into a single six-dimensional vector. The three-body problem in ordinary space becomes equivalent to a two-body problem in six dimensions, with a noncentral potential. A generalized partial wave expansion leads to an infinite set of coupled radial equations. In practice, however, a very good convergence is achieved with a few partial waves only. 

The hyperspherical method, from a formal viewpoint, is general and thus can be applied to any N-body Coulomb problem. Our analysis of the three body Coulomb problem exploits considerations on the symmetry of the seven-dimensional rotational group. The matrix elements which have to be calculated to set up the secular equation can be very compactly formulated. All intervals can be written in closed form as matrix elements corresponding to coupling, recoupling or transformation coefficients of hyper-angular momenta algebra. 

O.I. Tolstikhin and M. Matsuzawa, Exploring the separability of the three-body coulomb problem in hyperspherical elliptic coordinates. Phys. Rev. A, 63 062705/1-062705/23, 2001. 

This matrix was introduced by F. T. Smith [25] for the treatment of non-adiabatic (diabatic) couplings in atomic collisions. It is now familiar also in molecular structure problems, to indicate local breakdowns of the Born-Oppenheimer approximation. Within the hyperspherical formalism, it has been introduced in the three-body Coulomb problem [20] and in chemical reactions [21-24], see also Section 3. Also, from equation (A4)... 

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