The many-body problem

Fane U 1964 Liouville representation of quantum mechanics with application to relaxation processes Lectures on the Many Body Problem /o 2, ed E R Caianiello (New York Academic) pp 217-39... 

Simulations. In addition to analytical approaches to describe ion—soHd interactions two different types of computer simulations are used Monte Cado (MC) and molecular dynamics (MD). The Monte Cado method rehes on a binary coUision model and molecular dynamics solves the many-body problem of Newtonian mechanics for many interacting particles. As the name Monte Cado suggests, the results require averaging over many simulated particle trajectories. A review of the computer simulation of ion—soUd interactions has been provided (43). 

This potential is referred to in electromagnetism texts as the retarded potential. It gives a clue as to why a complete relativistic treatment of the many-body problem has never been given. A theory due to Darwin and Breit suggests that the Hamiltonian can indeed be written as a sum of nuclear-nuclear repulsions, electron-nuclear attractions and electron-electron repulsions. But these terms are only the leading terms in an infinite expansion. 

Brueckner, K. A., and Levinson, C. A., Phys. Rev. 97, 1344, Approximate reduction of the many-body problem for strongly interacting particles to a problem of SCF fields/ ... 

Rodberg, L. S., Ann. Phys. 2, 199, The many-body problem and the Brueckner approximation."... 

Tredgold, R. H., Phys. Rev. 105, 1421, "Density matrix and the many-body problem."... 

At the end of Section 8.16 we mentioned that the Fock representation avoids the use of multiple integrations of coordinate space when dealing with the many-body problem. We can see here, however, that the new method runs into complications of its own To handle the immense bookkeeping problems involved in the multiple -integrals and the ordered products of creation and annihilation operators, special diagram techniques have been developed. These are discussed in Chapter 11, Quantum Electrodynamics. The reader who wishes to study further the many applications of these techniques to problems of quantum statistics will find an ample list of references in a review article by D. ter Haar, Reports on Progress in Physics, 24,1961, Inst, of Phys. and Phys. Soc. (London). 

Use of the Born-Oppenheimer approximation is implicit for any many-body problem involving electrons and nuclei as it allows us to separate electronic and nuclear coordinates in many-body wave function. Because of the large difference between electronic and ionic masses, the nuclei can be treated as an adiabatic background for instantaneous motion of electrons. So with this adiabatic approximation the many-body problem is reduced to the solution of the dynamics of the electrons in some frozen-in configuration of the nuclei. However, the total energy calculations are still impossible without making further simplifications and approximations. 

These limitations, most urgently felt in solid state theory, have stimulated the search for alternative approaches to the many-body problem of an interacting electron system as found in solids, surfaces, interfaces, and molecular systems. Today, local density functional (LDF) theory (3-4) and its generalization to spin polarized systems (5-6) are known to provide accurate descriptions of the electronic and magnetic structures as well as other ground state properties such as bond distances and force constants in bulk solids and surfaces. 

Singer, Computational Methods in Classical and Quantum Physics, The Many-Body Problem in Statistical Mechanics, Ed. by M. B. Hooper, Advance Pub., London, 1976, p. 289. 

K. A. Brueckner, The Many-Body Problem. J. Wiley and Sons, Inc., New York, 1959. 

M. Moshinsky, in Gronp Theory and the Many-Body Problem (Gordon and Breach, New York, 1968). 

Next, the effect of z on A IT through the transition matrix element Hoj is considered as follows for rigorous determination of IToi, all electrons in the system should be treated. However, for the sake of simplicity, we devote our attention only to the transferring electron the other electrons would be regarded as forming the effective potential (x) for the transferring electron (x the coordinate of the electron given from the ion center). This enables us to reduce the many-body problem to a one-body problem ... 

A central issue in statistical thermodynamic modelling is to solve the best model possible for a system with many interacting molecules. If it is essential to include all excluded-volume correlations, i.e. to account for all the possible ways that the molecules in the system instantaneously interact with each other, it is necessary to do computer simulations as discussed above, because there are no exact (analytical) solutions to the many-body problems. The only analytical models that can be solved are of the mean-field type. 

The most essential step in a mean-field theory is the reduction of the many-body problem to a scheme that treats just a small number of molecules in an external field. The external field is chosen such that it mimics the effect of the other molecules in the system as accurately as possible. In this review we will discuss the Bragg Williams approach. Here the problem is reduced to behaviour of a single chain (molecule) in an external field. Higher order models (e.g. Quasi-chemical or Bethe approximations) are possible but we do not know applications of this for bilayer membranes. 

March, N.H., Young, W.H. and Sampanthar, S. (1967), The Many-Body Problem in Quantum Mechanics, Cambridge University Press, Cambridge. 

The purpose of this chapter is to show and discuss the connection between TD-DFT and Bohmian mechanics, as well as the sources of lack of accuracy in DFT, in general, regarding the problem of correlations within the Bohmian framework or, in other words, of entanglement. In order to be self-contained, a brief account of how DFT tackles the many-body problem with spin is given in Section 8.2. A short and simple introduction to TD-DFT and its quantum hydrodynamical version (QFD-DFT) is presented in Section 8.3. The problem of the many-body wave function in Bohmian mechanics, as well as the fundamental grounds of this theory, are described and discussed in Section 8.4. This chapter is concluded with a short final discussion in Section 8.5. 

MSN.71.1. Prigogine, Irreversibility as a symmetry-breaking process, Nature 246, 67-71 (1973). MSN.72. I. Prigogine, Irreversibility in the Many-Body Problem, J. Biel and J. Rae, eds., Book... 

But there is a more basic difficulty in the Hohenberg-Kohn formulation [19-21], which has to do with the fact that the functional iV-representability condition on the energy is not properly incorporated. This condition arises when the many-body problem is presented in terms of the reduced second-order density matrix in that case it takes the form of the JV-representability problem for the reduced 2-matrix [19, 22-24] (a problem that has not yet been solved). When this condition is not met, an energy functional is not in one-to-one correspondence with either the Schrodinger equation or its equivalent variational principle therefore, it can lead to energy values lower than the experimental ones. 

R. U. Ayres, Variational approach to the many-body problem. Phys. Rev. Ill, 1453 (1958). 

R. D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem, 2nd ed., McGraw-Hill, New York, 1976. 

The many-body problem is reformiilated here by using a system of equations involving only first order Reduced Density Matrices. These matrices correspond to all the states of the spectrum of the system and to the transitions among the different states. Some results concerning the correlation effects are also reported here. 

N.H. March, W.H. Young and S. Sampanthar, 1967, The many-body problem in quantum mechanics, Cambridge University Press... 

Before discussing these points in detail, it is worthwhile to consider how the diffusion equation for relative motion of two species is developed from a reduction of the diffusion equation describing the motion of both species separately. It introduces some of the complexities to the many-body problem and, at the same time, shows an interesting parallel to the theory of bimolecular reaction rates in the gas phase [475]. 

Reactions between species which are present in comparable, and large, concentrations are complicated to analyse because any one species may react with one of several reactants. This competitive effect is one form of the many-body problem and these cannot be solved exactly. 

The variational principle has not been widely used in diffusion kinetic problems. Nevertheless, it is such a powerful technique that it is suitable for discussing the many-body problems which have still to be tackled. Wherever approximate methods are necessary, the variational principle should be considered. The trial function(s) should be chosen with care, based on a good idea of the nature of the trial function from its behaviour in certain asymptotic limits. The only application known to the author of the variation principle to a numerical study of a diffusion kinetic problem on a molecular system is that of Delair et al. [377]. They used the variational principle to generate an implicit finite difference scheme for solving the Debye—Smoluchowski equation. Interesting comments have been made by Brykalski and Krason more in the context of heat diffusion [510]. 

Liquids are difficult to model because, on the one hand, many-body interactions are complicated on the other hand, liquids lack the symmetry of crystals which makes many-body systems tractable [364, 376, 94]. No rigorous solutions currently exist for the many-body problem of the liquid state. Yet the molecular properties of liquids are important for example, most chemistry involves solutions of one kind or another. Significant advances have recently been made through the use of spectroscopy (i.e., infrared, Raman, neutron scattering, nuclear magnetic resonance, dielectric relaxation, etc.) and associated time correlation functions of molecular properties. 

B. Zumino, Lectures on Field Theory and The Many Body Problem, E. R. Caraniello, Ed., Academic Press, London, 1961, p. 37. 

Before the advent of the high speed digital computer, the theoretical treatment of atomic motion was 1imited to systems whose dynamics admitted an approximate separation of the many-body problem into analytically tractable one- or two-body problems. Two approximations were the most useful in making this separation ... 

Unfortunately, the stationary Schrodinger equation (1.13) can be solved exactly only for a small number of quantum mechanical systems (hydrogen atom or hydrogen-like ions, etc.). For many-electron systems (which we shall be dealing with, as a rule, in this book) one has to utilize approximate methods, allowing one to find more or less accurate wave functions. Usually these methods are based on various versions of perturbation theory, which reduces the many-body problem to a single-particle one, in fact, to some effective one-electron atom. 

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