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This book covers the theory and applications of the Wigner phase space distribution function and its symmetry properties. The book explains why the phase space picture of quantum mechanics is needed, in addition to the conventional Schr”dinger or Heisenberg picture. It is shown that the uncertainty relation can be represented more accurately in this picture. In addition, the phase space picture is shown to be the natural representation of quantum mechanics for modern optics and relativistic quantum mechanics of extended objects.
Wigner's quasi-probability distribution function in phase space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence, quantum computing, and quantum chaos. It is also important in signal processing and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: it furnishes a third, alternative, formulation of quantum mechanics, independent of the conventional Hilbert space, or path integral formulations.In this logically complete and self-standing formulation, one need not choose sides ? coordinate or momentum space. It works in full phase space, accommodating the uncertainty principle, and it offers unique insights into the classical limit of quantum theory. This invaluable book is a collection of the seminal papers on the formulation, with an introductory overview which provides a trail map for those papers; an extensive bibliography; and simple illustrations, suitable for applications to a broad range of physics problems. It can provide supplementary material for a beginning graduate course in quantum mechanics.
This is a text on quantum mechanics formulated simultaneously in terms of position and momentum, i.e. in phase space. It is written at an introductory level, drawing on the remarkable history of the subject for inspiration and motivation. Wigner functions — density matrices in a special Weyl representation — and star products are the cornerstones of the formalism.The resulting framework is a rich source of physical intuition. It has been used to describe transport in quantum optics, structure and dynamics in nuclear physics, chaos, and decoherence in quantum computing. It is also of importance in signal processing and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: it furnishes a third, alternative way to formulate and understand quantum mechanics, independent of the conventional Hilbert space or path integral approaches to the subject.In this logically complete and self-standing formulation, one need not choose sides between coordinate or momentum space variables. It works in full phase space, accommodating the uncertainty principle; and it offers unique insights into the classical limit of quantum theory. The observables in this formulation are c-number functions in phase space instead of operators, with the same interpretation as their classical counterparts, only composed together in novel algebraic ways using star products.This treatise provides an introductory overview and supplementary material suitable for an advanced undergraduate or a beginning graduate course in quantum mechanics.
This is a text on quantum mechanics formulated simultaneously in terms of position and momentum, i.e. in phase space. It is written at an introductory level, drawing on the remarkable history of the subject for inspiration and motivation. Wigner functions density matrices in a special Weyl representation and star products are the cornerstones of the formalism. The resulting framework is a rich source of physical intuition. It has been used to describe transport in quantum optics, structure and dynamics in nuclear physics, chaos, and decoherence in quantum computing. It is also of importance in signal processing and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: it furnishes a third, alternative way to formulate and understand quantum mechanics, independent of the conventional Hilbert space or path integral approaches to the subject. In this logically complete and self-standing formulation, one need not choose sides between coordinate or momentum space variables. It works in full phase-space, accommodating the uncertainty principle; and it offers unique insights into the classical limit of quantum theory. The observables in this formulation are c-number functions in phase space instead of operators, with the same interpretation as their classical counterparts, only composed together in novel algebraic ways using star products. This treatise provides an introductory overview and supplementary material suitable for an advanced undergraduate or a beginning graduate course in quantum mechanics.
In this monograph, we shall present a new mathematical formulation of quantum theory, clarify a number of discrepancies within the prior formulation of quantum theory, give new applications to experiments in physics, and extend the realm of application of quantum theory well beyond physics. Here, we motivate this new formulation and sketch how it developed. Since the publication of Dirac's famous book on quantum mechanics [Dirac, 1930] and von Neumann's classic text on the mathematical foundations of quantum mechanics two years later [von Neumann, 1932], there have appeared a number of lines of development, the intent of each being to enrich quantum theory by extra polating or even modifying the original basic structure. These lines of development have seemed to go in different directions, the major directions of which are identified here: First is the introduction of group theoretical methods [Weyl, 1928; Wigner, 1931] with the natural extension to coherent state theory [Klauder and Sudarshan, 1968; Peremolov, 1971]. The call for an axiomatic approach to physics [Hilbert, 1900; Sixth Problem] led to the development of quantum logic [Mackey, 1963; Jauch, 1968; Varadarajan, 1968, 1970; Piron, 1976; Beltrametti & Cassinelli, 1981], to the creation of the operational approach [Ludwig, 1983-85, 1985; Davies, 1976] with its application to quantum communication theory [Helstrom, 1976; Holevo, 1982), and to the development of the C* approach [Emch, 1972]. An approach through stochastic differential equations ("stochastic mechanics") was developed [Nelson, 1964, 1966, 1967].
This volume demonstrates that the key to the modeling, diagnosis and control of the next generation manufacturing processes is to integrate knowledge-based systems with traditional techniques. An up-to-date study is given here of this relatively recent development.The book is for those working primarily with traditional techniques and those working in the knowledge-based systems field. Both sets of readers will find it to be a source of many specific ideas about the integration of knowledge-based systems with traditional techniques, and carrying a wealth of useful references.
Wigner's quasi-probability distribution function in phase space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence, quantum computing, and quantum chaos. It is also important in signal processing and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: it furnishes a third, alternative, formulation of quantum mechanics, independent of the conventional Hilbert space, or path integral formulations.In this logically complete and self-standing formulation, one need not choose sides — coordinate or momentum space. It works in full phase space, accommodating the uncertainty principle, and it offers unique insights into the classical limit of quantum theory. This invaluable book is a collection of the seminal papers on the formulation, with an introductory overview which provides a trail map for those papers; an extensive bibliography; and simple illustrations, suitable for applications to a broad range of physics problems. It can provide supplementary material for a beginning graduate course in quantum mechanics.
In learning quantum theory, intuitions developed for the classical world fail, and the equations to be solved are sufficiently complex that they require a computer except for the simplest situations. This book represents an attempt to jump the hurdle to an intuitive understanding of wave mechanics by using illustrations to present the time evolution and parameter dependence of wave functions in a wide variety of situations. Most of the illustrations are computer-generated solutions of the Schrödinger equation for one- and three-dimensional systems, with the situations discussed ranging from the simple particle in a box through resonant scattering in one dimension to the hydrogen atom and Regge classification of resonant scattering. Thoroughly revised and expanded to include a discussion of spin and magnetic resonance.
The aim of this book is to explain the basic concepts and phenomena of quantum mechanics by means of visualization. Computer-generated illustrations are used extensively throughout the text, helping to establish the relation between quantum mechanics-wave functions, interference, atomic structure, and so forth-and classical physics-point mechanics, statistical mechanics, and wave optics. Even more important, by studying the pictures in parallel with the text, readers develop an intuition for such notoriously abstract phenomena as • the tunnel effect • excitation and decay of metastable states • wave packet motion within a well • systems of distinguishable and indistinguishable particles • free wave packets and scattering in three dimensions • angular momentum decomposition • stationary bound states in various three-dimensional potentials • Kepler motion of wave packets in the Coulomb field • spin and magnetic resonance Illustrations from experiments in a variety of fields, including chemistry, and molecular, atomic, nuclear, and particle physics, underline the basic as well as the practical importance of quantum mechanics. This third edition includes a CD-ROM with all illustrations from the book in full color, suitable for direct display or for printing out as posters or teaching aids. It also includes a new chapter on quantile motion and tunneling time, as well as some additional phenomena and examples.