The Few-Body Problem

Few-body problems can be handled by conventional integrators, such as Runge-Kutta or Adams-Moulton methods. Here one calculates the position and velocity for each particle and then the precise two-body interaction for that body with every other particle in the system. Both methods are predictor-corrector procedures in which the next step is computed and corrected iteratively. Leapfrog methods, which use the velocity from one step and the positions from the previous step to compute the new positions, are also computationally efficient and stable. The basic problem is to solve the equations of motion for a particle at position Fj, 

When the number of objects becomes large, more than a few dozen, then conventional integration techniques for orbit calculation become inadequate, and new procedures have to be introduced. The primary reason is not any change in the physics. Rather, it is the enormous number of individual quantities that must be tracked for the constituent particles (three coordinates, three velocities). Conventional few-body integrators require N N — 1) calculations per step to determine the motion of the particles. So the rate of calculation scales like For complex systems, like galaxies, this is prohibitively expensive and slow. 

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Reducing the degrees of freedom of the only nucleus is fruitful in the case of a heavy nucleus. In the positronium atom the nucleus has the same mass as the electron and it is useful to treat both particles symmetrically. It is well known that the a4m terms originate not only from relativistic effects, but also from annihilation contributions and the Fermi interaction. Due to that, the most useful approximation is a non-relativistic one and the final single-body equation is an effective Schrodinger equation with Coulomb interaction. This approach, based on an effective equation, was also developed for the few-body problem in nucleus physics. 

If we proceed systematically from small to large systems, then even in elementary quantum mechanics1 the few-body problem, beginning with just three particles, is unsolved. Many recent developments, including current studies of chaos and quantum chaos, serve to underline the fundamental significance of this fact, even for as simple an atom as helium. 

For many-electron atoms, since Coulomb interactions between just two particles are well understood, complexities arising from the nature of the forces do not arise. One can thus concentrate entirely on the few-body problem. The very simplicity of the Coulomb interaction, which made atoms seem uninteresting to many physicists twenty years ago, now serves to place atomic physics, once again, in a central position. 

Pistol, M. E. A -Representability of two-electron densities and density matrices and the application to the few-body problem. Chem. Phys. Lett. 2004,400, 548-552. 

Proc. Conf. on the Few-Body Problem (Eugene, Oregon, 1980) ed. P.S. Levin, Nucl. Phys. A 353 (1981) ... 

We have focused on the lower bound method, but density matrix research has moved forward on a much broader front than that. In particular, work on the contracted Schrodinger equation played an important role in developments. A more complete picture can be found in Coleman and Yukalov s book [23]. It has taken 55 years and work by many scientists to fulfill Coleman s 1951 claim at Chalk River that except for a few details which would be easily overcome in a couple of weeks—the A-body problem has been reduced to a 2.5-body problem ... 

The higher-order two-loop corrections are to be calculated within the so-called external filed approximation (i. e. neglecting by the nuclear motion), while the recoil effects require an essential two-body treatment. There are a few approaches to solve the two-body problem (see e.g. [31]). Most start with the Green function of the two-body system which has to have a pole at the energy of the bound state... 

The CT/ET free energy surface is the central concept in the theory of CT/ ET reactions. The surface s main purpose is to reduce the many-body problem of a localized electron in a condensed-phase environment to a few collective reaction coordinates affecting the electronic energy levels. This idea is based on the Born-Oppenheimer (BO) separation " of the electronic and nuclear time scales, which in turn makes the nuclear dynamics responsible for fluctuations of electronic energy levels (Eigure 1). The choice of a particular collective mode is dictated by the problem considered. One reaction coordinate stands out above all others, however, and is the energy gap between the two CT states as probed by optical spectroscopy (i.e., an experimental observable). 

In order to show the data collapse for quantum few-body problems, let us examine the main assumption we have made in Eq. (60) for the existence of a scaling function for each truncated magnitude (0) with a unique scaling exponent v. 

In this section we plan to review the analytical properties of the eigenvalues of the Hamiltonian for two-electron atoms as a function of the nuclear charge. This system, in the infinite-mass nucleus approximation, is the simplest few-body problem that does not admit an exact solution, but has well-studied ground-state properties. The Hamiltonian in the scaled variables [96] has the form... 

V. Aquilanti, S. Cavalli, C. Coletti, D. Di Domenico, and G. Grossi, Hyperspherical harmonics as sturmian orbitals in momentum space a systematic approach to the few-body coulomb problem. Int. Rev. in Phys. Chem., 20 673-709, 2001. 

The scattering of a proton with a He atom is at least a three-body problem involving the projectile-active-electron and the projectile-target-core interaction (the four-body problem is reduced to a three-body problem by application of the independent-electron frozen-core model). Therefore, the conversion from impact parameter to projectile-scattering angle should be done carefully. For incident energies above a few hundred eV/amu and for... 

Intriguing questions arise concerning the applicability of the correspondence principle in any quantum system whose underlying classical dynamics becomes chaotic. Also, the Pauli principle (which possesses no classical analogue) somehow contributes to making the quantum few-body problem simpler to handle than the classical one. 

This contribution examines current approaches to Coulomb few-body problems mainly from a methodological perspective, in contrast to recent reviews which have focused on the results obtained for benchmark problems. The methods under discussion here employ wavefunctions which explicitly involve all the interparticle coordinates and use functional forms appropriate to nonadiabatic systems in which all the particles are of comparable mass. The use of such wavefunctions for states of arbitrary angular symmetry is reviewed, and the kinetic-energy operator, written in the interparticle coordinates, is presented in a convenient form. Evaluation of the resultant angular matrix elements is discussed in detail. For exponentially correlated wavefunctions, problems of integral evaluation are surveyed, the relatively new analytical procedures are summarized, and relations among matrix elements are presented. The current status of Gaussian-orbital and Hylleraas methods is also reviewed. 

However, some of the recent experiments cast doubt on the applicability of this assumption. First, experiments done in the gas phase are few-body problems where taking the thermodynamic limit is not always appropriate. In other words, we have to take into account the fact that the size of the environment is finite. Second, initial states prepared by laser are so highly excited that the timescale for the energy redistribution would be comparable to that of the reaction. Third, the timescale for observing reactions can be much shorter than that for relaxation. Therefore, dynamical behavior of reactions should be studied without assuming local equilibrium. 

J.L. Friar, Invited Talk at the Theory Institute on The Nuclear Hamiltonian and Electromagnetic Current for the 90s, Argonne, IL, 1991 quoted in S.A. Coon and M.T. Pena, Proc. Xlllth European Conference on Few-Body Problems, Elba, Italy, 1991, Few-Body Systems, Suppl. 6 (1992) 242. 

The evaluation of the flux or random force correlation functions involves the many-body problem and is very difficult. However, the assumption that is rapidly decaying and insensitive to critical points allows much to be done. Many schemes for approximating A(/) have been proposed, all based on the idea that decays in a few molecular collision times. Furthermore,... 

The separation of polymer-polymer interactions into these two types is a division into qualitatively different kinds of interactions. Type a interactions involve only a few monomers at a time and therefore represent a few-body problem. The configurational statistics of polymers which have only short-range interactions can usually be treated exactly by using, say, the mathematical methods of the one-dimensional Ising model, or. 

In this chapter, we present a brief review of the two-body problem, as it enters into the nonrelativistic quark model of mesons. Experts can easily skip this chapter and read instead more advanced reviews on quarkonium [17-20]. Note, however, that we present some mathematical results on the level ordering, level spacing, and mass dependence of the energy, which will be useful for the generalization to the three-body case. Important concepts will be introduced for nonexperts Jacobi variables, scaling laws, numerical solutions, i.e., the basic tools for few-body calculations. 

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 Bom-Oppenheimer method is very often used in molecular physics and other few-body problems and always turns out to be very efficient and to actually work better than expected. This... 

W. Gloeckle, The Quantum Mechanical Few-Body Problem (Springer, Berlin, 1983) ... 

Threshold, to be published in the proceedings of the 12th Int. Conference on Few Body Problems in Physics, Vancouver, B.C. Canada, July 2-8, 1989. 

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