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Effective Evolution Equations From Quantum Mechanics
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Book Synopsis Effective Evolution Equations from Quantum Dynamics by : Niels Benedikter
Download or read book Effective Evolution Equations from Quantum Dynamics written by Niels Benedikter and published by Springer. This book was released on 2015-11-04 with total page 97 pages. Available in PDF, EPUB and Kindle. Book excerpt: These notes investigate the time evolution of quantum systems, and in particular the rigorous derivation of effective equations approximating the many-body Schrödinger dynamics in certain physically interesting regimes. The focus is primarily on the derivation of time-dependent effective theories (non-equilibrium question) approximating many-body quantum dynamics. The book is divided into seven sections, the first of which briefly reviews the main properties of many-body quantum systems and their time evolution. Section 2 introduces the mean-field regime for bosonic systems and explains how the many-body dynamics can be approximated in this limit using the Hartree equation. Section 3 presents a method, based on the use of coherent states, for rigorously proving the convergence towards the Hartree dynamics, while the fluctuations around the Hartree equation are considered in Section 4. Section 5 focuses on a discussion of a more subtle regime, in which the many-body evolution can be approximated by means of the nonlinear Gross-Pitaevskii equation. Section 6 addresses fermionic systems (characterized by antisymmetric wave functions); here, the fermionic mean-field regime is naturally linked with a semiclassical regime, and it is proven that the evolution of approximate Slater determinants can be approximated using the nonlinear Hartree-Fock equation. In closing, Section 7 reexamines the same fermionic mean-field regime, but with a focus on mixed quasi-free initial data approximating thermal states at positive temperature.
Book Synopsis Effective Evolution Equations from Quantum Mechanics by : Nikolai Leopold
Download or read book Effective Evolution Equations from Quantum Mechanics written by Nikolai Leopold and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Effective Evolution Equations from Many-body Quantum Mechanics by : Niels Benedikter
Download or read book Effective Evolution Equations from Many-body Quantum Mechanics written by Niels Benedikter and published by . This book was released on 2014 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Evolution Equations by : David Ellwood
Download or read book Evolution Equations written by David Ellwood and published by American Mathematical Soc.. This book was released on 2013-06-26 with total page 587 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is a collection of notes from lectures given at the 2008 Clay Mathematics Institute Summer School, held in Zürich, Switzerland. The lectures were designed for graduate students and mathematicians within five years of the Ph.D., and the main focus of the program was on recent progress in the theory of evolution equations. Such equations lie at the heart of many areas of mathematical physics and arise not only in situations with a manifest time evolution (such as linear and nonlinear wave and Schrödinger equations) but also in the high energy or semi-classical limits of elliptic problems. The three main courses focused primarily on microlocal analysis and spectral and scattering theory, the theory of the nonlinear Schrödinger and wave equations, and evolution problems in general relativity. These major topics were supplemented by several mini-courses reporting on the derivation of effective evolution equations from microscopic quantum dynamics; on wave maps with and without symmetries; on quantum N-body scattering, diffraction of waves, and symmetric spaces; and on nonlinear Schrödinger equations at critical regularity. Although highly detailed treatments of some of these topics are now available in the published literature, in this collection the reader can learn the fundamental ideas and tools with a minimum of technical machinery. Moreover, the treatment in this volume emphasizes common themes and techniques in the field, including exact and approximate conservation laws, energy methods, and positive commutator arguments. Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).
Book Synopsis C*-Algebras and Mathematical Foundations of Quantum Statistical Mechanics by : Jean-Bernard Bru
Download or read book C*-Algebras and Mathematical Foundations of Quantum Statistical Mechanics written by Jean-Bernard Bru and published by Springer Nature. This book was released on 2023-06-16 with total page 497 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook provides a comprehensive introduction to the mathematical foundations of quantum statistical physics. It presents a conceptually profound yet technically accessible path to the C*-algebraic approach to quantum statistical mechanics, demonstrating how key aspects of thermodynamic equilibrium can be derived as simple corollaries of classical results in convex analysis. Using C*-algebras as examples of ordered vector spaces, this book makes various aspects of C*-algebras and their applications to the mathematical foundations of quantum theory much clearer from both mathematical and physical perspectives. It begins with the simple case of Gibbs states on matrix algebras and gradually progresses to a more general setting that considers the thermodynamic equilibrium of infinitely extended quantum systems. The book also illustrates how first-order phase transitions and spontaneous symmetry breaking can occur, in contrast to the finite-dimensional situation. One of the unique features of this book is its thorough and clear treatment of the theory of equilibrium states of quantum mean-field models. This work is self-contained and requires only a modest background in analysis, topology, and functional analysis from the reader. It is suitable for both mathematicians and physicists with a specific interest in quantum statistical physics.
Book Synopsis Partially Integrable Evolution Equations in Physics by : R. Conte
Download or read book Partially Integrable Evolution Equations in Physics written by R. Conte and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 609 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the many physical phenomena ruled by partial differential equations, two extreme fields are currently overcrowded due to recent considerable developments: 1) the field of completely integrable equations, whose recent advances are the inverse spectral transform, the recursion operator, underlying Hamiltonian structures, Lax pairs, etc 2) the field of dynamical systems, often built as models of observed physical phenomena: turbulence, intermittency, Poincare sections, transition to chaos, etc. In between there is a very large region where systems are neither integrable nor nonintegrable, but partially integrable, and people working in the latter domain often know methods from either 1) or 2). Due to the growing interest in partially integrable systems, we decided to organize a meeting for physicists active or about to undertake research in this field, and we thought that an appropriate form would be a school. Indeed, some of the above mentioned methods are often adaptable outside their original domain and therefore worth to be taught in an interdisciplinary school. One of the main concerns was to keep a correct balance between physics and mathematics, and this is reflected in the list of courses.
Book Synopsis Nonlinear, Deformed And Irreversible Quantum Systems - Proceedings Of The International Symposium On Mathematical Physics by : Heinz-dietrich Doebner
Download or read book Nonlinear, Deformed And Irreversible Quantum Systems - Proceedings Of The International Symposium On Mathematical Physics written by Heinz-dietrich Doebner and published by World Scientific. This book was released on 1995-08-31 with total page 494 pages. Available in PDF, EPUB and Kindle. Book excerpt: In recent years nonlinear and irreversible quantum mechanics is becoming increasingly important because of the availability of precision experiments. There are new and successful attempts to understand quantum irreversibility. The development of generalized symmetries has to led to new families of evolution equations for pure and mixed states. On the one hand, this timely symposium covers nonlinear and irreversible quantum mechanics, the theory of quantization methods, causality and various problems important in this context. On the other hand, it reports the development of quantum group symmetries, and of methods to construct deformed quantum mechanical evolution equations like the q-deformed Schrödinger equations.
Book Synopsis Advances in Quantum Mechanics by : Alessandro Michelangeli
Download or read book Advances in Quantum Mechanics written by Alessandro Michelangeli and published by Springer. This book was released on 2017-08-01 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume collects recent contributions on the contemporary trends in the mathematics of quantum mechanics, and more specifically in mathematical problems arising in quantum many-body dynamics, quantum graph theory, cold atoms, unitary gases, with particular emphasis on the developments of the specific mathematical tools needed, including: linear and non-linear Schrödinger equations, topological invariants, non-commutative geometry, resonances and operator extension theory, among others. Most of contributors are international leading experts or respected young researchers in mathematical physics, PDE, and operator theory. All their material is the fruit of recent studies that have already become a reference in the community. Offering a unified perspective of the mathematics of quantum mechanics, it is a valuable resource for researchers in the field.
Book Synopsis Dynamical Systems and Evolution Equations by : John A. Walker
Download or read book Dynamical Systems and Evolution Equations written by John A. Walker and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book grew out of a nine-month course first given during 1976-77 in the Division of Engineering Mechanics, University of Texas (Austin), and repeated during 1977-78 in the Department of Engineering Sciences and Applied Mathematics, Northwestern University. Most of the students were in their second year of graduate study, and all were familiar with Fourier series, Lebesgue integration, Hilbert space, and ordinary differential equa tions in finite-dimensional space. This book is primarily an exposition of certain methods of topological dynamics that have been found to be very useful in the analysis of physical systems but appear to be well known only to specialists. The purpose of the book is twofold: to present the material in such a way that the applications-oriented reader will be encouraged to apply these methods in the study of those physical systems of personal interest, and to make the coverage sufficient to render the current research literature intelligible, preparing the more mathematically inclined reader for research in this particular area of applied mathematics. We present only that portion of the theory which seems most useful in applications to physical systems. Adopting the view that the world is deterministic, we consider our basic problem to be predicting the future for a given physical system. This prediction is to be based on a known equation of evolution, describing the forward-time behavior of the system, but it is to be made without explicitly solving the equation.
Book Synopsis Singularly Perturbed Evolution Equations with Applications to Kinetic Theory by : J. R. Mika
Download or read book Singularly Perturbed Evolution Equations with Applications to Kinetic Theory written by J. R. Mika and published by World Scientific. This book was released on 1995 with total page 332 pages. Available in PDF, EPUB and Kindle. Book excerpt: In recent years there appeared a large number of papers as well as chapters in more general monographs devoted to evolution equations containing small (or large) parameters. In this book it is intended to gather the existing results as well as to introduce new ones on the field of initial value problems for singularly perturbed evolution equations of the resonance type. Such equations are of great interest in the applied sciences, particularly in the kinetic theory which is chosen as the main field of application for the asymptotic theory developed in the monograph.
Book Synopsis XVIth International Congress on Mathematical Physics by : Pavel Exner
Download or read book XVIth International Congress on Mathematical Physics written by Pavel Exner and published by World Scientific. This book was released on 2010 with total page 709 pages. Available in PDF, EPUB and Kindle. Book excerpt: The International Congress on Mathematical Physics is the flagship conference in this exciting field. Convening every three years, it gives a survey on the progress achieved in all branches of mathematical physics. It also provides a superb platform to discuss challenges and new ideas. The present volume collects material from the XVIth ICMP which was held in Prague, August 2009, and features most of the plenary lectures and invited lectures in topical sessions as well as information on other parts of the congress program. This volume provides a broad coverage of the field of mathematical physics, from dominantly mathematical subjects to particle physics, condensed matter, and application of mathematical physics methods in various areas such as astrophysics and ecology, amongst others.
Book Synopsis Macroscopic Limits of Quantum Systems by : Daniela Cadamuro
Download or read book Macroscopic Limits of Quantum Systems written by Daniela Cadamuro and published by Springer. This book was released on 2018-10-26 with total page 257 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on the workshop of the same name, this proceedings volume presents selected research investigating the mathematics of collective phenomena emerging from quantum theory at observable scales. Featured contributions from leading scientists provide a thorough overview of current and active research. Methods from functional analysis, spectral theory, renormalization group theory, and variational calculus are used to prove rigorous results in quantum physics. Topics include superconductivity and mathematical aspects of the BCS theory, the Jellium model and Bose-Einstein condensation, among others. Presenting technical details in an accessible way, this book serves as an introduction to research for advanced graduate students and is suitable for specialists in mathematical physics. The workshop “Macroscopic Limits of Quantum Systems” was held over three days in the spring of 2017 at the Technical University of Munich. The conference celebrated the achievements of Herbert Spohn and his reception of the Max Planck Medal.
Book Synopsis A Mathematical Primer on Quantum Mechanics by : Alessandro Teta
Download or read book A Mathematical Primer on Quantum Mechanics written by Alessandro Teta and published by Springer. This book was released on 2018-04-17 with total page 265 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a rigorous yet elementary approach to quantum mechanics that will meet the needs of Master’s-level Mathematics students and is equally suitable for Physics students who are interested in gaining a deeper understanding of the mathematical structure of the theory. Throughout the coverage, which is limited to single-particle quantum mechanics, the focus is on formulating theory and developing applications in a mathematically precise manner. Following a review of selected key concepts in classical physics and the historical background, the basic elements of the theory of operators in Hilbert spaces are presented and used to formulate the rules of quantum mechanics. The discussion then turns to free particles, harmonic oscillators, delta potential, and hydrogen atoms, providing rigorous proofs of the corresponding dynamical properties. Starting from an analysis of these applications, readers are subsequently introduced to more advanced topics such as the classical limit, scattering theory, and spectral analysis of Schrödinger operators. The main content is complemented by numerous exercises that stimulate interactive learning and help readers check their progress.
Book Synopsis Mathematical Problems in Quantum Physics by : Federico Bonetto
Download or read book Mathematical Problems in Quantum Physics written by Federico Bonetto and published by American Mathematical Soc.. This book was released on 2018-10-24 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the QMATH13: Mathematical Results in Quantum Physics conference, held from October 8–11, 2016, at the Georgia Institute of Technology, Atlanta, Georgia. In recent years, a number of new frontiers have opened in mathematical physics, such as many-body localization and Schrödinger operators on graphs. There has been progress in developing mathematical techniques as well, notably in renormalization group methods and the use of Lieb–Robinson bounds in various quantum models. The aim of this volume is to provide an overview of some of these developments. Topics include random Schrödinger operators, many-body fermionic systems, atomic systems, effective equations, and applications to quantum field theory. A number of articles are devoted to the very active area of Schrödinger operators on graphs and general spectral theory of Schrödinger operators. Some of the articles are expository and can be read by an advanced graduate student.
Book Synopsis XVIIth International Congress on Mathematical Physics by : Arne Jensen
Download or read book XVIIth International Congress on Mathematical Physics written by Arne Jensen and published by World Scientific. This book was released on 2014 with total page 743 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is an in-depth study of not just about Tan Kah-kee, but also the making of a legend through his deeds, self-sacrifices, fortitude and foresight. This revised edition sheds new light on his political agonies in Mao's China over campaigns against capitalists and intellectuals.
Book Synopsis Evolution Equations: Applications to Physics, Industry, Life Sciences and Economics by : Mimmo Iannelli
Download or read book Evolution Equations: Applications to Physics, Industry, Life Sciences and Economics written by Mimmo Iannelli and published by Birkhäuser. This book was released on 2012-12-06 with total page 419 pages. Available in PDF, EPUB and Kindle. Book excerpt: The international conference on which the book is based brought together many of the world's leading experts, with particular effort on the interaction between established scientists and emerging young promising researchers, as well as on the interaction of pure and applied mathematics. All material has been rigorously refereed. The contributions contain much material developed after the conference, continuing research and incorporating additional new results and improvements. In addition, some up-to-date surveys are included.
Book Synopsis An Introduction to Semilinear Evolution Equations by : Thierry Cazenave
Download or read book An Introduction to Semilinear Evolution Equations written by Thierry Cazenave and published by Oxford University Press. This book was released on 1998 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents in a self-contained form the typical basic properties of solutions to semilinear evolutionary partial differential equations, with special emphasis on global properties. It has a didactic ambition and will be useful for an applied readership as well as theoretical researchers.
