Matematisk fysik

The book is intended as an advanced undergraduate or first-year graduate course for students from various disciplines, including applied mathematics, physics and engineering. It has evolved from courses offered on partial differential equations (PDEs) over the last several years at the Politecnico di Milano. These courses had a twofold purpose: on the one hand, to teach students to appreciate the interplay between theory and modeling in problems arising in the applied sciences, and on the other to provide them with a solid theoretical background in numerical methods, such as finite elements. Accordingly, this textbook is divided into two parts. The first part, chapters 2 to 5, is more elementary in nature and focuses on developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. In turn the second part, chapters 6 to 11, concentrates on the development of Hilbert spaces methods for the variational formulation and the analysis of (mainly) linear boundary and initial-boundary value problems.

Quantum mechanics is one of the most fascinating elements of the physics curriculum, but its conceptual nuances and mathematical complexity can be daunting for beginning students. This user-friendly text is designed for a one-semester course which bridges the gap between sophomore-level treatments and advanced undergraduate/lower-graduate courses. Qualitative explanations and descriptions of historical background are combined with detailed mathematical analyses to help students establish a firm foundation for further study. Classical problems such potential wells, barrier penetration, alpha decay, the harmonic oscillator, and the hydrogen atom are examined in detail, and formalisms and techniques such as operators, expectation values, commutators, perturbation theory, numerical solutions, and the variational theorem are also covered. Particular emphasis is placed on providing numerous worked examples and exercises.

Das Buch wendet sich an alle, egal ob im Studium, in technischen Berufen, oder in der Wissenschaft, die die sich für die analytische Transmissionselektronenmikroskopie interessieren und einen Überblick über diese Methode erhalten möchten. Insbesondere betrifft dies Personen, die an einem Transmissionselektronenmikroskop arbeiten wollen oder müssen, die aber noch keine spezielle elektronenmikroskopische Ausbildung durchlaufen haben. Das Buch basiert auf den Erfahrungen der Autoren bei der Unterrichtung von Studierenden, Promovierenden und in technischen Berufen Tätigen. Der Überblick über die analytische Transmissionselektronenmikroskopie umfasst die Schwerpunkte Optische Abbildung, Elektronenwellen, magnetische Linsen, Abbildungsfehler, Aufbau eines Transmissionselektronenmikroskops, Präparation dünner Proben, Justage des Mikroskops, Elektronenbeugung, Kontrastentstehung, Höchstauflösungselektronenmikroskopie, Rastertransmissionselektronenmikroskopie sowie Analytik mittels energiedispersiver Röntgenspektroskopie und Elektronenenergieverlust-Spektroskopie. Ein mathematischer Anhang erklärt grundlegende Formalismen zur Thematik.

Unifying a range of topics that are currently scattered throughout the literature, this book offers a unique and definitive review of mathematical aspects of quantization and quantum field theory. The authors present both basic and more advanced topics of quantum field theory in a mathematically consistent way, focusing on canonical commutation and anti-commutation relations. They begin with a discussion of the mathematical structures underlying free bosonic or fermionic fields, like tensors, algebras, Fock spaces, and CCR and CAR representations (including their symplectic and orthogonal invariance). Applications of these topics to physical problems are discussed in later chapters. Although most of the book is devoted to free quantum fields, it also contains an exposition of two important aspects of interacting fields: diagrammatics and the Euclidean approach to constructive quantum field theory. With its in-depth coverage, this text is essential reading for graduate students and researchers in departments of mathematics and physics.

This book introduces and critically appraises the main proposals for how to understand quantum mechanics, namely the Copenhagen interpretation, spontaneous collapse, Bohmian mechanics, many-worlds, and others. The author makes clear what are the crucial problems, such as the measurement problem, related to the foundations of quantum mechanics and explains the key arguments like the Einstein-Podolsky-Rosen argument and Bell's proof of nonlocality. He discusses and clarifies numerous topics that have puzzled the founding fathers of quantum mechanics and present-day students alike, such as the possibility of hidden variables, the collapse of the wave function, time-of-arrival measurements, explanations of the symmetrization postulate for identical particles, or the nature of spin. Several chapters are devoted to extending the different approaches to relativistic space-time and quantum field theory. The book is self-contained and is intended for graduate students and researchers who want to step into the fundamental aspects of quantum physics. Given its clarity, it is accessible also to advanced undergraduates and contains many exercises and examples to master the subject.

The Mathieu series is a functional series introduced by Émile Léonard Mathieu for the purposes of his research on the elasticity of solid bodies. Bounds for this series are needed for solving biharmonic equations in a rectangular domain. In addition to Tomovski and his coauthors, Pogany, Cerone, H. M. Srivastava, J. Choi, etc. are some of the known authors who published results concerning the Mathieu series, its generalizations and their alternating variants. Applications of these results are given in classical, harmonic and numerical analysis, analytical number theory, special functions, mathematical physics, probability, quantum field theory, quantum physics, etc. Integral representations, analytical inequalities, asymptotic expansions and behaviors of some classes of Mathieu series are presented in this book. A systematic study of probability density functions and probability distributions associated with the Mathieu series, its generalizations and Planck¿s distributionis also presented. The book is addressed at graduate and PhD students and researchers in mathematics and physics who are interested in special functions, inequalities and probability distributions.

This book introduces the scattering theory of nonrelativistic systems, a standard tool for interpreting collision experiments with quantum particles at energies not too high. The goal is to explore the interaction between particles and their properties. The authors cover the basics of the theory through a detailed discussion of elastic scattering using the stationary Schrodinger equation and the Lippmann-Schwinger equation. These remarks are supplemented by a consideration of the time-dependent formulation of scattering theory. Selection rules for effective cross sections due to symmetries conditioned by the structure of the interparticle forces and the scattering of spin-polarized particles are discussed. The foundations for the treatment of inelastic processes are laid and explained by application to three-body and nucleotransfer processes.In all chapters, the more technical, mathematical aspect and the more physics-oriented explanations are separated as far as possible. The explanations are well comprehensible and suitable to introduce the reader to the physics of impact processes.This book is a translation of the original German 1st edition Streutheorie in der nichtrelativistischen Quantenmechanik by Reiner M. Dreizler, Tom Kirchner & Cora S. Ludde, published by Springer-Verlag GmbH Germany, part of Springer Nature in 2018. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). The present version has been revised extensively with respect to technical and linguistic aspects by the authors. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors.

It is generally believed that collisions of particles reduce the self-diffusion coefficient. In this book, Erik Kalz shows that in classical systems under the effect of Lorentz force, which are characterized by diffusion tensors with antisymmetric elements, collisions surprisingly can enhance self-diffusion. In these systems, due to an inherent curving effect, the motion of particles is facilitated, instead of hindered by collisions. Consistent with this the author finds that the collective diffusion remains unaffected. Using a geometric model, he theoretically predicts a magnetic field governed crossover from a reduced to an enhanced self-diffusion. The physical interpretation is quantitatively supported by the force autocorrelation function, which turns negative with increasing the magnetic field. Using Brownian-dynamics simulations, he validates the predictions.

Die mathematischen Sätze gelernt? Die Formeln auch? Nun fehlt noch das Wichtigste: die Anwendung des Gelernten auf Probleme, die über ein einfaches Beispiel hinausgehen. Die mathematischen Werkzeuge, die man sich in jedem naturwissenschaftlichen Studium aneignet, lernt man schließlich erst bei der Bearbeitung von Problemen wirklich kennen. Jedoch kann das Lernen oft frustrierend werden, wenn man die Lösung nicht findet, die Lektüre diese als trivial abtut oder nur schlicht durch eine nackte Zahl ohne erläuternden Kommentar angibt.Das vorliegende Lehrbuch macht die Lesenden mit zentralen mathematischen Werkzeugen für den Physiker vertraut. Dabei werden wichtige Standardaufgaben sowie physikalische Problemstellungen ausführlich gelöst und parallel dazu die Lösungswege diskutiert. Der Schwerpunkt liegt insbesondere auf der Verknüpfung der verschiedenen Aufgaben, und zwar unabhängig vom Schwierigkeitsgrad, so dass das Lösen immer komplexerer Aufgaben schrittweise erleichtert wird. Infolgedessen sind die Lösungen keine bloße Ergänzung, sondern ein zentraler Bestandteil dieses Lehrbuchs.

This book delivers a comprehensive and up-to-date treatment of practical applications of metamaterials, structured media, and conventional porous materials. With increasing levels of urbanization, a growing demand for motorized transport, and inefficient urban planning, environmental noise exposure is rapidly becoming a pressing societal and health concern. Phononic and sonic crystals, acoustic metamaterials, and metasurfaces can revolutionize noise and vibration control and, in many cases, replace traditional porous materials for these applications.In this collection of contributed chapters, a group of international researchers reviews the essentials of acoustic wave propagation in metamaterials and porous absorbers with viscothermal losses, as well as the most recent advances in the design of acoustic metamaterial absorbers. The book features a detailed theoretical introduction describing commonly used modelling techniques such as plane wave expansion, multiple scattering theory, and the transfer matrix method. The following chapters give a detailed consideration of acoustic wave propagation in viscothermal fluids and porous media, and the extension of this theory to non-local models for fluid saturated metamaterials, along with a description of the relevant numerical methods. Finally, the book reviews a range of practical industrial applications, making it especially attractive as a white book targeted at the building, automotive, and aeronautic industries.

Because of the correspondences existing among all levels of reality, truths pertaining to a lower level can be considered as symbols of truths at a higher level and can therefore be the "foundation" or support leading by analogy to a knowledge of the latter. This confers to every science a superior or "elevating" meaning, far deeper than its own original one. - R. GUENON, The Crisis of Modern World Having been interested in the Kepler Problem for a long time, I have al­ ways found it astonishing that no book has been written yet that would address all aspects of the problem. Besides hundreds of articles, at least three books (to my knowledge) have indeed been published al­ ready on the subject, namely Englefield (1972), Stiefel & Scheifele (1971) and Guillemin & Sternberg (1990). Each of these three books deals only with one or another aspect of the problem, though. For example, En­ glefield (1972) treats only the quantum aspects, and that in a local way. Similarly, Stiefel & Scheifele (1971) only considers the linearization of the equations of motion with application to the perturbations of celes­ tial mechanics. Finally, Guillemin & Sternberg (1990) is devoted to the group theoretical and geometrical structure.

¿ A popular edition, in a digestible, edited format.¿ New series featuring the greatest works of science and philosophy.¿ New introduction highlights the influence of Newton and the contemporary response to his work.

"This modern text describes the remarkable developments in quantum condensed matter physics following the experimental discoveries of quantum Hall effects and high temperature superconductivity in the 1980s. After a review of the phases of matter amenable to an independent particle description, entangled phases of matter are described in an accessible and unified manner. The concepts of fractionalization and emergent gauge fields are introduced using the simplest resonating valence bond insulator with an energy gap, the Z2 spin liquid. Concepts in band topology and the parton method are then combined to obtain a large variety of experimentally relevant gapped states. Correlated metallic states are described, beginning with a discussion of the Kondo effect on magnetic impurities in metals. Metals without quasiparticle excitations are introduced using the Sachdev-Ye-Kitaev model, followed by a discussion of critical Fermi surfaces and strange metals. Numerous end-of-chapter problems expand readers' comprehension and reinforce key concepts"--