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CRC Press eBooks

Individuals, Ensembles, States

Bernard d’Espagnat

Published Mar 5, 2018• 0 citations
Publisher

Abstract

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.

DOI
10.1201/9780429501449-13

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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.
It has always been the ideal of science to discover laws of nature on which we can all agree. Agreement requires evidence independent of individual judgment, i.e., objective evidence. We apply the term objective to such unbiased scientific evidence because we associate it with an external world of objects that we conceive to be the origin of our information. This external world is instinctively invented by each of us as a means of dealing with invariant patterns in his private world of experience. The invention is validated and elaborated as a consequence of the fact that it turns out to be sharable. In fact the common content of all our invented external worlds is so unambiguous and independent of individual bias that it can form the universally accepted basis for communication and action in an organized society. The consistency of the patterns of experience associated with the external world is such that we are under severe psychological compulsion to assign to this world an existence independent of our experience and a reality superior to that of any small part of our experience. Physical science is devoted to the study of the “laws” of the external physical universe, i.e., to the behaviour of the objects in it. Hence there is a strong tendency to project into the external world the information gathered by scientifically objective methods and even the theoretical concepts in terms of which the information is interpreted. Whatever is so projected is objective in a second sense of the term, i.e., assigned by us to the external world of objects.
Introduction In our treatment of quantum mechanics we discussed the notion of the state of a given system. The state was specified by giving the eigenvalues of a maximal set of mutually commuting observables. This represented the most information we could have about a given system in principle. In practice, of course, we do not really have all of the necessary information about the eigenvalues available. No experimentalist sets out with the primary intention of obtaining results which correspond to a maximal set of mutually compatible observables for any macroscopic system, such as a gas. Even a very small volume of gas contains an enormous number of particles. The quantum mechanical state would require us to know a great deal of information about the internal states of the electrons and nuclei of the gas molecules. The experimentalist may only be concerned with gross properties such as the pressure, internal energy, or specific heat for example. Most of the detailed information about the detailed state of the system is neither observed, nor of direct interest. Instead we must content ourselves with a description of the average, or most likely, behaviour of the system in the probabilistic sense. The usual technique for doing this consists of the introduction of an ensemble, meaning a very large collection of systems whose measured macroscopic properties are the same as those for the system we are interested in, but whose unobserved properties can vary.
In quantum physics, we often need to consider ensembles that are not pure cases, that is, that cannot be described by state vectors. This is so, for instance, when an ensemble is composed of several subensembles that are described by nonequivalent state vectors. It also happens in most cases in which we consider ensembles of definite subsystems of some larger systems. The density matrix—or "statistical operator"—formalism is one that makes it possible to deal with such situations swiftly and efficiently. The rules apply only to ensembles that can be described by state vectors. These rules entail as consequences new expressions for quantum probabilities and mean values, which can be called the—derived— "statistical operator rules." However, it is also possible to formulate the statistical operator rules directly instead. The Heisenberg "uncertainty" relations imply that in general f(x,p) cannot be interpreted as really being a joint probability distribution.
What Niels Bohr called the `epistemological lesson' of `complementarity' was the result of reasoning analogically from the classical conception of a mechanical state to a new quantum mechanical conception of an `object' in a mechanical state. Bohr proposed to redefine the `objectivity' essential for scientific description in terms of the epistemological demand for unambiguously communicable descriptions of observational results, a move which has profound consequences for how we can understand the concept of the quantum mechanical state and the nature of the `system' which is `in' this state. Here it is argued that the old notion of the `object' which is in a classical mechanical state is drawn from a substance/property ontology derived from Aristotle's analysis of categorical propositions. In moving to describing a system in a quantum mechanical state, the system that is `in' such a state can no longer be so regarded as a substance possessing properties. Bohr argues that the concept refers to an interaction which has a feature of wholeness or `individuality' that implies that the distinction between `object system' and `observing system' is relative to the context of the description. This conclusion, in turn, implies the need for a combination of complementary modes of description; however, because of his reticence in making ontological claims, he failed to develop this dimension of his new framework of complementarity.
Quantum mechanics has been enormously successful in explaining the behavior of microsystems such as molecules, atoms, and nuclei. We have seen that quantum mechanics describes a quantum system by the wave function ψ, which is the solution of the Schro¨dinger equation. The wave function determines the probability amplitudes (whose modulus squared gives the probability) for all possible outcomes of a measurement corresponding to an observable on the system. In general, the wave function does not uniquely predict the result of a measurement; it only provides a probability distribution for all possible results of the measurement on a quantum system. Only if the system happens to be in an eigenstate corresponding to the observable being measured, is the result of a measurement of this observable completely predictable. On the other hand, the results of measurements corresponding to a conjugate or non-commuting observable, on the same system in the same state, are indeterminate. Quantum mechanical description of physical systems raises profound questions: How are the wave function and the probabilities for different outcomes of a measurement to be interpreted? Does the wave function represent a single system or an ensemble of systems? Does the physical system actually have the attribute corresponding to the measurement in question or does the measurement itself create the attribute? These questions led to different interpretations of quantum mechanics, which were vigorously debated during the years of its discovery as also in the following decades. The new idea of indeterminacy was especially difficult to reconcile with the long-held view that a physical theory should aim at predicting accurately, and without ambiguity, the result of any measurement on any system, big or small. These debates not only gave us a deeper understanding of quantum mechanics but also changed profoundly the way we view the micro-world.