Theory of Many-Particle Systems. I

Stephen J. Gustafson, Israel Michael Sigal

K. Kong Wan

A general physical system is a compound system consisting of many constituents. A constituent may be made up of more than one particle, e.g., a molecule is made of atoms which themselves are made up of many particles. In a superconductor at low temperatures the electrons form pairs. 1 Each pair can be treated as a constituent of a superconducting current. For convenience we also call these constituents particles . A general physical system is referred to as a many-particle system. The theory for many-particle systems can be built up from one-particle theory. The situation can be complicated since there are different types of many-particle systems: A compound system may consists of a definite number of particles, i.e., the number of particles in the system remains the same for all time. Such a system may consist of distinguishable particles, or identical, i.e., indistinguishable, particles. 560 A compound system may consists of an indefinite number of particles, i.e., the number of particles in the system may change in time due to interactions. The particles involved may or may not be identical.

Arjun Sharma

A comprehensive account of the quantum theory of many particle systems is reported in this chapter. Essential features of a system of N-indistinguishable Bosonic and Fermionic particles, the exchange operator, space and spin dependent parts of wave function, properties of ground and excited states of the helium atom with detailed derivations for direct and exchange integrals, Hartree and Hartree-Fock approximations, and Thomas-Fermi and density functional theories for systems of N-particles are the contents of the chapter. Carefully chosen solved examples and unsolved exercises, which demonstrate how to deal with a system of more than a particle are included towards the end of chapter. The elaborated discussions on Hartree and Hartree-Fock approximations and Thomas-Fermi and density functional theories for systems of N-particles are special feature of this chapter.

Stephen J. Gustafson, Israel Michael Sigal

I. I. Ivanchik

Margarete Mattes, M. Sorg

The relativistic Schrödinger theory (RST) is applied to a general 2-particle system with electromagnetic interactions. The emphasis of the investigation lies on the exchange and overlap effects in connection with the development of adequate single-particle and 2-particle concepts. It is possible to eliminate the 2-particle quantities and to formulate the 2-particle theory exclusively in terms of single-particle quantities but then there arises a new unobservable field degree of freedom: ` renormalization '.