This introductory chapter sets out a general structure of physical theories which is applicable to both classical and quantum mechanics. We start with measurable properties of a given physical system, be it classical or quantum. These properties are called observables . We then introduce a definition of the state of a physical system in terms of measured values of a sufficiently large set of observables. A theory to describe the system should consists of four basic components: Basic mathematical framework This comprises a set of elements endowed with some specific mathematical structure and properties. In mathematics such a set is generally known as a space . Description of states States are described by elements of the space in the chosen mathematical framework. For this reason the space is called the state space of the system. Description of observables Observables are to be described by quantities defined on the state space. The description should yield all possible values of observables. The relationship between observables and states should be explicitly stated. The following two cases are of particular interest: 2 For a deterministic theory like classical mechanics a state should determine the values of all observables. For a probabilistic theory like quantum mechanics a state should determine the probability distribution of the values of all observables. Description of time evolution (dynamics).
This introductory chapter sets out a general structure of physical theories which is applicable to both classical and quantum mechanics. We start with measurable properties of a given physical system, be it classical or quantum. These properties are called observables . We then introduce a definition of the state of a physical system in terms of measured values of a sufficiently large set of observables. A theory to describe the system should consists of four basic components: Basic mathematical framework This comprises a set of elements endowed with some specific mathematical structure and properties. In mathematics such a set is generally known as a space . Description of states States are described by elements of the space in the chosen mathematical framework. For this reason the space is called the state space of the system. Description of observables Observables are to be described by quantities defined on the state space. The description should yield all possible values of observables. The relationship between observables and states should be explicitly stated. The following two cases are of particular interest: 2 For a deterministic theory like classical mechanics a state should determine the values of all observables. For a probabilistic theory like quantum mechanics a state should determine the probability distribution of the values of all observables. Description of time evolution (dynamics).
A physical theory aims to give a description of the properties and behaviour of a physical system, both qualitatively and quantitatively. Qualitatively a theory should explain the behaviour of a given physical system in terms of some fundamental properties of the system. The study of the mathematical aspects of theoretical physics is often referred to as mathematical physics. Apart from the requirement of mathematical consistency, a theory is judged by how well its predictions agree with empirical observations, especially by its ability to make new and experimentally verifiable predictions. A physical system possesses many physical quantities which reflect the properties of the system. A physical quantity is a property of the system which can be physically observed and measured to yield numerical values. To emphasise their measurability, these quantities are called observables. The relationship between states and observables can be complicated, depending on whether we are dealing with classical systems or quantum systems.
This chapter presents a systematic treatment of quantum theory in terms of six groups of postulates on the fundamental properties of quantum systems: postulate on the description of pure states, postulate on the description of observables and their values, postulate on the transition from classical to quantum systems, postulates on the probabilistic relationship between states and observables, postulates on the time evolution of the system and postulate on states after measurement. It extends the basic theory to discuss mixed states, superselection rules and many-particle systems. The chapter presents a brief discussion of conceptual problems in quantum mechanics. A maximum amount of information about the system corresponds to a set of simultaneously measured values of a complete set of discrete observables of the system. Such an amount of information characterises a state of the system. States characterised by such a maximum amount of information about the system are called pure states.
It is a scientific truism nowadays that quantum mechanical predictions are of statistical or rather of probabilistic nature: a quantum theory assigns a probability distribution to each measurable quantity (observable) of a physical system prepared according to a given preparation prescription (state). Such sets of probability distributions represent the most general quantitative predictions of physical theories. The aim of this study is to determine all theoretical frameworks which yield such a kind of output taking the probabilistic nature of quantitative physical predictions as initial hypothesis. Astonishingly, this project succeeds with a minimal number of additional assumptions, which arise quite naturally in the course of the analysis, and which essentially amount to regularity conditions. The result is that there exist only two classes of physical theories providing quantitative probabilistic predictions: classical theories where observables are represented by real functions over a phase space and states are represented by probability measures upon the phase space, and quantum theories where observables are represented by selfadjoint operators acting on a complex Hilbert space and states are represented by state operators. On empirical grounds, classical theories cannot provide the all-encompassing, fundamental frame for the mathematical description of physical systems. Quantum theories remain, and there is no more complete formal frame for quantitative physical theories. As an additional result, it is shown that there exist physical systems which possess a quantum mechanical but no classical description; a prominent example is the spin system of two spin-1/2 particles. The classical hidden variables model, which J. Bell relates to this system in order to prove his theorem, cannot be a true description of it because no such classical model exists, and Bell's theorem loses its foundation.
From the outset statistical mechanics will be framed in the language of quantum theory. The typical macroscopic system is composed of multiple constituents, and hence described in some many-particle Hilbert space. In general, not much is known about such a system, certainly not the precise preparation of all its microscopic details. Thus, its description requires a more general notion of a quantum state, a so-called mixed state. This chapter begins with a brief review of the basic axioms of quantum theory regarding observables, pure states, measurements, and time evolution. Particular attention is paid to the use of projection operators and to the most elementary quantum system, a two-level system. The chapter then motivates the introduction of mixed states and examines in detail their mathematical representation and properties. It also dwells on the description of composite systems, introducing, in particular, the notions of statistical independence and correlations.
Vectors and operators in a Hilbert space are independent quantities and there is no obvious link between them. The mathematical description of the states and observables of a quantum system given by Postulates 25.1(PS) and 26.1(OV) in terms of vectors and operators does not tell us how a given state would determine the value of an observable. Physically the relationship between states and observables is about how the measurable values of an observable is related to a state. This relationship should lead to the quantum properties QMP5.3(1) to QMP5.3(3). First we must know how a state can give rise to a probability distribution of measurable values of an observable. This would require a prescription to relate a unit vector to the spectrum of selfadjoint operators. The discussion in Chapter 22 tells us how to obtain such a prescription. Because of the intricacy of the relationship both physically and mathematically we shall set out the relationship for discrete and continuous observables separately.
From this chapter we start our exposition of quantum mechanics. First, we lay down the typical setting of a quantum mechanical problem. The fundamental feature of quantum indeterminism and the probabilistic nature of quantum mechanics is emphasized. We proceed by constructing, from the single notion of measurement, all the essential conceptual ingredients of quantum mechanics (i.e., the concepts of observable, state and system). Then we formulate the target questions that quantum mechanics aims to solve. Next, we proceed to state the specific laws that constitute the logical premise of the subject. Finally, we demonstrate how the laws, most dominantly, the superposition principle (the fact that quantum probabilities interfere) provides an improved recipe for obtaining the answers to the target questions of quantum mechanics. This also leads to a more refined definition of the quantum state. In order to be able to quickly introduce the basic framework of quantum mechanics, we restrict the discussion to a simple scenario of “instantaneously subsequent measurements” (where there is no time evolution), and where states can be prepared by a single measurement (in the standard jargon, this would be called, “quantum mechanics without degeneracy”; these are descriptions involving CSCOs comprising just one observable).