Few-body problem

Y. Suzuki and K. Varga, Stochastic Variational Approach to Quantum Mechanical Few-Body Problems, Springer-Verlag, Berlin, 1998. 

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. 

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... 

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. 

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. 

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. 

Harris, F.E. Current methods for Coulomb few-body problems, Adv. Quant. Chem. 2004,47,129-... 

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. 

Current Methods for Coulomb Few-Body Problems interchanging I and l, we reach... 

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. 

Keywords Few-body problems Correlated wavefunctions Li atom... 

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. 

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. 

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. 

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... 

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

Proc. Conf. on Few-Body Problems in Physics (Karlsruhe, 1983) ed. B. Zeitnitz, Nucl. Phys. A 416 (1984) ... 

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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