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Pure States

K. Kong Wan

Published Jul 9, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.1201/b22504-25

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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.
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).
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.
This chapter starts with the definition of postulates. Here, in quantum mechanics, the postulates concern the atomic and molecular properties, which are quite far removed from everyday experience. The postulates described here make no claim to those happy mathematical adjectives such as ‘rigorous’, ‘complete’, or irreducible. The ‘proof’ of the postulates is their ability to explain the adequate experimental observations. Here, we have incorporated six postulates. The first postulate opines that ‘the state of quantum mechanical objects is described by a wavefunction’, whereas the second postulate says that ‘every physical observable is expressed by a linear operator’. It also describes how a quantum mechanical operator is constructed. The third postulate gives an idea that the measurement of a physical observable will yield a result, which is one of the Eigen values of the corresponding operator for that observable. The fourth postulate gives how the average of a sequence of measurement is connected with an operator. The fifth postulate is called the time-dependent postulate. Here, it gives a relationship between the time-dependent wave function and an operator, whereas the sixth postulate says that the wave function must be antisymmetric (or symmetric) for the exchange of identical fermions. It must be noted that each and every postulate has been explained clearly. The references, solved problems, and questions on concepts have been given at the end of the chapter.
The authors present briefly the basic notions of the quantum formalism: pure and mixed states; quantum observables; quantum probability; Born’s rule, superposition, and ‘state collapse’; the projection postulate (von Neumann–Lüders postulate); Dirac’s ket and bra-vector notations; elements of quantum information theory and quantum logic; Schrödinger’s and von Neumann’s equations; unitary dynamics; and positive operator valued measures. They also define the basic mathematical notions related to the quantum formalism: Hilbert space; scalar product; norm; Hermitian operator (matrix); projector; and unitary operator and adjoint operator. This chapter may be useful for newcomers to the field, but for those readers who have preliminary knowledge about quantum mechanics (QM) they can proceed directly to advanced chapters. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Chapter 2, Postulates of Quantum Mechanics, points to the complementarity and inseparability of quantum mechanics and spectroscopy. To elucidate the close relationship of spectroscopy and quantum mechanics and to arouse interest in the subject, the collaborative evolution of the theoretical and experimental ideas of the quantization of magnetic moment of particle spin is briefed first. The necessary quantum mechanical background is introduced by discussing the physical and statistical meaning of the postulates. The physical meaning of the observable of a measurement is also explained. The principle of superposition of eigenstates and the time evolution of a state function during system-radiation interactions are analyzed. It is mentioned that a system weakly perturbed by an impulse can often be analyzed by perturbation theory. The chapter ends with a discussion of perturbation approaches to degenerate and nondegenerate systems.
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).