The quantum theory of measurement has been widely regarded, since the 1940s, as an esoteric subject of little relevance to “real” physics. Quite the opposite was true in the 1920s and 1930s when Niels Bohr, John von Neumann, and others were struggling, by means of gedankin experiments, to develop an understanding of how measurements on single quantum objects were to be incorporated into quantum theory. However, the understanding they achieved turned out to be of little use in the quantum mechanical applications that commanded the attention of physicists between 1940 and 1980: the interaction of photons, atomic nuclei, and elementary particles, the theory of masers and lasers, the properties of matter (superfluidity, superconductivity, semiconductors), etc. In each of these applications, although the phenomena studied were quantum mechanical at heart, the measurements used to probe them were so far removed from the quantum domain that the only feature of the quantum theory of measurement which entered into the experiments was the probability interpretation of squared amplitudes. There was no need to invoke what soon came to be regarded as an esoteric, problematic, dubious “collapse (reduction) of the wave function.” Fundamentally, the reason for this irrelevance of the quantum theory of measurement was technological.
The quantum theory of measurement has been widely regarded, since the 1940s, as an esoteric subject of little relevance to “real” physics. Quite the opposite was true in the 1920s and 1930s when Niels Bohr, John von Neumann, and others were struggling, by means of gedankin experiments, to develop an understanding of how measurements on single quantum objects were to be incorporated into quantum theory. However, the understanding they achieved turned out to be of little use in the quantum mechanical applications that commanded the attention of physicists between 1940 and 1980: the interaction of photons, atomic nuclei, and elementary particles, the theory of masers and lasers, the properties of matter (superfluidity, superconductivity, semiconductors), etc. In each of these applications, although the phenomena studied were quantum mechanical at heart, the measurements used to probe them were so far removed from the quantum domain that the only feature of the quantum theory of measurement which entered into the experiments was the probability interpretation of squared amplitudes. There was no need to invoke what soon came to be regarded as an esoteric, problematic, dubious “collapse (reduction) of the wave function.” Fundamentally, the reason for this irrelevance of the quantum theory of measurement was technological.
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
The background for this book was a shift of interest among experimental physicists that began approximately 20 years ago. The shift was from quantum measurements on large ensembles of microscopic quantum objects, to experiments of very high precision on single, macroscopic objects. This shift was accompanied by developments in the “culture” of such measurements and the invention and elaboration of new experimental methods. In parallel with these experimental developments, there was a noticeable increase of theoretical interest in the foundations and interpretation of quantum theory, and even in attempts to revise its foundational principles. Although these efforts have not produced any significant changes in the fundamental structure of quantum theory, they have revealed a rather large number of “blank spots” and unsolved problems, to which the founders of quantum physics had paid no attention. A reasonable number of the blank spots have been filled in during the past decade. The reader has probably noticed that this book can be divided into two parts. In the first (chapters I―VII and XI), the authors have tried to present the modern approach to the quantum theory of measurement. This part can be regarded as a tutorial appendix to textbooks for advanced undergraduate or graduate-level courses in quantum mechanics, where little attention is usually paid to measurement theory.
This book is an up-to-date introduction to the quantum theory of measurement, a fast developing field of intense current interest to scientists and engineers for its potential high technology applications. It is also a subject of importance to students for its central role in the foundations of quantum mechanics. Although the main principles of the field were elaborated in the 1930s by Bohr, Schrödinger, Heisenberg, von Neumann and Mandelstam, it was not until the 1980s that technology became sufficiently advanced to allow its application in real experiments. Quantum measurement is now central to many ultra-high technology developments, such as squeezed light, single atom traps, and searches for gravitational radiation. It is also considered to have great promise for computer science and engineering, particularly for its applications in information processing and transfer. The book contains a pedagogical introduction to the relevant theory written at a level accessible to those with only a modest background in quantum mechanics. It then goes on to discuss aspects of the design of practical quantum measurement systems. This book is essential reading for all scientists and engineers interested in the potential applications of technology near the quantum limit. It will also serve as an ideal supplement to standard quantum mechanics textbooks at the advanced undergraduate or graduate level.
For many years the physical interpretation of quantum theory has been dominated by the “wave-particle duality” attitude of the “Copenhagen school.” This point of view is eloquently described in Bohr's collection of essays on the subject. Despite persistent concerns with apparent paradoxes and limitations to this interpretation (as exemplified by Schrodinger's cat, the paradox of Einstein, Podolsky and Rosen, among others), the Copenhagen view persists de facto in the daily life of the modern physicist. Traditional quantum theory has so successfully explained such a vast amount of data in atomic and molecular physics, solid state physics (and to a lesser extent, elementary particle physics) that little doubt can exist concerning its essential validity. As a consequence of the overwhelming practical success of the “Copenhagen interpretation, ” the latter has acquired the status of dogma. For many years, therefore, most physicists have found it expedient to relegate the puzzling aspects of the theory to philosophers, and mathematicians. Nevertheless, a persistent interest in this subject has produced a significant and fascinating literature, reviewed by Jammer. In recent years, concerns over the proper meaning to be ascribed to quantum theory have produced an increasingly deep and incisive series of investigations. These investigations fall generally into two categories: either (1) discussion of the philosophical content of the theory, or (2) analyses of the mathematical variants of the theory and their connection with differing interpretational schemes. Physicists tend to be detached from a commitment to philosophical issues, because of their realization of the transient character of the meanings attached to theories of the day. As a simple illustration of this we can mention the profound differences between nonrelativistic and relativistic quantum mechanics.
In the standard, textbook treatment of quantum mechanics, the contact with experiment is described in terms of probabilities for obtaining this, that, or another result, when identical experiments are performed on a huge number ( ensemble ) of identical objects. Such an ensemble description was adequate during the first half century of quantum theory, because the technology of that era was incapable of making a measurement without destroying or severely changing the measured object (photon, atom, …). In recent years, however, advances in technology and technique have made possible repetitive quantum measurements on a single quantum object, with each measurement influencing the object only minimally. Such measurements cannot be analyzed using solely the ensemble theory. Additional theoretical concepts are needed. The purpose of this book is to describe the methods and theory of such measurements (as well as the more elementary theory of ensembles of measurements). This book is written for two types of readers: those who have had only a little previous contact with quantum mechanics, and those who have had much. Readers of the first type are presumed to understand atomic physics at a qualitative level. The unstarred sections of this book offer such readers an overview of the present state-of-the-art of quantum measurements on single objects and prospects for the future, as well as an elementary overview of the theory of such measurements.
Abstract The concept of the collapse of the wave function was introduced into quantum theory by John von Neumann in the early 1930s and has since become an integral part of the theory’s orthodox interpretation. But if we were to ask ourselves what evidence we have that this collapse is a real physical phenomenon, we would have to admit that we have none. The notion of a collapse is necessary to explain how a quantum system initially present in a linear superposition state before the process of measurement is converted into a quantum system present in one, and only one, of the measurement eigenstates after the process has occurred. It was introduced into the theory because it is our experience that pointers point in only one direction at a time.