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States and Observables

Huzihiro Araki

Published Oct 21, 1999• 0 citations
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Abstract

Abstract The states and the observables are basic concepts in the description of a physical system and their description has undergone a drastic fundamental change in the transition from the classical theory to the quantum theory.
DOI
10.1093/oso/9780198517733.003.0001

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The quantities in physics that can be measured are conventionally denoted observables : entities that can be observed (directly or indirectly) and to which a value in appropriate units can be assigned. Examples include position, momentum, and energy. Unlike in classical physics, the concept of an observable in quantum theory possesses a significance beyond that of the measurable: to each observable there corresponds a unique operator in an abstract, linear vector space. These operators (and others) govern all aspects of quantum systems. Quantum operators act on abstract vectors in the linear space. Especially important are the eigenvectors and eigenvalues of these operators, in particular because some of the eigenvectors are the quantum states mentioned in the preceding chapters. Furthermore, for any operator that is the mathematical image of an observable, its eigenvalues constitute the entire set of values that measurement of the observable can yield . Finally, from the quantum states one can obtain the probabilities for the occurrence of events in the microscopic world, for example, the most likely value that measurement of an observable will yield. The concepts of operators and eigenvalue problems, as well as those of quantal probabilities, of Hermiticity, and of complete sets, were encountered and exemplified in the preceding chapter using the language of differential equations. While much of quantum mechanics could be presented in terms of this language alone, certain fundamental concepts (spin, for instance) cannot be, and it eventually becomes necessary to consider quantum mechanics as a theoretical framework embedded in that abstract linear vector space known as a Hilbert space.
Quantum theory has created a fundamental change in our ideas as relativity theory had done before. But while relativity theory is essentially classical in its conceptions—it is in fact the culmination of classical physics—quantum mechanics is founded on some new principles. The quantum laws are very different in character from the laws we find in classical theory. In particular, quantum mechanics is a strictly statistical type of theory : from this arises the different conception of causality that marks the new outlook of modern physics. Classical theory describes the so-called macroscopic phenomena, that is, things and events which are more or less of the same size and kind as we find them in ordinary life; and so a description can be given in terms of concepts close to the ones we use in ordinary language. The language of quantum mechanics is based upon a duality principle ; and most of our ordinary language is equally dualistic.
The basic concepts of space and time have undergone radical changes in the course of history, which render the Newtonian notions of elementary particles obsolete. This was inherent in the formulations of quantum mechanics, as the zitterbewegung of quantum-mechanical free particles are not the same as that of classical ones. The quantum field theories do not fare any better as particles and the corresponding fields are inseparable concepts. As physical concepts are becoming fuzzy, the definition of what constitutes an experimental "discovery" of an elementary particle is being revised. In these attempts, the quasi-particles, which are simple mathematical conveniences, tend to be reclassified as physical entities. The current model depictions of physical vacuum as superconducting medium do not accommodate a picture of elementary particles as individual entities. It is a situation where interactions and interactants become entangled and description of either one independent of the other is unattainable.
One of the main purposes of a physical theory is to describe and predict events observed in physical experiments. In this chapter we introduce the key physical concepts used in the description of quantum experiments and present their mathematical formalization as Hilbert space operators. Duality of states and effects Fundamental in quantum theory is the concept of a state, which is usually understood as a description of an ensemble of similarly prepared systems. An effect , on the other hand, is a measurement apparatus that produces either ‘yes’ or ‘no’ as an outcome. The concept of an effect was coined by Ludwig [96] and made popular by Kraus [89]. The duality of states and effects means, essentially, that when a state and an effect are specified we can calculate a probability distribution. This can be compared with the outcomes of real experiments, and it gives a physical meaning to the mathematical Hilbert space machinery. Basic statistical framework A basic situation in physics is the following: we have an object system under investigation, and we are trying to learn something about it by doing an experiment. As a result, a measurement outcome is registered. A statistical theory, such as quantum theory, does not in general predict which individual outcomes will occur in any particular measurement; it merely predicts their probabilities of occurrence. Hence, we take the output of an experiment (which consists of many measurement runs) to be a probability distribution on a set Ω of the possible measurement outcomes.