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Many-Particle Systems

K. Kong Wan

Published Jul 9, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.1201/b22504-33

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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.
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An Individual and Its Possibilities: The State Space The first question in any scientific research is its subject matter: What are we studying? The most general answer is a certain kind of system. A system, no matter how complex, is an individual , which is logically a particular, a unit with characters describable in general terms and a unique identity that enables us to single it out and refer to it unambiguously without invoking its characters. We must delineate the system before we delve into its properties. Thus the first general concept in the formulation of a theory is that of an individual. Individuals are represented in scientific theories by the state spaces , which form the keystone of the theories by defining their topics. The notion of a composite system as an integral individual with its own state space is the synthetic conceptual framework of many-body theories. The general concept of a concrete individual in the real world is much more complicated than that of an abstract individual, which may be merely an unanalyzable x over which predicate variables range. The complication is most apparent in many-body theories, where both the composite system and its constituents are explicitly treated as individuals. An individual is isolated if it is not encumbered by any relation and not guaranteed to be relatable; situated, if it is an interactive part of a system; composite, if analyzable into parts; simple, if not.
Although many of the principles of quantum mechanics can be adequately illustrated by considering systems that consist of only one particle subject to external forces, there are a number of important phenomena that are manifest only in systems containing two or more particles, and these will be the subject of the present chapter. We shall begin with some general statements and go on to consider the case of two interacting particles subject to no external forces, when we shall find that, in this case, the problem can be separated into one describing the behaviour of the centre of mass of the system and another describing the relative motion. We shall then consider the case of two noninteracting particles and show that these can often be treated independently as would be expected. In the case of indistinguishable particles, however, we shall find that a symmetry is imposed on the wave function that ensures that the behaviour of such particles is coupled even when they do not interact. We shall introduce the problem of interacting systems containing more than two identical particles, explaining what is meant by “fermions” and “bosons” and how their properties compare. These results will be illustrated by discussing the energy levels and optical spectra of the helium atom, starting with the approximation that the two electrons do not interact and extending the discussion to the realistic case using perturbation theory. The chapter closes with a consideration of the effects of particle indistinguishability on scattering theory.
Liquids and solids, atoms and molecules, nuclei—all these clearly are interacting many-body systems. Even a nucleon may be regarded as a manyparticle system, not just because it is now known to consist of three quarks interacting via gluons, but because of the possibility in quantum field theory of virtual excitation of many particles from the vacuum. in the table page 56 we list some of the many-particle systems we encounter in the physical world at length scales that range from a few centimeters to a few fermis. Many of these systems exhibit phenomena—superconductivity in solids and fission in nuclei, for example—whose understanding does not follow immediately from knowledge of the constituents of the system and the interactions among the constituents, but requires new concepts and ideas. Many-body physics is the branch of theoretical physics that studies the new phenomena or “emergent properties” that arise from interactions among “elementary” constituents of a many-particle system and provides means and methods for carrying out precise calculations of such characteristic properties of these systems as may be compared with experimental results to verify hypotheses about the nature of the constituents and their interactions.
Bare essentials of statistical mechanics Atoms are examples of many-particle systems, but atoms are extraordinarily simpler than macroscopic systems consisting of 1020-1030 atoms. Despite their great size, many properties of macroscopic systems depend intimately on the microscopic behavior of their microscopic constituents. The proper quantum mechanical description of an N -particle system is a wavefunction that depends on 3N coordinates (3 ways of moving, in general, for every particle) and 4N quantum numbers (3 motional quantum numbers and 1 spin quantum number for every particle). (If the “particles” are molecules there might be additional quantum numbers describing rotations and internal vibrations.) The latter is referred to as a microstate. In a macroscopic solid or liquid there are about 1023 atoms per cm3; in a gas there are about 1020. Thus, the amount of information required to specify the wavefunction for such a system is scandalously large. A fundamental theory of macroscopic systems cannot be based on all of the microscopic details. Instead, a new way of thinking is required. This is known as statistical mechanics.
An atom consists of a nucleus, containing protons and neutrons, with various electrons swarming around the outskirts. The electrons are held in the vicinity of the nucleus as they are negatively charged and hence attracted by the positively charged protons. Although atoms are complicated objects, there are certain situations where they can be treated as simple lumps of matter. Their internal structure has no relevance to the details of what is happening. This sort of situation is exemplified by experiments in which atoms are deliberately smashed into one another at comparatively low energy and as such provide an ideal start in a study of the quantum physics of identical particles. When we are dealing with atoms, there is something more to being ‘identical’ than we can appreciate with larger-scale objects. Atoms of the same type are really identical, measurably identical if you like, which can make a considerable difference in some physically important situations.