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Cambridge University Press eBooks

Compound System

Enrico G. Beltrametti, Gianni Cassinelli, Peter Carruthers

Published Dec 28, 1984• 0 citations
Publisher

Abstract

Composition of Different and of Identical Systems It is a general tendency of physics to interpret physical systems as composed of simpler, more elementary subsystems. The question “What is it made of?” is a very tempting and popular one. Though this attitude does not reflect a logical necessity, it has proved to be successful in significant circumstances: we think, e.g., of macroscopic matter as composed of atoms and molecules, of molecules as composed of atoms, of atoms as composed of nuclei and electrons, and of nuclei as composed of protons and neutrons. Loosely speaking, one might expect that this attitude will be successful whenever the interaction that packs the constituent subsystems together is not strong enough to make them lose their identity. Here, we are concerned with the problem of constructing the quantum-mechanical formalism of a compound system knowing the formalisms of the constituent subsystems. Thus our first problem is to specify the Hilbert space of the compound system. For the sake of simplicity, let us restrict ourselves to the case of a system composed of only two subsystems. The prescriptions of quantum theory vary according to whether one is faced with the composition of identical or nonidentical subsystems.

DOI
10.1017/cbo9781107340725.012

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Composition of Different and of Identical Systems It is a general tendency of physics to interpret physical systems as composed of simpler, more elementary subsystems. The question “What is it made of?” is a very tempting and popular one. Though this attitude does not reflect a logical necessity, it has proved to be successful in significant circumstances: we think, e.g., of macroscopic matter as composed of atoms and molecules, of molecules as composed of atoms, of atoms as composed of nuclei and electrons, and of nuclei as composed of protons and neutrons. Loosely speaking, one might expect that this attitude will be successful whenever the interaction that packs the constituent subsystems together is not strong enough to make them lose their identity. Here, we are concerned with the problem of constructing the quantum-mechanical formalism of a compound system knowing the formalisms of the constituent subsystems. Thus our first problem is to specify the Hilbert space of the compound system. For the sake of simplicity, let us restrict ourselves to the case of a system composed of only two subsystems. The prescriptions of quantum theory vary according to whether one is faced with the composition of identical or nonidentical subsystems.
THE HIERARCHY OF QUANTUM SYSTEMS It is frequently noted that some quantum systems are composed of others – their subsystems: For example, a hydrogen atom is composed of a proton and an electron, and an α-particle is composed of two protons and two neutrons. It is less frequently noted that all such quantum systems may also be thought of as subsystems of more extensive systems: Thus, any ten α-particles constitute a quantum system, no matter how widely dispersed – and indeed this system has the important property that its quantum state is required to be symmetric under exchange of any pair of α-particles. In the context of nonrelativistic quantum mechanics, it is reasonable to assume that there is a fixed set of quantum systems, none of whose members has any nontrivial subsystems (i.e., none except itself and the null subsystem Ø , which is a subsystem of every quantum system); any system in this set will be called atomic (note that atoms are not atomic quantum systems in this sense – even a hydrogen atom is composed of an electron and a proton: An electron may be atomic in this sense, but the assumption that it is forms no part of quantum mechanics). All atomic systems together compose the universal quantum system ω , which may be finite or infinite, depending on the total number of atomic systems.
Abstract Various topics concerning the entanglement of composite quantum systems are considered with particular emphasis concerning the strict relations of such a problem with the one of attributing objective properties to the constituents. In particular we will focus our attention to composite quantum systems composed of identical constituents, with the purpose of dealing with subtle issues, which have never been adequately discussed in the literature, originating from the true indistinguishability of the subsystems involved.
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.
In this and the following chapter I apply the present interpretation of quantum mechanics to coupled systems of the type studied by Einstein et al. (1935), Bohm (1951), Bell (1964), and Aspect et al. (1982a, 1982b) (to name just a few seminal contributors to the large and growing literature on this topic). The main conclusions were already anticipated in Chapter 1. In the present interpretation, quantum mechanics offers a detailed theoretical account of the physical events and processes that underlie the correlations exhibited in measurements on such systems. This account depends both on the postulation of irreducible properties of compound systems, and also on the treatment of measurement as a quantum mechanical interaction. In the present chapter I present technical details of this account, and in the next chapter I explore its metaphysical aspects and implications, especially for the notions of holism and causal explanation. For concreteness, I restrict attention here to systems composed of two spin-½ atomic systems: The generalization to other pairs of systems with different spins is straightforward. Such systems may result from a prior interaction between the two component systems, or they may be produced de novo (as a positron-electron pair may be produced by annihilation of a γ-ray). Call the component systems A, B and the compound system A⊕B . The system representative of A⊕B will be a subspace of the tensor product Hilbert space H A ⊗H B .
The early twentieth century was a revolutionary time in the history of physics. People often think of Einstein's special theory of relativity of 1905, which changed our conceptions of time and space. But among physicists, quantum mechanics is usually regarded as an even more radical change in our thinking about the physical world. Quantum mechanics, which was developed between 1900 and 1926, began as a theory of atoms and light but has now become the framework in terms of which all basic physical theories are expected to be cast. We need quantum ideas not only to understand atoms, molecules, and elementary particles, but also to understand the electronic properties of solids and even certain astronomical phenomena such as the stability of white dwarf stars. The theory was radical in part because it introduced probabilistic behavior as a fundamental aspect of the world, but even more because it seems to allow mutually exclusive situations to exist simultaneously in a “quantum superposition.” We will see later that the possibility of quantum superposition is largely responsible for a quantum computer's distinctive advantage over an ordinary computer. The present chapter is devoted to introducing the basic principles of quantum mechanics. There are essentially four components of the mathematical structure of quantum mechanics. We need to know how to represent (i) states, (ii) measurements, (iii) reversible transformations, and (iv) composite systems. We will develop the first three in stages, starting with a very simple case – linear polarization of photons – and working toward the most general case.