This tome is a formal presentation of the unsharp observable approach to quantum mechanics using the positive operator valued (POV) concept of an observable. It is intended for philosophers and mathematicians as well as physicists. This is a very formalistic book. There are, however, portions that should be read by all experimentalists performing quantum mechanical studies as well as graduate students completing their first graduate year quantum mechanics course. If we interpret quantum mechanics as making statistical predictions on the results of states prepared in identical fashions, we will often find that the observed probabilities are nearly, but not exactly, the expected 1 or 0 for measurements that are expected to always give the same result. Hence, rather than demanding sharp measures with probabilities of exactly 1 or 0, quantum mechanics allows a reasonable interpretation of unsharp results of nearly 1 or nearly 0. As the authors point out in chapter II, POV measures and the associated unsharp measures are not essential for an interpretation of quantum mechanical results. However, more sharp variables, and hence a larger Hilbert space, will be needed than are required by the unsharp approach. Also, the theoretical interpretation will often not be as straightforward with only sharp observables. As I have previously stated, all graduate students, both those of a theoretical and those of an experimental bent, should read parts of this book (after they have completed a normal introduction using sharp variables). They should at least read the first chapter which explains the unsharp variable measure concept as well as giving some examples, one of which is the Stern - Gerlach experiment. In chapter VII, the last chapter, the Stern - Gerlach experiment photon polarization and other quantum optical experiments and wave-particle duality experiments for photons are examined in detail using the POV formalism. This would also be an excellent chapter for all graduate students to read. Some sections may send them back to some of the intermediate chapters, hopefully to peruse in greater detail. The intermediate chapters contain the formal development of the POV formalism and compare it to the normal method. There is also a brief introduction to measurement theory using unsharp observables. As the authors state, this is not intended to be a complete solution to the problem of measurement theory. The next two chapters deal with uncertainty and phase space formalism as considered by the unsharp observables. Philosophers of physics and mathematicians will find this an excellent introduction to the POV formalism and the distinction between sharp and unsharp measurements. Many texts will gloss over both the philosophical and mathematical consequences inherent in the full statistical formalism of quantum mechanics. This monograph is not, by the authors' admission, a full treatment of quantum measurement theory. Also, although in the introduction the authors discuss the importance of applying quantum mechanics to individual objects, this book mainly considers the interpretations as statistical measures. There is little attempt to deal with the quantum mechanical interpretations of the behaviour of a single particle. That is, questions about measurements on a single particle or the interpretation of the wavefunction for a single particle are not fully covered. The authors claim that the POV approach is also the best for individual systems, but do little in applications. The major example given is in the last chapter where they consider interference measurements for single photons. They might have referenced more of the current alternative approaches to measurement theory of single particle wavefunctions. The readers of this journal will be disappointed, but not surprised, that the covariant nature of observables is restricted almost entirely to the Galilei group. The only Poincare group material is the example of photon localization in the last chapter. This is a reflection of the great difficulty of applying the special or the general theories of relativity to the Hilbert space formalism.
This tome is a formal presentation of the unsharp observable approach to quantum mechanics using the positive operator valued (POV) concept of an observable. It is intended for philosophers and mathematicians as well as physicists. This is a very formalistic book. There are, however, portions that should be read by all experimentalists performing quantum mechanical studies as well as graduate students completing their first graduate year quantum mechanics course. If we interpret quantum mechanics as making statistical predictions on the results of states prepared in identical fashions, we will often find that the observed probabilities are nearly, but not exactly, the expected 1 or 0 for measurements that are expected to always give the same result. Hence, rather than demanding sharp measures with probabilities of exactly 1 or 0, quantum mechanics allows a reasonable interpretation of unsharp results of nearly 1 or nearly 0. As the authors point out in chapter II, POV measures and the associated unsharp measures are not essential for an interpretation of quantum mechanical results. However, more sharp variables, and hence a larger Hilbert space, will be needed than are required by the unsharp approach. Also, the theoretical interpretation will often not be as straightforward with only sharp observables. As I have previously stated, all graduate students, both those of a theoretical and those of an experimental bent, should read parts of this book (after they have completed a normal introduction using sharp variables). They should at least read the first chapter which explains the unsharp variable measure concept as well as giving some examples, one of which is the Stern - Gerlach experiment. In chapter VII, the last chapter, the Stern - Gerlach experiment photon polarization and other quantum optical experiments and wave-particle duality experiments for photons are examined in detail using the POV formalism. This would also be an excellent chapter for all graduate students to read. Some sections may send them back to some of the intermediate chapters, hopefully to peruse in greater detail. The intermediate chapters contain the formal development of the POV formalism and compare it to the normal method. There is also a brief introduction to measurement theory using unsharp observables. As the authors state, this is not intended to be a complete solution to the problem of measurement theory. The next two chapters deal with uncertainty and phase space formalism as considered by the unsharp observables. Philosophers of physics and mathematicians will find this an excellent introduction to the POV formalism and the distinction between sharp and unsharp measurements. Many texts will gloss over both the philosophical and mathematical consequences inherent in the full statistical formalism of quantum mechanics. This monograph is not, by the authors' admission, a full treatment of quantum measurement theory. Also, although in the introduction the authors discuss the importance of applying quantum mechanics to individual objects, this book mainly considers the interpretations as statistical measures. There is little attempt to deal with the quantum mechanical interpretations of the behaviour of a single particle. That is, questions about measurements on a single particle or the interpretation of the wavefunction for a single particle are not fully covered. The authors claim that the POV approach is also the best for individual systems, but do little in applications. The major example given is in the last chapter where they consider interference measurements for single photons. They might have referenced more of the current alternative approaches to measurement theory of single particle wavefunctions. The readers of this journal will be disappointed, but not surprised, that the covariant nature of observables is restricted almost entirely to the Galilei group. The only Poincare group material is the example of photon localization in the last chapter. This is a reflection of the great difficulty of applying the special or the general theories of relativity to the Hilbert space formalism.
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.
The topics of this book are the mathematical foundations of non-relativistic quantum mechanics and the mathematical theory they require. The main characteristic of the book is that the mathematics is developed assuming familiarity with elementary analysis only. Moreover, all the proofs are carried out in detail. These features make the book easily accessible to readers with only the mathematical training offered by undergraduate education in mathematics or in physics, and also ideal for individual study. The principles of quantum mechanics are discussed with complete mathematical accuracy and an effort is made to always trace them back to the experimental reality that lies at their root. The treatment of quantum mechanics is axiomatic, with definitions followed by propositions proved in a mathematical fashion. No previous knowledge of quantum mechanics is required. This book is designed so that parts of it can be easily used for various courses in mathematics and mathematical physics, as suggested in the Preface. The book is of interest to researchers and graduate students in functional analysis, who can see how closely an important part of their chosen field is linked with quantum mechanics, and also to physicists, who can see how the abstract language of functional analysis brings unity to the apparently distinct approaches employed in quantum theory.
This is a well written book; it makes easy and pleasant reading for both experts and physicists working in other areas. The book is almost self-contained. It comprises six chapters. In the first three the author introduces the subject and deals with basic results and analytic tools from quantum optics: quantum theory of light and quasi-probability functions. In the fourth chapter he discusses the quantum-mechanical description of simple optical instruments. The last two chapters are the core of the book; they are devoted to a discussion of quantum tomography, joint measurement of position and momentum and the quantum-optical phase. The author was directly involved in the development of the field, which guarantees a fresh and clear exposition. At the same time he makes an open effort for a general (re)formulation of the subject and to present the `classic' solutions to a number of problems in the area. Overall, the presentation of quantum tomography is shaped by a personal view but it accounts sufficiently for the different approaches discussed in the literature. My only reservation is the way in which connections between classical and quantum tomography are presented. In particular, I find that an explicit statement that classical tomography (inverse Radon transform) cannot be straightforwardly applied to the quantum case should be made. The book was published in 1997. Since then the field has been growing rapidly and impetuously and the book shows its age. Nevertheless, it is still a useful reference tool for active researchers and a nice and valuable introduction for (graduate) students who are entering this area. It is worth noting that only a few books have been published on this subject; Leonhardt's volume is well written and to the point. The book is dedicated to Harry Paul, the supervisor of Leonhardt's postdoctoral thesis and the coauthor of most of his papers in the area. M G A Paris
Designed for a two-semester advanced undergraduate or graduate level course, this distinctive and modern textbook provides students with the physical intuition and mathematical skills to tackle even complex problems in quantum mechanics with ease and fluency. Beginning with a detailed introduction to quantum states and Dirac notation, the book then develops the overarching theoretical framework of quantum mechanics, before explaining physical quantum mechanical properties such as angular momentum and spin. Symmetries and groups in quantum mechanics, important components of current research, are covered at length. The second part of the text focuses on applications, and includes a detailed chapter on quantum entanglement, one of the most exciting modern applications of quantum mechanics, and of key importance in quantum information and computation. Numerous exercises are interspersed throughout the text, expanding upon key concepts and further developing students' understanding. A fully worked solutions manual and lecture slides are available for instructors.
After a short review of some essential concepts of classical physics (with a discussion of the so-called `scientific method', which is rarely found in textbooks of this kind), including the classical description of particles and waves, together with a sketch of statistical thermodynamics, the basics of quantum mechanics are introduced. The third chapter then treats systems of identical particles and their statistics, as well as the statistics of discrete energy levels. This is followed by an extensive presentation of the behaviour of quantum particles in periodic potentials with obvious applications to crystalline materials, including metals, insulators, semiconductors and superconductors. The band structure of some special materials, intrinsic carrier concentrations and electrons in metals are further topics of this main chapter. In most quantum mechanics textbooks the subjects of the fifth chapter, quantum wells, harmonic oscillators and the hydrogen atom, are usually introduced before a discussion of periodic potentials, but the author's exposition makes it quite plausible to place them afterwards. The concept of tunnelling through potential barriers and important application examples, ranging from ohmic contacts to scanning tunnelling microscopy and even radioactivity, make up the next chapter. Some approximation methods, i.e. stationary perturbation theory with applications to coupled wells, and the variational method, are treated in the next chapter. The final chapter dealing with quantum mechanics then is concerned with scattering and collision problems, again, of course, with examples from solid state physics. This last chapter, however, looks somehow merely added on, since it introduces only quite briefly the theory of special relativity -almost certainly in order not to leave out this important revolution of modern physics. Three appendices, on Fermi's Golden Rule, Boltzmann's transport theory and quantum interference devices, round off the contents. The book abounds with links to modern technologies that have evolved out of quantum mechanics (with, of course, the exception of the last chapter on special relativity). To the reviewer this constitutes its outstanding value as a textbook on quantum mechanics for engineering students. In addition, about 100 solved numerical examples in the text illustrate very lucidly some of the more abstract concepts, and at the end of the chapters altogether 200 further problems are posed. Although some of the concepts and results of quantum mechanics are introduced in a seemingly ad hoc manner, nevertheless the book as a whole certainly represents a valuable teaching tool at engineering schools; even physics majors may profit from the wealth of technical applications not usually found in their textbooks on quantum mechanics.
Designed for a two-semester advanced undergraduate or graduate level course, this distinctive and modern textbook provides students with the physical intuition and mathematical skills to tackle even complex problems in quantum mechanics with ease and fluency. Beginning with a detailed introduction to quantum states and Dirac notation, the book then develops the overarching theoretical framework of quantum mechanics, before explaining physical quantum mechanical properties such as angular momentum and spin. Symmetries and groups in quantum mechanics, important components of current research, are covered at length. The second part of the text focuses on applications, and includes a detailed chapter on quantum entanglement, one of the most exciting modern applications of quantum mechanics, and of key importance in quantum information and computation. Numerous exercises are interspersed throughout the text, expanding upon key concepts and further developing students' understanding. A fully worked solutions manual and lecture slides are available for instructors.