Lectures on Morse Homology, by Augustin Banyaga and David Hurtubise
This book presents in great detail all the results one needs to prove the Morse Homology Theorem using classical techniques from algebraic topology and homotopy theory. Most of these results can be found scattered throughout the literature dating from the mid to late 1900′s in some form or other, but often the results are proved in different contexts with a multitude of different notations and different goals. This book collects all these results together into a single reference with complete and detailed proofs. The core material in this book includes CW-complexes, Morse theory, hyperbolic dynamical systems (the Lamba-Lemma, the Stable/Unstable Manifold Theorem), transversality theory, the Morse-Smale-Witten boundary operator, and Conley index theory. More advanced topics include Morse theory on Grassmann manifolds and Lie groups, and an overview of Floer homology theories. With the stress on completeness and by its elementary approach to Morse homology, this book is suitable as a textbook for a graduate level course, or as a reference for working mathematicians and physicists.

Dimension theory of general spaces, by A. R. Pears
A complete and self-contained account of the dimension theory of general topological spaces, with particular emphasis on the dimensional properties of non-metrizable spaces. It makes the subject accessible to beginning graduate students and will also serve as a reference work for general topologists.

Dimension Theory, by Witold Hurewicz and Henry Wallman
The standard treatise on classical dimension theory. Dimension theory is that area of topology concerned with giving a precise mathematical meaning to the concept of the dimension of a space. An active area of reasearch in the early 20th century, but one that has fallen into disuse in topology, dimension theory has experienced a revitalization due to connections with fractals and dynamical systems, but none of those developments are in this 1948 book. Instead,this book is primarily used as a reference today for its proof of Brouwer’s Theorem on the Invariance of Domain.

Homotopy theoretic methods in Group Cohomology, by William G. Dwyer and Hans-Werner Henn
This 89-pages-book looks at group cohomology with tools that come from homotopy theory. These tools give both decomposition theorems (which rely on homotopy colimits to obtain a description of the cohomology of a group in terms of the cohomology of suitable subgroups) and global structure theorems (which exploit the action of the ring of topological cohomology operations). The approach is expository and thus suitable for graduate students and others who would like an introduction to the subject that organizes and adds to the relevant literature and leads to the frontier of current research. The book should also be interesting to anyone who wishes to learn some of the machinery of homotopy theory (simplicial sets, homotopy colimits, Lannes’ T-functor, the theory of unstable modules over the Steenrod algebra) by seeing how it is used in a practical setting.

A compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab, by William E. Schiesser and Graham W. Griffiths
A Compendium of Partial Differential Equation Models presents numerical methods and associated computer codes in Matlab for the solution of a spectrum of models expressed as partial differential equations (PDEs), one of the mostly widely used forms of mathematics in science and engineering. The authors focus on the method of lines (MOL), a well-established numerical procedure for all major classes of PDEs in which the boundary value partial derivatives are approximated algebraically by finite differences. This reduces the PDEs to ordinary differential equations (ODEs) and thus makes the computer code easy to understand, implement, and modify.

Pseudodifferential operators and nonlinear PDEs, by Michael Taylor
For the past 25 years the theory of pseudodifferential operators has played an important role in many exciting and deep investigations into linear PDE. Over the past decade, this tool has also begun to yield interesting results in nonlinear PDE. This book is devoted to a summary and reconsideration of some uses of pseudodifferential operator techniques in nonlinear PDE.

A history of Non-Euclidean Geometry, by Boris A. Rosenfeld
This book is an investigation of the mathematical and philosophical factors underlying the discovery of the concept of noneuclidean geometries, and the subsequent extension of the concept of space. Chapters one through five are devoted to the evolution of the concept of space, leading up to chapter six which describes the discovery of noneuclidean geometry, and the corresponding broadening of the concept of space. The author goes on to discuss concepts such as multidimensional spaces and curvature, and transformation groups. The book ends with a chapter describing the applications of nonassociative algebras to geometry.

Topology of 4-manifolds, by Michael H. Freedman and Frank Quinn
This book introduces the reader to a fascinating branch of topology and has the clearest proof of the 4-dimensional Poincare conjecture. In addition, the authors do not hesitate to employ diagrams as needed to illustrate the main points and to assist the reader in visualizing 4-dimensional objects. The authors give a fine discussion as to the reasons why four dimensions is harder to deal with topologically than dimensions five or greater, this being essentially due to the behavior of 2-dimensional disks: mapping 2-disks into 3-manifolds results (generically) with 1-dimensional self-intersections; in 4-dimensions the intersections are isolated points, and in 5 dimensions or more the 2-disks can be embedded.

Interpolation and definability: Modal and Intuitionistic Logics, by Dov M. Gabbay and Larisa Maksimova

Tensors in Image Processing and Computer Vision, by Santiago Aja-Fernández, Rodrigo de Luis García, Dacheng Tao and Xuelong Li
Tensor signal processing is an emerging field with important applications to computer vision and image processing. This book presents the state of the art in this new branch of signal processing, offering a great deal of research and discussions by leading experts in the area.

Perspectives on the history of Mathematical Logic, by Thomas Drucker
This volume offers insights into the development of mathematical logic over the last century. Arising from a special session of the history of logic at an American Mathematical Society meeting, the chapters explore technical innovations, the philosophical consequences of work during the period, and the historical and social context in which the logicians worked.

Function Theory in the unit ball of $\mathbb{C}^n$, by Walter Rudin

Algebraic K-Theory (second edition), by V. Srinivas

Computation of special functions, by Shanjie Zhang and Jianming Jin

The concentration-compactness principle in the Calculus of Variations, by Pierre-Louis Lions
The author of this paper was the first to give a complete solution with proof to the Boltzmann equation. He was awarded the Fields Medal in 1994.

The geometry of Topological Stability, by Andrew du Plessis and Terry Wall
In presenting a detailed study of the geometry and topology of numerous classes of “generic” singularities, Geometry of Topological Stability bridges the gap between algebraic calculations and continuity arguments to detail the necessary and sufficient conditions for a C (infinity) to be C0-stable. Throughout, the authors masterfully examine this important subject using results culled from a broad range of mathematical disciplines, including geometric topology, stratification theory, algebraic geometry, and commutative algebra. Packed with original research, much of which is presented here for the first time, the book will be welcomed by students and researchers interested in singularity theory and related areas.

Spectra of graphs, by Andries E. Brouwer and Willem H. Haemers
An introduction to the interesting subject of graph spectra: the analysis of the relation between the properties of a (maybe directed, with loops and multiedges) graph and the eigendecomposition of its adjacency matrix.

Projective Differential Geometry of curves and ruled surfaces, by E. J. Wilczynski

And in Physics:

Bose-Condensed gases at finite temperatures, by Allan Griffin, Tetsuro Nikuni and Eugene Zaremba
The discovery of Bose Einstein condensation (BEC) in trapped ultracold atomic gases in 1995 has led to an explosion of theoretical and experimental research on the properties of Bose-condensed dilute gases. The first treatment of BEC at finite temperatures, this book presents a thorough account of the theory of two-component dynamics and nonequilibrium behaviour in superfluid Bose gases.

Brownian movement and molecular reality, by Jean Perrin
Early studies by Einstein and Perrin provided some of the first evidence for the existence of molecules. Perrin, a Nobel Laureate, wrote this classic to explain his measurements of displaced particles of a resin suspended in water. It introduced the concept of Avogadro’s number, along with other groundbreaking work. 1910 edition.

The value of Science (first book), by Henri Poincaré

Experiment and theory in Physics, by Max Born

The philosophy of Physics, by Max Planck

Popular scientific lectures, by Hermann von Helmholtz

Physics and beyond: encounters and conversations, by Werner Heisenberg

Contextual approach to Quantum Formalism, by Andrei Khrennikov
The aim of this book is to show that the probabilistic formalisms of classical statistical mechanics and quantum mechanics can be unified on the basis of a general contextual probabilistic model. By taking into account the dependence of (classical) probabilities on contexts (i.e. complexes of physical conditions), one can reproduce all distinct features of quantum probabilities such as the interference of probabilities and the violation of Bell’s inequality. Moreover, by starting with a formula for the interference of probabilities (which generalizes the well known classical formula of total probability), one can construct the representation of contextual probabilities by complex probability amplitudes or, in the abstract formalism, by normalized vectors of the complex Hilbert space or its hyperbolic generalization. Thus the Hilbert space representation of probabilities can be naturally derived from classical probabilistic assumptions.

Brownian Motion: Fluctuations, dynamics, and applications, by Robert M. Mazo

Many-Body problems and Quantum Field Theory: An introduction, by Philippe A. Martin and Francois Rothen
The book gives an introduction to the concepts and methods of many-body problems and quantum fields for graduate students and researchers. The formalism is developed in close conjunction with the description of a number of physical systems: cohesion and dielectric properties of the electron gas, superconductivity, superfluidity, nuclear matter and nucleon pairing, matter and radiation, interaction of fields by particle exchange and mass generation. Emphasis is put on analogies between the various systems.

Introduction to Statistical Physics, by Silvio Salinas
A text for graduate-level students, covering the statistical basis of thermodynamics, with examples from solid-state physics. Reviews statistical methods and classical thermodynamics, moving into statistical mechanics, with discussion of quantum statistical mechanics. Includes coverage of blackbody radiation, phonons, and magnons.

Electronic properties of materials, by Rolf E. Hummel
This book on electrical, optical, magnetic and thermal properties of materials differs from other introductory texts in solid state physics. First, it is written for engineers, particularly materials and electrical engineers, who want to gain a fundamental understanding of semiconductor devices, magnetic materials, lasers, alloys, and so forth. Second, it stresses concepts rather than mathematical formalism. Third, it is not an encyclopedia: The topics are restricted to material considered to be essential and which can be covered in one 15-week semester. The book is divided into five parts. The first part, “Fundamentals of Electron Theory,” introduces the essential quantum mechanical concepts needed for understanding materials science; the other parts may be read independently of each other.

Complex Variables: A physical approach with applications and MATLAB tutorials, by Steven G. Krantz
From the algebraic properties of a complete number field, to the analytic properties imposed by the Cauchy integral formula, to the geometric qualities originating from conformality, Complex Variables: A Physical Approach with Applications and MATLAB explores all facets of this subject, with particular emphasis on using theory in practice. The first five chapters encompass the core material of the book. These chapters cover fundamental concepts, holomorphic and harmonic functions, Cauchy theory and its applications, and isolated singularities. Subsequent chapters discuss the argument principle, geometric theory, and conformal mapping, followed by a more advanced discussion of harmonic functions. The author also presents a detailed glimpse of how complex variables are used in the real world, with chapters on Fourier and Laplace transforms as well as partial differential equations and boundary value problems. The final chapter explores computer tools

Quantum Field Theory: A tourist guide for mathematicians, by Gerald B. Folland
Quantum field theory has been a great success for physics, but it is difficult for mathematicians to learn because it is mathematically incomplete. Folland, who is a mathematician, has spent considerable time digesting the physical theory and sorting out the mathematical issues in it. Fortunately for mathematicians, Folland is a gifted expositor. The purpose of this book is to present the elements of quantum field theory, with the goal of understanding the behavior of elementary particles rather than building formal mathematical structures, in a form that will be comprehensible to mathematicians. Rigorous definitions and arguments are presented as far as they are available, but the text proceeds on a more informal level when necessary, with due care in identifying the difficulties. The book begins with a review of classical physics and quantum mechanics, then proceeds through the construction of free quantum fields to the perturbation-theoretic development of interacting field theory and renormalization theory, with emphasis on quantum electrodynamics. The final two chapters present the functional integral approach and the elements of gauge field theory, including the Salam-Weinberg model of electromagnetic and weak interactions.

Musimathics, volume 1: The mathematical foundations of Music, by Gareth Loy
In this volume, Loy presents the materials of music (notes, intervals, and scales); the physical properties of music (frequency, amplitude, duration, and timbre); the perception of music and sound (how we hear); and music composition. Musimathics is carefully structured so that new topics depend strictly on topics already presented, carrying the reader progressively from basic subjects to more advanced ones.

Path integral methods, by T. Kashiwa, Y. Ohnuki and M. Suzuki
Path integrals are useful in quantum as well as statistical mechanics, providing more intuitive and easier methods of handling equations than other treatments. This book addresses fundamental issues in the field and covers more recent topics, such as the lattice fermion problem in quantum theory and the Monte Carlo method in statistical mechanics.

Quantum Mechanics, by Ernest S. Abers
A comprehensive book that intends to cover all subjects of Quantum Mechanics with a modern, quite rigorous mathematical exposition.

Quantum: The Quantum Theory of Particles, Fields and Cosmology, by Edgar Elbaz
This book gives a new insight into the interpretation of quantum mechanics (stochastic, integral paths, decoherence), a completely new treatment of angular momentum (graphical spin algebra) and an introduction to Fermion fields (Dirac equation) and Boson fields (e.m. and Higgs) as well as an introduction to QED (quantum electrodynamics), supersymmetry and quantum cosmology.

Special Relativity and Quantum Mechanics, by Francis Halpern
Special Relativity and Quantum Mechanics in just 138 pages.

[English]

Set Theory, by Felix Hausdorff

From the Preface (1937): “The present book has as its purpose an exposition of the most important theorems of the theory of sets, along with complete proofs, so that the reader should not find it necessary to go outside this book for supplementary details while, on the other hand, the book should enable him to undertake a more detailed study of the voluminous literature on the subject”.

Rings of operators, by Irving Kaplansky

Harmonic analysis on symmetric spaces and applications, by Audrey Terras

Fredholm theory in Banach spaces, by Anthony Francis Ruston

In this book, Dr Ruston presents analogues for operators on Banach spaces of Fredholm’s solution of integral equations of the second kind. Much of the presentation is based on research carried out over the last twenty-five years and has never appeared in book form before. Dr Ruston begins with the construction for operators of finite rank, using Fredholm’s original method as a guide. He then considers formulae that have structure similar to those obtained by Fredholm, using, and developing further, the relationship with Riesz theory. In particular, he obtains bases for the finite-dimensional subspaces figuring in the Riesz theory. Finally he returns to the study of specific constructions for various classes of operators. Dr Ruston has made every effort to keep the presentation as elementary as possible, using arguments that do not require a very advanced background. Thus the book can be read with profit by graduate students as well as specialists working in the general area of functional analysis and its applications.

Banach algebras, by Wieslaw Tadeusz Zelazko

An introductory book to Banach algebras.

And in Physics:

Global Warming: The science of climate change, by Frances Drake

Extremely topical over recent years, global warming has been the subject of a huge and growing amount of literature. Current literature however tends to fall into two camps: that which is highly scientific in nature and inaccessible to the average student, and that which is directed to the “lay” reader and lacks detail required by students. This book successfully bridges this gap, providing an accessible explanation of the physical mechanisms of global warming–discussed within the wider context of climate change.

Computational Statistics in Climatology, by Iliak Polyak

Scientific descriptions of the climate have traditionally been based on the study of average meteorological values taken from different positions around the world. In recent years however it has become apparent that these averages should be considered with other statstics that ultimately characterize spatial and temporal variability.

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Set Theory, de Felix Hausdorff

Del Prefacio (1937): “Este libro tiene como propósito la exposición de los teoremas más importantes de la teoría de conjuntos junto con sus pruebas completas, de tal forma que el lector no necesite acudir a otras fuentes fuera del libro en busca de detalles suplementarios, mientras que, por otra parte, debería capacitarle para llevar a cabo un estudio más detallado de la voluminosa literatura sobre el tema”.

Rings of operators, de Irving Kaplansky

Harmonic analysis on symmetric spaces and applications, de Audrey Terras

Fredholm theory in Banach spaces, de Anthony Francis Ruston

En este libro, Ruston presenta operadores de espacios de Banach análogos de la solución de Fredholm de ecuaciones integrales de segundo tipo.

Banach algebras, de Wieslaw Tadeusz Zelazko

Un libro introductorio a las álgebras de Banach.

And in Physics:

Global Warming: The science of climate change, by Frances Drake

El calentamiento global es un tópico ampliamente debatido en años recientes y ha sido objeto de mucha y creciente literatura. La actual, sin embargo, tiende a caer en dos campos: o bien es altamente especializada e inaccesible al estudiante medio, o bien está dirigida al laico en la materia y obvia los detalles necesarios para el estudiante. Este libro rellena el hueco de forma satisfactoria.

Computational Statistics in Climatology, de Iliak Polyak

Las descripciones científicas del clima se han basado tradicionalmente en el estudio de los valores meteorológicos medios tomados desde distintas posiciones alrededor del globo. En años recientes se ha entendido que estas medias deberían considerarse junto con otros estadísticos que caracterizan en última instancia la variabilidad espacial y temporal.

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[English]

Noncommutative localization in algebra and topology, by Andrew Ranicki
Noncommutative localization is a powerful algebraic technique for constructing new rings by inverting elements, matrices and more generally morphisms of modules. Originally conceived by algebraists (notably P.M. Cohn) it is now an important tool not only in pure algebra but also in the topology of non-simply-connected spaces, algebraic geometry and noncommutative geometry. This volume consists of 9 articles on noncommutative localization in algebra and topology. The aricles include basic definitions, surveys, historical background and applications, as well as presenting new results. The book is an introduction to the subject, an account of the state of the art, and also provides many references for further material.

Fundamental algorithms for permutation groups, by Gregory Butler
This is the first-ever book on computational group theory. It provides extensive and up-to-date coverage of the fundamental algorithms for permutation groups with reference to aspects of combinatorial group theory, soluble groups, and p-groups where appropriate. The book begins with a constructive introduction to group theory and algorithms for computing with small groups, followed by a gradual discussion of the basic ideas of Sims for computing with very large permutation groups, and concludes with algorithms that use group homomorphisms, as in the computation of Sylow subgroups. No background in group theory is assumed. The emphasis is on the details of the data structures and implementation which makes the algorithms effective when applied to realistic problems. The algorithms are developed hand-in-hand with the theoretical and practical justification.All algorithms are clearly described, examples are given, exercises reinforce understanding, and detailed bibliographical remarks explain the history and context of the work. Much of the later material on homomorphisms, Sylow subgroups, and soluble permutation groups is new.

Christian Huygens and the development of science in the seventeenth century, by Arthur Bell

Handbook of boolean algebras (volume 1), by J. Donald Monk and Robert Bonnet
Handbook of boolean algebras (volume 2), by J. Donald Monk and Robert Bonnet
Handbook of boolean algebras (volume 3), by J. Donald Monk and Robert Bonnet
This Handbook treats those parts of the theory of Boolean algebras of most interest to pure mathematicians: the set-theoretical abstract theory and applications and relationships to measure theory, topology, and logic. It is divided into two parts (published in three volumes). Part I (volume 1) is a comprehensive, self-contained introduction to the set-theoretical aspects of the theory of Boolean Algebras. It includes, in addition to a systematic introduction of basic algebra and topological ideas, recent developments such as the Balcar-Franek and Shelah-Shapirovskii results on free subalgebras. Part II (volumes 2 and 3) contains articles on special topics describing – mostly with full proofs – the most recent results in special areas such as automorphism groups, Ketonen’s theorem, recursive Boolean algebras, and measure algebras.

And in Physics:

From c-numbers to q-numbers, the classical analogy in the history of quantum theory, by Olivier Darrigol
The history of quantum theory is a maze of conceptual problems, through which Olivier Darrigol provides a lucid and learned guide, tracking the role of formal analogies between classical and quantum theory. From Planck’s first introduction of the quantum of action to Dirac’s formulation of quantum mechanics, Darrigol illuminates not only the history of quantum theory but also the role of analogies in scientific thinking and theory change. Unlike previous works, which have tended to focus on qualitative, global arguments, Darrigol’s study follows the lines of mathematical reasoning and symbolizing and so is able to show the motivations of early quantum theorists more preciselyand provocativelythan ever before. Erudite and original, From c- Numbers to q-Numbers sets a new standard as a philosophically perceptive and mathematically precise history of quantum mechanics.

Selected papers of M. Ohya, by N. Watanabe
This volume is a collection of articles written by Professor M Ohya over the past three decades in the areas of quantum teleportation, quantum information theory and quantum computers.

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Noncommutative localization in algebra and topology, de Andrew Ranicki
La localización no conmutativa es una técnica algebraica poderosa para construir nuevos anillos mediante matrices, y más generalmente, mediante morfismos de módulos. Concebida originalmente por algebristas (en particular por P.M. Cohn), ahora es una herramienta importante también en la topología de espacios no simplemente conexos, en geometría algebraica y en geometría no conmutativa.

Fundamental algorithms for permutation groups, de Gregory Butler
El primer libro escrito sobre teoría de grupos computacional. Cubre de forma extensa y actualizada los algoritmos fundamentales de los grupos de permutación con respecto a la teoría combinatoria de grupos, grupos solubles, y p-grupos.

Christian Huygens and the development of science in the seventeenth century, de Arthur Bell

Handbook of boolean algebras (volumen 1), de J. Donald Monk y Robert Bonnet
Handbook of boolean algebras (volumen 2), de J. Donald Monk y Robert Bonnet
Handbook of boolean algebras (volumen 3), de J. Donald Monk y Robert Bonnet
Este Handbook comprende aquellas partes de la teoría de álgebras Booleanas de mayor interés para los matemáticos: teoría de conjuntos abstracta y sus aplicaciones a y relaciones con la teoría de la medida, topología y lógica. Se divide en tres partes, publicadas en tres volúmenes: La primera parte (volumen 1) es una introducción exhaustiva y autocontenida a los aspectos conjuntistas de la teoría de álgebras de Boole. Incluye, además de una introducción sistemática de las ideas algebraicas y topológicas básicas, desarrollos recientes como los resultados de Balcar-Franek y Shelah-Shapirovskii en subálgebras libres. La segunda parte (volúmenes 2 y 3) contienen artículos sobre temas especializados y describen (en su mayor parte con demostraciones completas) los resultados más recientes en áreas especializadas como los grupos de automorfismos, el teorema de Ketonen, álgebras de Boole recursivas y and álgebras de medida.

Y en Física:

From c-numbers to q-numbers, the classical analogy in the history of quantum theory, de Olivier Darrigol
La historia de la teoría cuántica es un laberinto de problemas conceptuales, a través de los cuales Olivier Darrigol proporciona una guía lúcida. Desde la introducción inicial del cuanto de acción de Planck hastea la formulación de la mecánica cuántica de Dirac, Darrigol no sólo ilumina la historia de la teoría cuántica, también lo consigue con el papel de las analogías en el pensamiento científico y el cambio de teorías.

Selected papers of M. Ohya, de N. Watanabe
Este volumen es una colección de artículos escritos por el profesor en las últimas tres décadas en teletransporte cuántico, teoría cuántica de la información y computadores cuánticos.

[English]

Essays in Group Theory, by Gersten
Contains five papers on topics of current interest which were presented in a seminar at MSRI, Berkeley in June, 1985. Special mention should be given to Gromov`s paper, one of the most significant in the field in the last decade. It develops the theory of hyperbolic groups to include a version of small cancellation theory sufficiently powerful to recover deep results of Ol’shanskii and Rips. Each of the remaining papers, by Baumslag and Shalen, Gersten, Shalen, and Stallings contains gems. For example, the reader will delight in Stallings’ explicit construction of free actions of orientable surface groups on R-trees. Gersten’s paper lays the foundations for a theory of equations over groups and contains a very quick solution to conjugacy problem for a class of hyperbolic groups. Shalen’s article reviews the rapidly expanding theory of group actions on R-trees and the Baumslag-Shalen article uses modular representation theory to establish properties of presentations whose relators are pth-powers.

Ergodic Theory, by Karl Petersen
A classic. The author presents the fundamentals of the ergodic theory of point transformations and several advanced topics of intense research. Each of the basic aspects of ergodic theory–examples, convergence theorems, recurrence properties, and entropy–receives a basic and a specialized treatment. The author’s accessible style and the profusion of exercises, references, summaries, and historical remarks make this a useful book for graduate students or self study.

Begriffsschrift, a Formula Language, Modeled Upon that of Arithmetic, for Pure Thought, by Gottlob Frege
This is the first work that Frege work in the field of logic, and, although a mere booklet of 88 pages, it is perhaps the most important single work ever written in logic. Its fundamental contributions, among lesser points, are the truth-functional propositional calculus, the analysis of the propositon into function and arguments instead of subject and predicate, the theory of quantification, a system of logic in which derivations are carried out exclusively according to the form of the expressions, and a logical definition of the notion of mathematical sequence.

Sur la puissance des ensembles measurables B, by Paul S. Alexandroff (Alexandrov)
The birth of the operation (A), called also “Suslin (Souslin) schemes”.

And in Physics:

Fundamentals of photonic crystal guiding, by Maksim Skorobogatiy and Jianke Yang.
For anyone looking to understand photonic crystals, this systematic, rigorous, and pedagogical introduction is a must. Here you’ll find intuitive analytical and semi-analytical models applied to complex and practically relevant photonic crystal structures. You will also be shown how to use various analytical methods borrowed from quantum mechanics, such as perturbation theory, asymptotic analysis, and group theory, to investigate many of the limiting properties of photonic crystals which are otherwise difficult to rationalize using only numerical simulations. An introductory review of nonlinear guiding in photonic lattices is also presented, as are the fabrication and application of photonic crystals.

Principles of radiation interaction in matter and detection, by Claude Leroy and Pier-Giorgio Rancoita.
Data analysis and instrumentation applications require an excellent knowledge of the interactions between radiation and matter, radiation and particle detectors, the principles and conditions of detector operation, as well as the limitations and advantages. This book provides abundant information about the energy deposition in detecting systems, the performance and optimization of detectors. It also addresses the situation where detectors (scanners of various types, etc.) have to be modified and improved to full optimization by the users. Furthermore, the book offers a description of detectors and techniques used in medical physics and covers physics principles and instrumentation knowledge needed in radioprotection and nuclear engineering.

Theory of elasticity (second edition), by Timoshenko and Goodier

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Essays in Group Theory, de Gersten
Contiene cinco artículos de interés actual que fueron presentados en un seminario en el MSRI, Berkeley, en Junio de 1985. Mención especial merece el artículo de Gromov, uno de los más significativos de las últimas décadas. Desarrolla la teoría de los grupos hiperbólicos para incluir una versión de la teoría de cancelación pequeña lo suficientemente potente como para recuperar resultados profundos de Ol’shanskii y Rips.

Ergodic Theory, de Karl Petersen

Un clásico. El autor presenta los fundamentos de la teoría ergódica de trasnformaciones punto a punto y diversos tópicos de investigación avanzada.

Begriffsschrift, a Formula Language, Modeled Upon that of Arithmetic, for Pure Thought, de Gottlob Frege
El primer trabajo de Frege en el campo de la lógica y, probablemente, el artículo más importante de la historia en este campo.

Sur la puissance des ensembles measurables B, de Paul S. Alexandroff (Alexandrov)
El nacimiento de la operación (A), también llamada “esquemas de Suslin (Souslin)”.

Y en Física:

Fundamentals of photonic crystal guiding, de Maksim Skorobogatiy y Jianke Yang.
Introducción sistemática y rigurosa a los cristales fotónicos.

Principles of radiation interaction in matter and detection, de Claude Leroy y Pier-Giorgio Rancoita.
El análisis de datos y las aplicaciones instrumentales requieren un conocimiento excelente de las interaciones entre radiación y materia, radiación y detectores de partículas y los principios y condiciones de operación de los detectores.

Theory of elasticity (segunda edición), de Timoshenko y Goodier

A lot of new Gigapedia items for today.

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Hoy Gigapedia nos trae un montón de novedades.

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[English]

Higher Algebra, by Hall and Knight
A classic.

Paraconsistency: The logical way to inconsistency, by Walter A. Carnielli and Marcelo E. Coniglio

Paraconsistency involves “negating” the principle of explosion and taking a logic weaker than classical logic in order to allow for “some inconsistencies” without the whole system blowing-up.

The history of statistics: The measurement of uncertainty before 1900, by Stephen M. Stigler

Modern mathematics in the light of the Fields Medal, by Michael Monastyrsky

A mathematician survival guide: Graduate school and early career development, by Steven G. Krantz

Entertaining mathematical puzzles, by Martin Gardner

Sur la classification de M. Baire, by Nicolas Lusin
Sur une definition des ensembles measurables B sans nombres transfinis, by M. Souslin
The birth of analytical sets.

Fundamentos de las matemáticas, by David Hilbert
Anthology of texts by Hilbert from 1899 to 1930 about his investigations in the Foundations of Mathematics. In Spanish.

…plus all the following titles from the “Lecture Notes in Pure and Applied Mathematics” series by CRC (I will not put the links nor mention the authors, you can check them in Gigapedia!):

• Number theory and its applications
• Algebraic geometry
• Analysis, algebra and computers in mathematical research
• Fourier analysis: analytic and geometric aspects
• Moduli of vector bundles
• P-adic function analysis
• Graphs, matrices and designs
• Computational algebra
• Finite fields, coding theory and advances in communications and computing
• Number theory with an emphasis on the Markoff spectrum
• Finite or infinite dimensional complex analysis
• A mathematician’s survival guide: graduate school and earlier career development
• Differential equations and control theory

Also, in Physics:

The Early Earth: physical, chemical and biological development, by the Geological Society

Very special relativity: an illustrated guide, by Sander Bais
Special relativity just with elementary geometry.

Contemporary problems in mathematical physics, by J. Govaerts

Advanced Quantum Mechanics, 4th edition, by Franz Schwabl

Enjoy!

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[Español]

Higher Algebra, de Hall y Knight
Un clásico.

Paraconsistency: The logical way to inconsistency, de Walter A. Carnielli y Marcelo E. Coniglio

La paraconsistencia implica la “negación” del principio de explosión y tomar una lógica más débil que la clásica para poder permitir “algunas inconsistencias” dentro del sistema sin que éste “explote”.

The history of statistics: The measurement of uncertainty before 1900, de Stephen M. Stigler

Modern mathematics in the light of the Fields Medal, de Michael Monastyrsky

A mathematician survival guide: Graduate school and early career development, de Steven G. Krantz

Entertaining mathematical puzzles, de Martin Gardner

Sur la classification de M. Baire, de Nicolas Lusin
Sur une definition des ensembles measurables B sans nombres transfinis, de M. Souslin
El nacimiento de los conjuntos analíticos.

Fundamentos de las matemáticas, by David Hilbert
Antología de textos de Hilbert, desde 1899 hast 1930, tratando sus disquisiciones sobre los Fundamentos de las Matemáticas (¡en español!).

…más todos los títulos siguientes de la colección “Lecture Notes in Pure and Applied Mathematics” de CRC (no pondré los links ni mencionaré a sus autores, ¡podéis buscarlos en Gigapedia!):

• Number theory and its applications
• Algebraic geometry
• Analysis, algebra and computers in mathematical research
• Fourier analysis: analytic and geometric aspects
• Moduli of vector bundles
• P-adic function analysis
• Graphs, matrices and designs
• Computational algebra
• Finite fields, coding theory and advances in communications and computing
• Number theory with an emphasis on the Markoff spectrum
• Finite or infinite dimensional complex analysis
• A mathematician’s survival guide: graduate school and earlier career development
• Differential equations and control theory

Además, en Física tenemos:

The Early Earth: physical, chemical and biological development, de la Geological Society

Very special relativity: an illustrated guide, de Sander Bais
La relatividad especial explicada mediante geometría elemental.

Contemporary problems in mathematical physics, de J. Govaerts

Advanced Quantum Mechanics, 4th edition, de Franz Schwabl

¡Que los disfrutéis!

In this new section, we will daily echo the most relevant updates to Gigapedia in Math and Physics (if any!).

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[Español]

En esta nueva sección nos haremos eco diariamente de las novedades más relevantes de Gigapedia en Matemáticas y Física (¡si es que hay alguna!).

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[English]

For today we have:

1. Number Fields, by Daniel A. Marcus

This book is a real gem on basic Algebraic Number Theory (I know because I own it!). It is carefully written, with a pedagogical effort and interesting, productive exercises. It is 279 pages long and was originally published by Springer in 1977.

Its contents are:

• A special case of Fermat’s conjecture
• Number fields and number rings
• Prime decomposition in number rings
• Galois theory applied to prime decomposition
• The ideal class group and the unit group
• The distribution of ideals in a number ring
• The Dedekind zeta function and the class number formula
• The distribution of primes and an introduction to class field theory
• Appendix 1: Commutative rings and fields
• Appendix 2: Galois theory for subfields of
• Appendix 3: Finite fields and rings
• Appendix 4: Two pages of primes

2. Lectures on Fibre Bundles and Differential Geometry, by Jean-Louis Koszul

This book by the father of the Koszul complex explores the theory of connections via the covariant derivation and the connection forms on principal bundles.

3.Sur les fonctions représentable analytiquement, by Henri Lebesgue

The seminal paper which founded descriptive set theory, and where Lebesgue made also its famous mistake, corrected 12 years later by Suslin.

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[Español]

Para hoy tenemos:

1. Number Fields, de Daniel A. Marcus

Este libro es una verdadera joya de la Teoría de Números Algebraica básica (¡lo sé porque lo tengo!). Esta escrito cuidadosamente, con énfasis pedagógico y muchos ejercicios interesantes y productivos. Tiene 279 páginas y fue publicado originalmente por Springer en 1977.

Su tabla de contenidos es:

• A special case of Fermat’s conjecture
• Number fields and number rings
• Prime decomposition in number rings
• Galois theory applied to prime decomposition
• The ideal class group and the unit group
• The distribution of ideals in a number ring
• The Dedekind zeta function and the class number formula
• The distribution of primes and an introduction to class field theory
• Appendix 1: Commutative rings and fields
• Appendix 2: Galois theory for subfields of
• Appendix 3: Finite fields and rings
• Appendix 4: Two pages of primes

2. Lectures on Fibre Bundles and Differential Geometry, de Jean-Louis Koszul

Este libro, escrito por el padre del complejo de Koszul, explora la teoría de las conexiones a través de los conceptos de derivación covariante y forma conectiva en fibrados principales.

3.Sur les fonctions représentable analytiquement, de Henri Lebesgue

El artículo que fundó la teoría descriptiva de conjuntos. También en el que Lebesgue cometió su más famoso error, corregido 12 años después por Suslin.

Are you looking for a proof lost in an old mathematical book? Are you desperate to have a copy of some out-of-print science book? Do you want to have a look to some chapters of a prohibitively expensive book?

Then Gigapedia is your site!

You just have to register to get access to over:

• 1600 Astronomy/Astrophysics books
• 12000 Biology books
• 9000 Chemistry books
• 26000 Computer/IT books
• 14000 Math books
• 10000 Physics books

Of course, you must get them just for personal use and never use them to teach or for getting richer than their publishers.

Enjoy it!

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[Español]

¿Estás buscando una demostración perdida en un antiguo libro de matemáticas? ¿Quizás desesperado por tener algún libro científico descatalogado? ¿Te gustaría echarle un vistazo a algún capítulo de un libro prohibitivamente caro?

¡Gigapedia está hecha para ti!

No tienes más que registrarte para poder acceder a:

• 1.600 libros de Astronomía/Astrofísica
• 12.000 libros de Biología
• 9.000 libros de Química
• 26.000 libros de Computación/Tecnologías de la Información
• 14.000 libros de Matemáticas
• 10.000 libros de Física

Por supuesto, debes darles únicamente un uso personal y jamás dar clases con ellos o utilizarlos para hacerte más rico que los editores.

¡Que la disfrutes!

[English]

Today we have the privilege of inaugurating our Book Summaries section with a Physics classic: A Quantum Mechanics Primer, by Daniel T. Gillespie. This well-thought book serves as a quick and amicable, but also rigorous introduction to Quantum Mechanics (QM from now on) for the pure layperson with a mathematical knowledge.

Gillespie presents us a simple, non-relativistic, non-degenerate, one-dimensional scenario to teach us the fundamentals of the theory. QM mathematical foundations (the Hilbert space) and postulates (about states, observables and measurement) are his main aims. That’s why he -intentionally- does not enter the slippery realm of interpretations and sticks to the Copenhagen interpretation, that he considers “orthodox” for being the most commonly accepted between physicists. He does not develop any applications of the theory either (but he proposes a general exercise at the end that we will try to solve in some detail). Nevertheless, he compares QM with Classical Mechanics (CM) all over the work and performs mathematical deductions to satisfactorily explain the most famous QM “unintuitive” results (namely Heisenberg’s Uncertainty Principle, the Wave-Particle Duality and CM as the comfortable macroscopic limit for QM).

In this post we’ll cover the two first chapters of this work, which stand for a popular introduction to the subject and an explanation of the basic mathematical tools we’ll be needing in the following posts. If we get apart from Gillespie’s path, it will be just in rewordings and briefings, but all the ideas (at least at a pedagogical level) and the spirit of the work are due to him and for that he deserves full credit.

[Continues below]

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[Español]

Tenemos hoy el privilegio de inaugurar nuestra sección Book Summaries [Resúmenes de libros] con un clásico de la Física: Introducción a la Mecánica Cuántica, de Daniel T. Gillespie. Este bien estructurado libro sirve como introducción rápida y amigable, a la par que rigurosa, a la Mecánica Cuántica (MQ a partir de ahora) para el verdadero lego en la materia que posea ciertos conocimientos matemáticos.

Gillespie nos presenta un escenario sencillo, no relativista, no degenerado y unidimensional para mostrarnos las bases de la teoría. Sus objetivos principales son los fundamentos matemáticos de la MQ (el espacio de Hilbert) y sus postulados (sobre estados, observables y medidas). Éste es el motivo de que nos deje fuera (intencionadamente) del resbaladizo terreno de las distintas interpretaciones de la MQ y se contente con adherirse a la interpretación de Copenhague, que considera ortodoxa por ser la más aceptada en la comunidad física. Tampoco desarrolla ninguna aplicación de la teoría (pero al final propone un ejercicio general que trataremos de resolver con cierto detalle). No obstante, nos presenta una comparación entre la MQ y la Mecánica Clásica (MC), además de deducciones satisfactorias con las que explicar las resultados mecánicocuánticos más contraintuitivos (en concreto, el Principio de Incertidumbre de Heisenberg, la Dualidad Onda-Partícula y el hecho de que la MQ se transforme en nuestra reconfortable MC a nivel macroscópico).

En este post atacaremos los dos primeros capítulos de esta obra, que consisten en una introducción divulgativa a la materia y una exposición de las herramientas matemáticas básicas que pondremos en práctica en posts siguientes. Si nos apartamos algo del camino trazado por Gillespie, será únicamente al refrasear sus textos y resumirlos; pero tanto las ideas vertidas (a nivel pedagógico) como el espíritu de la obra se deben por entero a él y por ello le otorgamos todo el crédito.

[Continúa pulsando en el enlace]

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[English]

A Quantum Mechanics Primer

Daniel T. Gillespie

Motivation: Why Quantum Mechanics?

Toward the end of the nineteenth century it seemed quite apparent to all physicists that the general concepts of what we now call Classical Physics were adequate to describe all physical phenomena. CM, first formulated by Isaac Newton in the late seventeenth century, evidently provided a completely valid framework for the treatment of the dynamics of material bodies. Moreover, Classical Electrodynamics, finalized by James Clerk Maxwell in the latter half of the nineteenth century, described all the properties of the electromagnetic field and gave an intelligible account of the wave nature of light.

During the first quarter of the twentieth century, as physicists turned from their successful treatment of the macroscopic world to an examination of the microscopic world, a number of unexpected difficulties arose, which can be broadly divided into two general categories:

First was the discovery of instances in nature in which certain physical variables assumed only quantized or discrete values, in contrast to the continuum of values expected on the basis of Classical Physics. For example, in order to explain the black-body radiation, i.e., the observed intensity spectrum of electromagnetic radiation inside a constant-temperature cavity, Max Planck in 1900 found it necessary to permit each atomic oscillator in the walls of the cavity to radiate energy only in the discrete amounts

Here, is the intrinsic frequency of the radiating oscillator (the cavity walls were assumed to contain oscillators of all frequencies), and is Planck’s constant, which value is

joules·sec

There were several other instances of such quantum effects uncovered in the early part of the twentieth century, as the quantization of the angular momentum of hydrogen atom electrons postulated by Niels Bohr in 1913. In each case, the quantization of the appropriate variable amounted to an ad hoc hypothesis, and was without precedent in earlier applications of Classical Physics.

The second category of difficulties which beset Classical Physics concerned the distinction between waves and particles. By 1900 it was generally believed that light was a wave, while the electron was a particle. However, in 1905 Albert Einstein put forth his theory of the photoelectric effect, which indicated that a light beam of frequency behaves as though it were a collection of particles, each with an energy

Einstein’s hypothesis was a bold extrapolation of Planck’s theory of blackbody radiation, but it was subsequently borne out in great detail by precise experimental investigations.

In addition, experiments by C. Davison and L. Germer showed in 1927 that from a beam of electrons one can obtain diffraction patterns virtually identical to those which result from the crystal scattering of X-rays.

In short, light was found to behave sometimes as a wave and sometimes as a particle, and the electron was found to behave sometimes as a particle and sometimes as a wave! These results evidently implied some sort of wave-particle duality in nature which was quite unintelligible in terms of purely classical concepts.

With things like this, a radically different approach was needed. Such a new approach was not long in coming: by 1930, through the efforts of W. Heisenberg, E. Schrodinger, M. Born, N. Bohr, P. A. M. Dirac, and many other physicists, a bold new system of mechanics called QM had been devised. The basic tenets of QM are in many respects quite foreign to the concepts and attitudes of classical physics . However, there is no denying the fact that QM, in its present form, has been amazingly successful from an operational point of view;  that is, its predictions, no matter how unusual, have always been very much in accord with experimental observations. This is the reason for the acceptance of modern quantum theory by the overwhelming majority of physicists today.

The mathematical language of Quantum Mechanics

CM is formulated in terms of the mathematical language of differential and integral calculus. For example, velocity and acceleration are defined in terms of the derivative, work and impulse are defined in terms of the integral, and the conservation principles of energy and momentum find their rigorous justifications in certain elementary theorems of calculus.

QM has a mathematical language too, that involves not only calculus but also complex variable, linear algebra and probability theory, and we will see that the fundamental principles of the theory are also justified by elementary theorems. In this section we present briefly all those mathematical concepts which are essential to a meaningful understanding of QM. The necessity for achieving a reasonable degree of fluency in this mathematical language is even greater in the case of QM than CM; for quantum theory unfortunately does not readily lend itself to nonmathematical clarifications in terms of notions familiar to us from everyday experience.

Probability

We will just refresh the definitions and basic concepts of probability theory in a schematical way.

We say is a set of probabilities defined over a set of similar objects (with labels ) if these two relations hold:

1. for every
2. 

The sum and product rules state that

1. 
2. if the events are independent

The expected mean value and expected root-mean-square (rms) deviation are

Note that a probability distribution has zero rms deviation if and only if , i.e., if and only if it is constant (not random at all).

Finally, for any function of its expected mean value is computed as

Complex numbers

We will do here the same as with probabilities.

A complex number is a number , where and are real numbers and . Recall that are called respectively the real and imaginary parts of , and we write .

The complex conjugate of is the complex number

Conjugates carry the following properties:

1. ,
2. if and only if
3. ,

The modulus of is the real, nonnegative number , which satisfies:

1. 
2. ,
3. 
4. 

In exact analogy with the foregoing, we can define a complex function of a real variable to be a function of the form

where and are ordinary real functions of the real variable . All the preceding equations hold for complex functions, provided that we replace with and by .

The complex function can be differentiated and integrated with respect to its argument . The rules for carrying out these two operations are just what one would expect:

The Euclidean space

The language of QM is mainly the language of vector spaces. The reader is assumed to be familiar with the elementary properties of “ordinary vectors” in three-dimensional Euclidean space .  Actually, the notion of a vector space is much more general than this. In fact, QM is formulated in terms of an infinite-dimensional vector space called the Hilbert space . A complete development of the mathematics of is beyond our reach; however, at the expense of a little mathematical rigor and generality, we shall come to a fairly good understanding of the Hilbert space by drawing suitable analogies with the simpler, more familiar properties of .

A vector in can be defined as a directed line segment, possessing the properties of magnitude and direction.

Two operations common to all vector spaces are scalar multiplication and vector addition. Scalars in are simply the set of all real numbers . The multiplication of a vector by a scalar yields a new vector , whose direction is the same as that of but whose magnitude is times the magnitude of . Negative scalar multipliers reverse the direction. The addition of two vectors and yields a new vector , obtained by placing the tail of at the head of  and constructing the directed line segment from the tail of to the head of .

Another important feature of many (but not all) vector spaces is the existence of an operation called the inner product. In the inner product of and is, by definition,

where is the angle between and when they are placed tail-to-tail. The inner product of two vectors is always a scalar (in this case, a real number). In particular, the inner product of a vector with itself, called its norm, is always nonnegative:

[Note that we don't follow here the usual mathematical convention, which establishes the norm as the positive square root of the inner product of a vector with itself.]

The inner product of satisfies:

1. 
2. 
3. 
4. (Schwarz inequality)

Two vectors and are said to be orthogonal if . The set is orthonormal if , and is complete if for every we can find a set of scalars such that (in , any set of three or more noncoplanar vectors happens to be complete). Of particular interest are those sets of vectors which are both orthonormal and complete; such a set is called an orthonormal basis. In there are infinitely many different orthonormal basis sets (that differ by simple rotations), and all have exactly three vectors: that’s why is said to be three-dimensional.

If is an orthonomal basis, then for any we have

And if are vectors with components in the orthonomal basis , then:

The Hilbert space

We define a vector in to be a complex function of a single real variable . Not all such functions are truly vectors in , but only those that satisfy a certain condition; we shall state and discuss it a bit later. The scalars in are by definition the set of all complex numbers. The two operations of scalar multiplication and vector adition are defined by the usual rules for adding and multiplying complex quantities.

The inner product of and , which is always a scalar, is defined as

and the norm of , which is always a real number, is

The inner product of satisfies:

1. 
2. 
3. 
4. (Schwarz inequality)

We have just seen that, if we adopt certain well-defined rules for obtaining the scalar product, vector sum and inner product for complex functions of a real variable,  we arrive at properties that are essentially identical to those of ; consequently, we are entirely justified in regarding complex functions as “vectors” in a vector space.  Our definition of the inner product, which probably seemed peculiar to the reader, was chosen simply because it was a way of obtaining a unique scalar from two vectors such that those equations were satisfied. If we could conjure up a different set of rules for forming linear combinations and inner products which still satisfied all the conditions above, then we would have constructed another perfectly valid vector space of complex functions; however, that one would probably not turn out to be as relevant for describing physical phenomena as our Hilbert space turns out to be.

The condition that must satisfy to be a vector in is to have a finite norm:

An analogous condition was implicitly imposed on vectors, through their definition as directed line segments (i.e., lines of finite length). This condition insures the following important results:

1. If and are in , then their inner product “exists” (is a complex number, not infinity). This result follows from the Schwarz inequality.
2. If and are in , then so is any linear combination of them. This is proved using elemental properties of complex numbers and (1) above.

Two vectors and are said to be orthogonal if . The set is orthonormal if , and is complete if for every we can find a set of scalars such that . Special use will be made of sets of vectors which are both orthonormal and complete; such a set is called an orthonormal basis. In there are also infinitely many, but all of them contain infinitely many vectors: for this reason it is said to be infinite-dimensional.

If is an orthonomal basis, then for any we have

And if are vectors with components in the orthonomal basis , then:

In the remainder of this book, we shall be concerned only with vectors in and not in . However, the correspondences which we have traced between the two will often allow us to visualize by analogy just what we are doing in . This will help us to keep our feet on the ground as we proceed through the rather abstract theory of QM.

Hilbert space operators

An operator in the Hilbert space specifies a correspondence which associates with each vector in another vector (i.e., it is a “function” over vectors). We write . The product of times , and the sum and product of and are by definition such that the following equations are valid for all vectors:

It is not necessarily true that ; if this equality holds for all vectors , we say that and commute (for example, and do not commute).

In QM virtually all operators of interest possess a property called linearity. is said to be a linear operator if for any vectors and any scalars we have

(for example, is a linear operator). If are linear, then so are their product and any linear combination of them.

Another property which many operators in QM possess is hermiticity. is Hermitian if for any vectors ,

(for example, the simple operator is Hermitian if ). If are Hermitian, then so is any real linear combination of them, and their product will be Hermitian if they commute.
We turn now to one final aspect of operators which will prove to be very essential to the mathematical formulation of QM. If the effect of a given operator on some particular vector is to multiply it by an scalar , then we say that is an eigenvector (or eigenfunction) of , and is its corresponding eigenvalue:

(for example,  () is an eigenfunction of with eigenvalue ). We can now establish two important results concerning the eigenvectors of Hermitian operators:

1. The eigenvalues of an Hermitian operator are real, because implies .
2. The eigenvectors corresponding to two unequal eigenvalues of an Hermitian operator are orthogonal to each other, because implies ( are real).

We shall now prove a theorem that is almost the converse of the preceding two results. Suppose is a linear operator which possesses a complete, orthonormal set of eigenvectors and a corresponding set of real eigenvalues . Then is Hermitian.

Proof:

Let be two arbitrary vectors, and their components in the orthonomal basis . Then,

and analogously we get

Note that any operator in the conditions of the theorem above is completely specified by its sets of eigenvectors and eigenvalues.

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Up to here the introduction to the mathematical language of QM. Look forward to the next post, featuring a brief review of CM and the first three postulates of QM!

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Introducción a la Mecánica Cuántica

Daniel T. Gillespie

¿Por qué era necesaria la Mecánica Cuántica?

Hacia finales del siglo XIX, los físicos creían que los conceptos generales de lo que hoy llamamos Física Clásica eran adecuados para describir todos los fenómenos físicos. La MC, formulada por Isaac Newton a finales del siglo XVII proporcionaba, evidentemente, un marco completamente válido para el tratamiento de la dinámica de los cuerpos materiales. Más allá aún, la Electrodinámica Clásica, finalizada por James Clerk Maxwell en la segunda mitad del siglo XIX, describía todas las propiedades del campo electromagnético y daba una expliación inteligible de la naturaleza ondulatoria de la luz.

Sin embargo durante el primer cuarto del siglo XX, cuando los físicos pasaron de su tratamiento satisfactorio del mundo macroscópico a examinar el mundo microscópico, surgieron una serie de dificultades inesperadas, que pueden ser clasificadas de forma general en dos categorías:

Primeramente se descubrieron en la naturaleza ejemplos en los que ciertas variables físicas asumían sólo valores cuantizados o discretos, en contraste con la continuidad de valores que se desprendía de la Física Clásica. Por ejemplo, para explicar la radiación del cuerpo negro, es decir, el espectro observado de intensidades de la radiación electromagnética emergente del interior de una cavidad a temperatura constante, Max Planck encontró necesario en 1900 permitir que cada oscilador atómico de las paredes de la cavidad radiara energía solamente en cantidades discretas iguales a

donde es la frecuencia intrínseca del oscilador radiante  (se suponía que las paredes de la cavidad contenían osciladores de todas las frecuencias) y es la constante de Planck, cuyo valor es

julios·seg

Se descubrieron más instancias de tales efectos cúanticos en la primera parte del siglo XX, como la cuantización del momento angular de los electrones del átomo de hidrógeno postulada por Niels Bohr en 1913. En cada caso, la cuantización de la variable adecuada se convertía en una hipótesis añadida a posteriori que no tenía precedente en las aplicaciones anteriores de la Física Clásica.

La segunda categoría de dificultades que infestaron la Física Clásica concernía a la distinción entre ondas y partículas. En 1900 se creía, en general, que la luz era una onda mientras que el electrón era una partícula. Sin embargo, Albert Einstein  presentó en 1905 su teoría del efecto fotoeléctrico, en la que indicaba que un rayo de luz de frecuencia se comporta como si fuese una colección de partículas, cada una de ellas con una energía

La hipótesis de Einstein era una extrapolación atrevida de la teoría de Planck de la radiación del cuerpo negro, pero fue confirmada enseguida en gran detalle por estudios experimentales precisos.

En añadidura, experimentos llevados a cabo en 1927 por C. Davison y L. Germer mostraron que a partir de un haz de electrones se pueden obtener patrones de difracción virtualmente idénticos a los que resultan de la dispersión de rayos X mediante cristales.

En resumen, ¡se encontró que la luz se comportaba a veces como una onda y a veces como una partícula, y que el electrón se comportaba a veces como una partícula y a veces como una onda! Estos resultados implicaban ciertamente un tipo de dualidad onda-partícula en la naturaleza que no podía explicarse mediante los conceptos puramente clásicos.

Así las cosas, era necesario un punto de vista radicalmente diferente. Dicho punto de vista no tardó en llegar: hacia 1930, gracias a los esfuerzos de W. Heisenberg, E. Schrodinger, M. Born, N. Bohr, P. A. M. Dirac y muchos otros físicos, surgió un nuevo y audaz sistema de Mecánica llamado MQ. Los principios fundamentales de la MQ son en muchos aspectos bastante extraños a los conceptos y actitudes de la Física Clásica. Aún así, no puede negarse el hecho de que la MQ, en su forma actual, ha tenido un éxito abrumador desde un punto de vista operativo; es decir, sus predicciones,  por muy inusuales que sean, han estado siempre en estrecho acuerdo con las observaciones experimentales. Esta es la razón de la aceptación de la teoría cuántica moderna por parte de la inmensa mayoría de los físicos actuales.

El lenguaje matemático de la Mecánica Cuántica

La MC se formula con el lenguaje matemático del cálculo diferencial e integral. Por ejemplo, la velocidad y la aceleración se definen como derivadas, el trabajo y el impulso como integrales, y los principios de conservación de la energía y de la cantidad de movimiento hallan sus justificaciones rigurosas en ciertos teoremas elementales del cálculo.

La MQ también tiene un lenguaje matemático, en el que intervienen no sólo el cálculo, sino también la variable compleja, el álgebra lineal y la teoría de la probabilidad, y como veremos, los principios fundamentales de la teoría también se justifican mediante teoremas elementales. En esta sección presentamos brevemente todos aquellos conceptos matemáticos que son esenciales para comprender el significado de la MQ. La necesidad de lograr un grado de fluidez razonable en este lenguaje matemático es aún mayor en el caso de la MQ que en el de la MC ya que, desgraciadamente, la teoría cuántica no se presta fácilmente a aclaraciones no matemáticas mediante nociones que nos sean familiares por nuestra experiencia cotidiana.

Probabilidad

Refrescaremos de forma esquemática las definiciones y los conceptos básicos de la teoría de la probabilidad.

Decimos que es un conjunto de probabilidades asociadas a un conjunto de objetos similares (con etiquetas ) si se tienen las dos relaciones siguientes:

1. for every
2. 

Las reglas de la suma y el producto establecen que

1. 
2. si los sucesos son independientes

El valor medio esperado y la desviación cuadrática media son, respectivamente,

Obsérvese que una distribución de probabilidad tiene desviación cuadrática media cero si y sólo si , es decir, si y sólo si es constante (no aleatoria).

Finalmente, el valor medio esperado de cualquier función de se calcula como

.

Números complejos

Haremos aquí lo mismo que con las probabilidades.

Un número complejo es un número , donde y son números reales e . Recordemos que se llaman respecivamente las partes real e imaginaria de y que escribimos .

El complejo conjugado de es el número complejo

Los conjugados poseen las siguientes propiedades:

1. ,
2. si y sólo si
3. ,

El módulo de es el número real no negativo , que satisface:

1. 
2. ,
3. 
4. 

En analogía exacta con lo anterior, podemos definir una función compleja de variable real como función de la forma

donde y son funciones reales ordinarias de variable real .  Todas las igualdades precedentes siguen siendo válidas para las funciones complejas, sin más que sustituir p por y por .

La función compleja puede derivarse e integrarse respecto de su argumento . Las reglas para efectuar estas dos operaciones son precisamente las que cabía esperar:

El espacio euclídeo

El lenguaje de la MQ es esencialmente el lenguaje de los espacios vectoriales. Se supone al lector familiarizado con las propiedades elementales de los “vectores ordinarios” en el espacio euclídeo tridimensional . En realidad, la noción de espacio vectorial es mucho más general. De hecho, la MQ se formula en función de un espacio vectorial de dimensión infinita, el espacio de Hilbert . El desarrollo completo de la matemática de se sale del ámbito de esta obra; sin embargo, sacrificando un poco el rigor matemático y la generalidad, podremos comprender bastante bien el espacio de Hilbert  utilizando analogías con las propiedades familiares y más sencillas de .

Un vector en puede definirse como un segmento orientado, y como tal posee las propiedades de magnitud, dirección y sentido.

Dos operaciones comunes a todos los espacios vectoriales son la multiplicación por un escalar y la suma de vectores. Los escalares de son simplemente el conjunto de todos los números reales . La multiplicación de un vector por un escalar genera un nuevo vector , cuya dirección es la misma de pero cuya magnitud es veces la magnitud de . La multiplicación por escalares negativos invierte el sentido. La suma de dos vectores y devuelve un nuevo vector , que se obtiene colocando el origen de en el extremo de y construyendo el segmento orientado que tiene por origen el de y por extremo el de .

Otra característica importante de muchos espacios vectoriales (pero no de todos)  es la existencia de una operación llamada producto escalar. En el producto escalar de y es, por definición,

donde es el ángulo que forman y cuando se hacen coincidir sus orígenes. El producto escalar de dos vectores es siempre un escalar (en este caso, un número real). En particular, el producto escalar de un vector por sí mismo, denominado su norma, es siempre no negativo:

[Note el lector que en este texto no se sigue la convención matemática habitual, que define la norma como la raíz cuadrada postiva del producto escalar de un vector por sí mismo.]

El producto escalar de satisface:

1. 
2. 
3. 
4. (desigualdad de Schwarz)

Dos vectores y se dicen ortogonales si . El conjunto es ortonormal si , y es completo si para cada podemos encontrar un conjunto de escalares tal que (en , todo conjunto de tres o más vectores no coplanarios resulta ser completo). Presentan especial interés aquelos conjuntos de vectores que son a la vez ortonormales y completos; a un tal conjunto se le denomina base ortonormal. En existen infinitas bases ortonormales distintas (se pasa de una a otra mediante rotaciones simples), y todas tienen exactamente tres vectores: por eso decimos que es tridimensional.

Si es una base ortonormal, para cualquier se tiene

Y si son vectores con componentes en la base ortonormal , entonces:

El espacio de Hilbert

Definimos un vector en como una función compleja de una variable real . No todas estas funciones son realmente vectores de , sino solamente aquellas que satisfagan cierta condición que enunciaremos y discutiremos un poco más adelante. Los escalares de son por definición el conjunto de los números comlejos. Las dos operaciones de multiplicación por escalares y suma de vectores se definen mediante las reglas usuales para sumar y multiplicar cantidades complejas.

El producto escalar de y , que es siempre un escalar, se define como

y la norma de , que es siempre un número real, es

El producto escalar de cumple:

1. 
2. 
3. 
4. (Desigualdad de Schwarz)

Acabamos de ver que si adoptamos ciertas reglas bien definidas para la obtención de un producto por escalares, una suma de vectores y un producto escalar para funciones complejas de una variable real,  llegamos a propiedades que son esencialmente idénticas a las de ; en consecuencia, está plenamente justificado que consideremos las funciones complejas como “vectores”  de un espacio vectorial. Nuestra definición del producto escalar, que probablemente pueda parecer peculiar al lector, fue elegida sencillamente porque era una manera de obtener un escalar único a partir de dos vectores de manera que se cumpliesen dichas ecuaciones. Si pudiéramos establecer un conjunto diferente de reglas para formar combinaciones lineales y productos escalares que también cumpliesen las condiciones anteriores,  habríamos construido otro espacio vectorial de funciones complejas perfectamente válido; sin embargo, éste probablemente no se mostraría tan relevante para describir los fenómenos físicos como lo es nuestro espacio de Hilbert.

La condición que debe satisfacer para ser un vector de consiste en tener norma finita:

Implícitamente, se impuso una condición análoga sobre los vectores de al definirlos como segmentos orientados (es decir, líneas orientadas de longitud finita). Esta condición nos asegura los dos importantes resultados siguientes:

1. Si and están en , entonces su producto “existe” (es un número complejo, no infinito). Este resultado se sigue de la desigualdad de Schwarz.
2. Si y están en , entonces cualquier combinación lineal de ellos también lo está. Esto se prueba usando propiedades elementales de los números complejos y el resultado (1) anterior.

Dos vectores y se dicen ortogonales si . El conjunto es ortonormal si , y es completo si para cada podemos encontrar un conjunto de escalares tal que . Especialmente útiles serán los conjuntos de vectores que son a la vez ortonormales y completos; los llamaremos bases ortonormales. En también hay infinitas de estas bases, pero todas contienen un número infinito de vectores: por eso se dice que es infinito-dimensional.

Si es una base ortonormal, entonces para cualquier se tiene

Y si son vectores con componentes en la base ortonormal , entonces:

En el resto de este libro sólo nos preocuparemos de , no de . Sin embargo, las correspondencias wue hemos establecido entre los dos nos permitirán a menudo visualizar por analogía lo que estamos haciendo en . Esto nos ayudará a tener los pies en el suelo al avanzar a través de la teoría, algo abstracta, de la MQ.

Operadores del espacio de Hilbert

Un operator del espacio de Hilbert especifica una correspondencia que asocia a cada vector de   otro vector (es decir, es una “función” de vectores). Escribiremos . El producto de por y la suma y el producto de y son por definición tales que sean válidas las siguientes relaciones para todos los vectores:

No es necerasiamente cierto que ; si se tiene esta igualdad para todos los vectores , decimos que y conmutan (por ejemplo, y no cnmmutan).

En MQ, virtualmente todos los operadores de interés poseen una propiedad llamada linealidad. Se dice que es un operador lineal si para cada par de vectores y cada par de escalares tenemos

(por ejemplo, es un operador lineal). Si son lineales, también lo son su producto y cualquier combinación lineal de ellos.

Otra propiedad que poseen muchos operadores en la MQ es la hermiticidad. es hermítico si para cada par de vectores ,

(por ejemplo, el sencillo operador es hermítico si ). Si son hermíticos, también lo será cualquier combinación lineal real de ellos, y su producto será hermítico si conmutan.

Pasemos ahora a un aspecto final de los operadores que resultará realmente esencial para la formulación matemática de la MQ. Si el efecto de un operador sobre un vector particular es el de multiplicarlo por un escalar , decimos que es un autovector de , y es el correspondiente autovalor:

(por ejemplo,  () es un autovector de con autovalor ). Podemos ahora establecer dos importantes resultados referentes a los autovectores y autovalores de los operadores hermíticos:

1. Los autovalores de un operador hermítico son reales, porque implica .
2. Los autovectores correspondientes a dos autovalores diferentes de un operador hermítico son ortogonales, porque implica ( son reales).

Vamos a demostrar un teorema que es casi el recíproco de los dos resultados anteriores. Supongamos que es un operador lineal que posee un conjunto ortonormal completo de vectores propios con autovalores reales asociados . Entonces es hermítico.

Demostración:

Sean dos vectores arbitrarios y sus componentes en la base ortonormal . Entonces

y análogamente conseguimos

Nótese que cualquier operador que esté en las condiciones del teorema anterior queda completamente especificado por sus conjuntos de autovalores y autovectores.

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Hasta aquí la introducción al lenguaje matemático de la MQ. ¡No dejéis de leer el próximo post,  que incluirá un breve resumen de la MC y el desarrollo de los tres primeros postulados de la MQ!

]]> http://josebrox.wordpress.com/2009/04/26/gillespies-a-quantum-mechanics-primer-i/feed/ 0 JoseBrox Cover Cover http://josebrox.wordpress.com/2009/04/21/tricki-lives/ http://josebrox.wordpress.com/2009/04/21/tricki-lives/#comments Tue, 21 Apr 2009 09:06:17 +0000 JoseBrox ]]> [English]

Gower’s Tricki, a Wiki about “mathematical tricks” with a great scope, has finally been launched on the net. It’s available at

www.tricki.org

Its fundamental aim consists on joining in one place all techniques and metatechniques that are useful as general rules to solve problems – or at least some kind of problems in some branch of mathematics. There’s a broad spectrum about what qualifies as a trick: from the general rule “divide an conquer” to the pidgeonhole principle in Combinatorics or a density argument in Topology, going through fairly technical general lemmas in Real Analysis. Any approach likely to solve a whole bunch of problems at the same time will find its place at the Tricki.

In this author’s opinion, this effort is crucial, because nowadays a mathematician needs a good amount of “math engineering” skills in order to confront the difficult, complex problems that arise in his area of expertise. Math tricks will help him in two main ways:

1. First, they boost the analysis process providing techniques to divide problems into simpler steps.
2. Second, they abstract some of these steps making them to be thought on a more general equivalence class of (solved) problems, managing to relate an a priori different problem to already known ideas.

Just as a student has more chances of winning a Math Olympiad if he has some well-known tricks onto his pocket, the proffesional mathematician can win much more if he has the correct Tricki trick at hand. One cannot help but think of Terence Tao, a magician at solving problems with his wonderful vision for identifying tricks. 

Although this kind of mentality has been around for a while in specific branches like Combinatorics or Harmonic Analysis, a general exercise of this class of techniques by the whole mathematical community could well mean a revolution in terms of number of solved problems and of our own perception about mathematical interrelations. How many separated aspects will converge together?

Do you imagine having a Terence Tao inside yourself? Well, at last you will have “it” at hand, at the Tricki.

More information available at:
Tricki now fully live – Gowers
Tricki now live – Tao

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[Español]

Por fin ha salido a la red la Tricki de Tim Gowers, una ambiciosa wiki sobre “trucos” (tricks) matemáticos. Se encuentra en

www.tricki.org

Su objetivo fundamental es aunar en un sólo lugar todas las técnicas y metatécnicas conocidas que sean útiles a la hora de resolver problemas en general (o al menos, ciertos problemas dentro de alguna rama de las matemáticas). El espectro de lo que califica como trick es muy amplio: desde la regla general “divide y vencerás” hasta el principio del palomar en Combinatoria o el argumento de densidad en Topología, pasando por lemas realmente técnicos del Análisis Real. Cualquier enfoque que en lugar de un único problema determinado ataque clases completas de problemas será bien recibido en la Tricki.

En la opinión de quien escribe, este esfuerzo es crucial, porque hoy en día cualquier matemático necesita una buena dosis de habilidad en “ingeniería matemática” si quiere enfrentarse con éxito a los complejos problemas que aparecen en el área en que es experto. Los tricks matemáticos le ayudarán de dos formas principalmente:

1. En primer lugar, aceleran el proceso de análisis ayudando a dividir el problema en pasos más simples;
2. En segundo lugar, permiten abstraer algunos de estos pasos identificándolos con toda una clase de equivalencia de problemas (resueltos), haciendo que un problema en principio diferente se asocie con ideas ya conocidas.

De la misma manera que un estudiante tiene más posibilidades de ganar una Olimpiada Matemática simplemente teniendo un par de trucos bien conocidos en el bolsillo, el matemático profesional puede conseguir mucho más si tiene a mano el trick correcto de la Tricki. Uno no puede evitar pensar en Terence Tao, un verdadero mago a la hora de resolver problemas con su espectacular capacidad para identificar tricks.

Aunque este tipo de mentalidad viene siendo habitual en ciertas ramas como la Combinatoria o el Análisis Armónico, el uso generalizado de esta clase de técnicas por parte de toda la comunidad matemática podría suponer una revolución en cuanto a la cantidad de problemas resueltos y a nuestra percepción de las interrelaciones matemáticas.  ¿Cuántos aspectos que hasta ahora parecían separados se revelarán como esencialmente los mismos?

¿Se imaginan llevando un Terence Tao dentro? Al menos podrán tenerlo a mano, en la Tricki.

Más información disponible en:
Tricki now fully live – Gowers
Tricki now live – Tao