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arXiv:1907.02953 · American Journal of Physics

Quantum Mechanics of Individual Systems

James B. Hartle

Published Aug 1, 1968• 331 citations
arXivHTMLPDF

Abstract

A formulation of quantum mechanics, which begins by postulating assertions for individual physical systems, is given. The statistical predictions of quantum mechanics for infinite ensembles are then derived from its assertions for individual systems. A discussion of the meaning of the “state” of an individual quantum mechanical system is given, and an application is made to the clarification of some of the paradoxical features of the theory.

DOI
10.1119/1.1975096

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A formulation of quantum mechanics, which begins by postulating assertions for individual physical systems, is given. The statistical predictions of quantum mechanics for infinite ensembles are then derived from its assertions for individual systems. A discussion of the meaning of the “state” of an individual quantum mechanical system is given, and an application is made to the clarification of some of the paradoxical features of the theory.
The Statistical Interpretation of quantum theory is formulated for the purpose of providing a sound interpretation using a minimum of assumptions. Several arguments are advanced in favor of considering the quantum state description to apply only to an ensemble of similarily prepared systems, rather than supposing, as is often done, that it exhaustively represents an individual physical system. Most of the problems associated with the quantum theory of measurement are artifacts of the attempt to maintain the latter interpretation. The introduction of hidden variables to determine the outcome of individual events is fully compatible with the statistical predictions of quantum theory. However, a theorem due to Bell seems to require that any such hidden-variable theory which reproduces all of quantum mechanics exactly (i.e., not merely in some limiting case) must possess a rather pathological character with respect to correlated, but spacially separated, systems.
Standard quantum mechanics—the theory that is made use of in practice and is, because of this, taught in university courses— is a theory that is weakly objective only. For this reason it is most conveniently and neatly expressed by means of a set of basic rules of calculation that enable physicists to make predictions concerning what will be observed. Classical and quantum physics both make considerable use of the expressions "physical system" and "state of a physical system." Quantum mechanics is essentially a probabilistic theory. Except in special cases it makes no predictions that bear on individual systems. Rather, it predicts statistical frequencies. In other words, it predicts, as a rule, the number n of times that a given event will be observed when a large number N of physical systems of the same type and satisfying specified conditions are subjected to a measurement process.
Our purpose in this chapter is to present the key concepts of quantum mechanics in the language of Hilbert spaces. The reader who has not previously met the physical ideas motivating quantum mechanics, and some of the more elementary applications of Schrödinger's equation, is encouraged to read any of a number of excellent texts on the subject such as [1–4]. Otherwise, the statements given here must to a large extent be taken on trust – not an altogether easy thing to do, since the basic assertions of quantum theory are frequently counterintuitive to anyone steeped in the classical view of physics. Quantum mechanics is frequently presented in the form of several postulates, as though it were an axiomatic system such as Euclidean geometry. As often presented, these postulates may not meet the standards of mathematical rigour required for a strictly logical set of axioms, so that little is gained by such an approach. We will do things a little more informally here. For those only interested in the mathematical aspects of quantum mechanics and the role of Hilbert space see [5–8]. Many of the standard applications, such as the hydrogen atom, will be omitted here as they can be found in all standard textbooks, and we leave aside the enormous topic of measurement theory and interpretations of quantum mechanics. This is not to say that we need be totally comfortable with quantum theory as it stands. Undoubtedly, there are some philosophically disquieting features in the theory, often expressed in the form of socalled paradoxes.
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.
Classical and quantum physics both make considerable use of the expressions "physical system" and "state of a physical system." Indeed, any sufficiently isolated thing—a voltmeter, an electron, or a molecule—is a physical system. The expression physical system is taken in its familiar, intuitive sense. For instance, a partially but sufficiently isolated atom is a physical system or, for short, a system. This chapter summarizes the rules of quantum mechanics. Quantum mechanics is essentially a statistical theory. Except in special cases it makes no prediction that bears on individual systems. The observables which have a continuous eigenvalue spectrum have eigenvectors that are not normalizable. As regards the measurement of these observables, some technical changes should therefore be made in expression. The chapter introduces vector space which has the structure of a Hilbert space for describing ensembles of physical systems of the same type.
Most of the predictions of quantum mechanics are of a statistical nature and therefore make sense only for ensembles. The axiomatics based on the quantum calculus of propositions might open the way toward a general definition of the quantum state of an individual system. It was soon recognized that this leads to separability and thereby to violation of some of the quantum-mechanical predictions. In contemporary literature, the concept of state is not wholly unambiguous. Some authors associate it with the notion of an individual system and say, for instance that a given system is in a given state. Others associate it exclusively with ensembles. The situation is complicated by the fact that for some theorists the concept of state refers only to ensembles that can be described by some state vector. For other authors it refers just as well to ensembles that cannot be associated with any state vector and for which only a density matrix description is therefore acceptable.