Overview

The second edition of this textbook presents the basic mathematical knowledge and skills that are needed for courses on modern theoretical physics, such as those on quantum mechanics, classical and quantum field theory, and related areas.  The authors stress that learning mathematical physics is not a passive process and include numerous detailed proofs, examples, and over 200 exercises, as well as hints linking mathematical concepts and results to the relevant physical concepts and theories.  All of the material from the first edition has been updated, and five new chapters have been added on such topics as distributions, Hilbert space operators, and variational methods. The text is divided into three parts: - Part I: A brief introduction to (Schwartz) distribution theory. Elements from the theories of ultra distributions and (Fourier) hyperfunctions are given in addition to some deeper results for Schwartz distributions, thus providing a rather comprehensive introduction to the theory of generalized functions. Basic properties and methods for distributions are developed with applications to constant coefficient ODEs and PDEs.  The relation between distributions and holomorphic functions is considered, as well as basic properties of Sobolev spaces. - Part II: Fundamental facts about Hilbert spaces. The basic theory of linear (bounded and unbounded) operators in Hilbert spaces and special classes of linear operators - compact, Hilbert-Schmidt, trace class, and Schrödinger operators, as needed in quantum physics and quantum information theory – are explored. This section also contains a detailed spectral analysis of all major classes of linear operators, including completeness of generalized eigenfunctions, as well as of (completely) positive mappings, in particular quantum operations. - Part III: Direct methods of the calculus of variations and their applications to boundary- and eigenvalue-problems for linear and nonlinear partial differential operators.  The authors conclude with a discussion of the Hohenberg-Kohn variational principle. The appendices contain proofs of more general and deeper results, including completions, basic facts about metrizable Hausdorff locally convex topological vector spaces, Baire’s fundamental results and their main consequences, and bilinear functionals. Mathematical Methods in Physics is aimed at a broad community of graduate students in mathematics, mathematical physics, quantum information theory, physics and engineering, as well as researchers in these disciplines.  Expanded content and relevant updates will make this new edition a valuable resource for those working in these disciplines.

Full Product Details

Author:   Philippe Blanchard ,  Erwin Brüning
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Edition:   2nd ed. 2015
Volume:   69
Dimensions:   Width: 15.50cm , Height: 3.30cm , Length: 23.50cm
Weight:   1.099kg
ISBN:  

9783319140445

ISBN 10:   3319140442
Pages:   598
Publication Date:   17 April 2015
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

Introduction.- Spaces of Test Functions.- Schwartz Distributions.- Calculus for Distributions.- Distributions as Derivatives of Functions.- Tensor Products.- Convolution Products.- Applications of Convolution.- Holomorphic Functions.- Fourier Transformations.- Distributions as Boundary Values of Analytic Functions.- Other Spaces of Generalized Functions.- Sobolev Spaces.- Hilbert Spaces: A Brief Historical Introduction.- Inner Product Spaces and Hilbert Spaces.- Geometry of Hilbert Spaces.- Separable Hilbert Spaces.- Direct Sums and Tensor Products.- Topological Aspects.- Linear Operators.- Quadratic Forms.- Bounded Linear Operators.- Special Classes of Linear Operators.- Elements of Spectral Theory.- Compact Operators.- Hilbert-Schmidt and Trace Class Operators.- The Spectral Theorem.- Some Applications of the Spectral Representation.- Spectral Analysis in Rigged Hilbert Spaces.- Operator Algebras and Positive Mappings.- Positive Mappings in Quantum Physics.- Introduction.- DirectMethods in the Calculus of Variations.- Differential Calculus on Banach Spaces and Extrema of Functions.- Constrained Minimization Problems (Method of Lagrange Multipliers).- Boundary and Eigenvalue Problems.- Density Functional Theory of Atoms and Molecules.- Appendices.- Index.  

Reviews

This text is a translated, considerably revised and extended version of the book Distributionen und Hilbertraumoperatoren: Mathematische Methoden der Physik.... The book is written in a very nice and understandable form and addresses mainly students with interest in the interaction between physics and mathematics. Mathematica Bohemica . . . [This] English version has been considerably revised and extended in order to improve the usefulness for students of physics and mathematics alike. For instance, proofs are now more detailed. Many added examples illustrate abstract mathematical concepts. Exercises were included to improve the skill of beginners. Some of these exercises hint to physical problems arising either in quantum mechanics or field theory. . . . Notably, there is one chapter at the end, dealing exclusively with physics applications, the density functional theory of atoms and molecules due to Hohenberg, Kohn, and Sham. In my view this is a nice way to illustrate the calculus of variations. . . . the book is self-contained, the only prerequisites being a solid background in analysis and linear algebra. Zentralblatt Math .. .many very nice and useful examples and applications are provided. ---Monatshefte fur Mathematik

“This book gives a detailed survey on mathematical methods in physics … . The book is very suitable for students of physics, mathematics or engineering with a good background in analysis and linear algebra. … All in all, the book has a high didactical and scientific quality so that it can be recommended for both graduate students and researchers.” (Michael Demuth, zbMATH 1330.46001, 2016)

Author Information

Philippe Blanchard is Professor of Mathematical Physics at Bielefeld University in Germany. Erwin Bruening is a Research Fellow at the University of KwaZulu-Natal in South Africa.

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