The Wigner Transform

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The Wigner Transform

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Author : Maurice de Gosson
language : en
Publisher: World Scientific Publishing Company
Release Date : 2017-03-17

The Wigner Transform written by Maurice de Gosson and has been published by World Scientific Publishing Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-03-17 with categories.

This book provides an in-depth and rigorous study of the Wigner transform and its variants. They are presented first within a context of a general mathematical framework, and then through applications to quantum mechanics. The Wigner transform was introduced by Eugene Wigner in 1932 as a probability quasi-distribution which allows expression of quantum mechanical expectation values in the same form as the averages of classical statistical mechanics. It is also used in signal processing as a transform in time-frequency analysis, closely related to the windowed Gabor transform. Written for advanced-level students and professors in mathematics and mathematical physics, it is designed as a complete textbook course providing analysis on the most important research on the subject to date. Due to the advanced nature of the content, it is also suitable for research mathematicians, engineers and chemists active in the field. Request Inspection Copy

The Wigner Transform

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Author : Maurice A. De Gosson
language : en
Publisher:
Release Date : 2017

The Wigner Transform written by Maurice A. De Gosson and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017 with categories.

The Wigner Distribution

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Author : W. Mecklenbräuker
language : en
Publisher:
Release Date : 1997-10-08

The Wigner Distribution written by W. Mecklenbräuker and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-10-08 with Computers categories.

Seven contributions discuss in depth several aspects of one of the methods for representing both the frequency domain and the temporal localization of signals, which has tremendous importance in signal analysis and processing. Among them are properties like positivity, spread, and interference-term geometry; signal synthesis methods and their application to signal design, time- frequency filtering, and signal separation; the analysis of non- stationary random processes; singular value decompositions and their application to detection and classification; and optical applications of the Wigner Distribution. Also includes a bibliography of published works on the subject from 1985 to 1992. Annotation copyrighted by Book News, Inc., Portland, OR

Weyl Transforms

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Author : M.W. Wong
language : en
Publisher: Springer Science & Business Media
Release Date : 1998-08-21

Weyl Transforms written by M.W. Wong and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-08-21 with Mathematics categories.

A study of the functional analytic properties of Weyl transforms as bounded linear operators on $ L2ü(äBbb Rünü) $ in terms of the symbols of the transforms. Further, the boundedness, the compactness, the spectrum and the functional calculus of the Weyl transform are proved in detail, while new results and techniques on the boundedness and compactness of the Weyl transforms in terms of the symbols in $ Lrü(äBbb Rü2nü) $ and in terms of the Wigner transforms of Hermite functions are given. The roles of the Heisenberg group and the symplectic group in the study of the structure of the Weyl transform are explained, and the connections of the Weyl transform with quantization are highlighted throughout the book. Localisation operators, first studied as filters in signal analysis, are shown to be Weyl transforms with symbols expressed in terms of the admissible wavelets of the localisation operators. The results and methods mean this book is of interest to graduates and mathematicians working in Fourier analysis, operator theory, pseudo-differential operators and mathematical physics.

The Physics Of Phase Space

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Author : Young S. Kim
language : en
Publisher: Springer
Release Date : 2005-09-13

The Physics Of Phase Space written by Young S. Kim and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-09-13 with Science categories.

The concept of phase space plays a decisive role in the study of the transition from classical to quantum physics. This is particularly the case in areas such as nonlinear dynamics and chaos, geometric quantization and the study of the various semi-classical theories, which are the setting of the present volume. Much of the content is devoted to the study of the Wigner distribution. This volume gives the first complete survey of the progress made by both mathematicians and physicists. It will serve as an excellent reference for further research.

Quantum Harmonic Analysis

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Author : Maurice A. de Gosson
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2021-07-05

Quantum Harmonic Analysis written by Maurice A. de Gosson and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-07-05 with Mathematics categories.

Quantum mechanics is arguably one of the most successful scientific theories ever and its applications to chemistry, optics, and information theory are innumerable. This book provides the reader with a rigorous treatment of the main mathematical tools from harmonic analysis which play an essential role in the modern formulation of quantum mechanics. This allows us at the same time to suggest some new ideas and methods, with a special focus on topics such as the Wigner phase space formalism and its applications to the theory of the density operator and its entanglement properties. This book can be used with profit by advanced undergraduate students in mathematics and physics, as well as by confirmed researchers.

Quantum Mechanics In Phase Space

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Author : Cosmas K Zachos
language : en
Publisher: World Scientific
Release Date : 2005-12-09

Quantum Mechanics In Phase Space written by Cosmas K Zachos and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-12-09 with Science categories.

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. Contents:The Wigner FunctionSolving for the Wigner FunctionThe Uncertainty PrincipleEhrenfest's TheoremIllustration: The Harmonic OscillatorTime EvolutionNondiagonal Wigner FunctionsStationary Perturbation TheoryPropagatorsCanonical TransformationsThe Weyl CorrespondenceAlternate Rules of AssociationThe Groenwold–van Hove Theorem and the Uniqueness of MBs and ∗-ProductsOmitted MiscellanySelected Papers: Brief Historical Outline Readership: Advanced undergraduates, beginning graduate students and researchers in physics, quantum computing, chemistry and information processing. Keywords:Phase Space Quantization;Wigner Functions;Star Products;DeformationsReviews:“… the authors have struck the right note in their choice of presentation and also their decision as to what to omit, since the subject matter covers a very broad range … the authors have performed an excellent job in presenting a timely and very useful resource for investigators, in potentially many areas requiring quantum physics, who wish to use quasi-probability functions, particularly the Wigner function. I highly recommend it.”International Journal of Quantum Information

The Weyl Operator And Its Generalization

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Author : Leon Cohen
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-13

The Weyl Operator And Its Generalization written by Leon Cohen and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-13 with Mathematics categories.

The discovery of quantum mechanics in the years 1925-1930 necessitated the consideration of associating ordinary functions with non-commuting operators. Methods were proposed by Born/Jordan, Kirkwood, and Weyl. Sometime later, Moyal saw the connection between the Weyl rule and the Wigner distribution, which had been proposed by Wigner in 1932 as a way of doing quantum statistical mechanics. The basic idea of associating functions with operators has since been generalized and developed to a high degree. It has found several application fields, including quantum mechanics, pseudo-differential operators, time-frequency analysis, quantum optics, wave propagation, differential equations, image processing, radar, and sonar. This book aims at bringing together the results from the above mentioned fields in a unified manner and showing the reader how the methods have been applied. A wide audience is addressed, particularly students and researchers who want to obtain an up-to-date working knowledge of the field. The mathematics is accessible to the uninitiated reader and is presented in a straightforward manner.

Proceedings Of The V Wigner Symposium

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Author : Peter Kasperkovitz
language : en
Publisher: World Scientific
Release Date : 1998-06-30

Proceedings Of The V Wigner Symposium written by Peter Kasperkovitz and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-06-30 with categories.

The Wigner Symposia deal with the most recent developments in those mathematical areas which were introduced to physics by E P Wigner, and also with related fields. The central themes of the proceedings of the 5th Wigner Symposium (WigSym5) are quantum algebras and groups, group-theoretical developments, quantum field theory and geometry, and phase space formulations of quantum mechanics. The proceedings also contain papers on the application of these techniques in various branches of physics, and many contributions in which fundamental mathematical and epistemological questions related to the foundations of quantum theory are discussed.

Linear Ray And Wave Optics In Phase Space

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Author : Amalia Torre
language : en
Publisher: Elsevier
Release Date : 2005-11-11

Linear Ray And Wave Optics In Phase Space written by Amalia Torre and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-11-11 with Science categories.

Ray, wave and quantum concepts are central to diverse and seemingly incompatible models of light. Each model particularizes a specific ''manifestation'' of light, and then corresponds to adequate physical assumptions and formal approximations, whose domains of applicability are well-established. Accordingly each model comprises its own set of geometric and dynamic postulates with the pertinent mathematical means.At a basic level, the book is a complete introduction to the Wigner optics, which bridges between ray and wave optics, offering the optical phase space as the ambience and the Wigner function based technique as the mathematical machinery to accommodate between the two opposite extremes of light representation: the localized ray of geometrical optics and the unlocalized wave function of wave optics.At a parallel level, the analogies with other branches of both classical and quantum physics, like classical and quantum mechanics, quantum optics, signal theory as well as magnetic optics, are evidenced by pertinent comments and/or rigorous mathematics. So, the Lie algebra and group methods are introduced and explained through the elementary optical systems within both the ray and wave optics contexts, the former being related to the symplectic group and the latter to the metaplectic group. In a like manner, the Wigner function is introduced by following the original issue to individualize a phase space representation of quantum mechanics, which is mirrored by the issue to individualize a local frequency spectrum within the signal theory context. The basic analogy with the optics of charged particles inherently underlying the ray-optics picture in phase space is also evidenced within the wave-optics picture in the Wigner phase space. · amalgamation of a great deal of contributions having witnessed the phase space picture of optics over the past 30 years· introduces abstract concepts through concrete systems· hosts of figures and logical diagrams to favour intuition and to introduce mathematics· emphasis on the interrelations with quantum optics, signal theory and magnetic optics · feeds a feeling for genuine issues in higher mathematics and theoretical physics