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Classical Systems

K. Kong Wan

Published Aug 27, 2020• 0 citations
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Abstract

This chapter sets out some general physical properties of classical systems which are divided into discrete and continuous: Discrete systems These are systems of discrete point particles. The specific structure of classical mechanics is presented with position, linear and angular momenta serving as basic observables. Continuous systems These systems are illustrated by a vibrating string. Continuous systems have different kinds of properties and observables, e.g., wave properties. In particular we have discussed: Description of states by solutions of the classical wave equation. The concept of eigenfunctions with orthonormality property and their superposition and interference. The concept of a complete set of states.

DOI
10.1201/9780429296475-2

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This chapter sets out some general physical properties of classical systems which are divided into discrete and continuous: Discrete systems These are systems of discrete point particles. The specific structure of classical mechanics is presented with position, linear and angular momenta serving as basic observables. Continuous systems These systems are illustrated by a vibrating string. Continuous systems have different kinds of properties and observables, e.g., wave properties. In particular we have discussed: Description of states by solutions of the classical wave equation. The concept of eigenfunctions with orthonormality property and their superposition and interference. The concept of a complete set of 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).
This chapter focuses on the notions of classical instruments and of classical measurements. It reviews the theories, based on the concept of macroscopic properties, that is, on the essential complexity of the measuring apparatus. One of the most distinctive features of quantum mechanics is the fact that it often assigns precise structures—or discrete "choices" among possible different structures—to the physical systems it describes. Hisenberg's uncertainty relations bear on the dispersions of ensembles as a whole, whereas the limitations that serve to define "classical measurements" essentially concern our possibilities of actually performing measurements in different elements of the ensemble. Thus from the uncertainty relations alone one can certainly not derive the conclusion that nonclassical measurements are impossible. The chapter considers the restricted set of observables. It is convenient to use the word "macrostate" to designate the whole set of microstates on which all the classical observables forming a definite complete set have definite values.
Chapter 1 was concerned with the need for a quantal description of microscopic phenomena. The thrust of Chapter 3 is an analysis and comparison of a classical and a quantum system; some mathematical generalizations of the analysis are presented as well. Chapter 4 provides an introduction to Hilbert-space concepts that are the mathematical framework of quantum theory, while, in Chapter 5, the axioms of quantum mechanics for pure states are stated and illustrated. The present chapter is intended to be a bridge between those chapters and Chapter 1. Here we review many of the concepts of classical mechanics that have relevance in quantum mechanics. Equally important, if not more so, we stress the lack of equivalence between a number of classical concepts and their quantal counterparts. This inequivalence is emphasized because of the vocabulary common to classical and quantum mechanics. Classical physics is, of course, the source for much of the vocabulary in quantum physics. However, the ways in which the words are used in the two domains often differ substantially, while the concepts are almost necessarily not identical. Some classical concepts have no direct quantal counterparts. For example, velocity and force are not primary quantal entities. This fact means that the quantal definitions of linear and angular momentum cannot be the same as those in classical mechanics. Similarly, the concept of a “state” in classical physics – the collection of the generalized positions and generalized velocities or momenta – does not carry over to quantum physics.
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.
Introduction In this chapter we discuss the basic physical concepts and equations related to the behavior of particles in the nanoworld. We introduce the Schrödinger wave equation for particles and determine the ways to calculate observable physical quantities. We find that, in wave mechanics, the motion of a particle confined to a finite volume is always characterized by discrete values of the energy and standing-wave-like wavefunctions, i.e., such a motion is quantized. While motion in an infinite space (i.e., free motion) is not quantized and is described by propagating waves, the energy of the particle is characterized by a continuous range of values. Keeping in mind the diverse variants of nanostructures, by using wave mechanics we analyze some particular examples, which highlight important quantum properties of particles. Many of the examples analyzed can serve as the simplest models of nanostructures and will be exploited in following chapters to understand the fundamentals of processes in nanoelectronics. The Schrödinger wave equation From the previous chapter, we conclude that nanosize physical systems are quantum-mechanical systems, inasmuch as their sizes are comparable to typical de Broglie wavelengths of the particles composing these systems. In dealing with quantum-mechanical systems, one aims at determining the wavefunction of a single particle or of the whole system. As we will demonstrate in the subsequent discussion, knowledge of the wavefunction in quantum mechanics is sufficient to describe completely a particle or even a system of particles.
In this chapter the basic concepts of quantum theory are formulated via a self-consistent set of six postulates. These six postulates deal with the following: operator images of physical observables; properties of state vectors and wave functions; the connection with experimental measurements; coordinate-space forms of those operators having classical analogs; the primary dynamical equation of quantum mechanics; and the completeness of the eigenvectors of observables. The postulates are the framework on which the edifice of quantum mechanics is constructed. A separate section is devoted to the statement of each postulate and to a few examples chosen to illustrate or apply it. A comprehensive set of applications is presented in Chapters 6 and 7, where various 1-D systems are used as the vehicles for illustrating the postulates. The basic theory of this chapter is extended and developed in Chapters 8 and 9. Since probability is a key interpretational concept, a brief introduction to this topic is given in an appendix. We emphasize that our formulation of quantum theory (in this chapter and throughout this book) is restricted to the case of pure states. This is not a major restriction. It means that, when we consider ensembles of individual quantum systems (e.g., single particles, atoms, nuclei, etc.), all of the systems in the ensemble are assumed to be in the same quantum state, not necessarily an eigenstate of a quantal operator. Ensembles for which this assumption does not hold are described by a density operator or density matrix rather than by a pure state; treatments of such cases can be found in references cited later.