From Spinors to Quantum Mechanics discusses group theory and its use in quantum mechanics. Chapters 1 to 4 offer an introduction to group theory, and it provides the reader with an exact and clear intuition of what a spinor is, showing that spinors are just a mathematically complete notation for group elements. Chapter 5 contains the first rigorous derivation of the Dirac equation from a simple set of assumptions. The remaining chapters will interest the advanced reader who is interested in the meaning of quantum mechanics. They propose a novel approach to the foundations of quantum mechanics, based on the idea that the meaning of the formalism is already provided by the mathematics. In the traditional approach to quantum mechanics as initiated by Heisenberg, one has to start from a number of experimental results and then derive a set of rules and calculations that reproduce the observed experimental results. In such an inductive approach the underlying assumptions are not given at the outset. The reader has to figure them out, and this has proven to be difficult. The book shows that a different, bottom-up approach to quantum mechanics is possible, which merits further investigation as it demonstrates that with the methods used, the reader can obtain the correct results in a context where one would hitherto not expect this to be possible. Contents: IntroductionIntroduction to GroupsSpinors in the Rotation GroupSpinors in the Homogeneous Lorentz GroupThe Dirac Equation from ScratchTowards a Better Understanding of Quantum MechanicsThe Hidden-Variables Issue and the Bell InequalitiesEquivalence of the Bohr-Sommerfeld and Dirac Theories for the Hydrogen AtomThe Problem of the Electron Spin within a Magnetic FieldThe Double-Slit Experiment and the Superposition PrincipleA Caveat About the Limitations of Group TheorySpin and Angular Momentum as Vector and Bi-Vector Concepts Readership: Graduate students and researchers in the field of quantum mechanics. Key Features:Offers an excellent introduction to the group theory coveredContains a new approach to the foundations of quantum mechanics, with some interesting resultsCorrects a number of mathematical errors in the existing literature, e.g. in relation with the anomalous g-factor of the electronKeywords:Group Theory;Quantum Mechanics;SU(2);SL(2,C);Dirac Equation
Spinors in Physics
Author: Jean Hladik
Publisher: Springer Science & Business Media
Invented by Dirac in creating his relativistic quantum theory of the electron, spinors are important in quantum theory, relativity, nuclear physics, atomic and molecular physics, and condensed matter physics. Essentially, they are the mathematical entities that correspond to electrons in the same way that ordinary wave functions correspond to classical particles. Because of their relations to the rotation group SO(n) and the unitary group SU(n), this discussion will be of interest to applied mathematicians as well as physicists.
The Theory of Spinors
Author: Élie Cartan
Publisher: Courier Corporation
Describes orthgonal and related Lie groups, using real or complex parameters and indefinite metrics. Develops theory of spinors by giving a purely geometric definition of these mathematical entities.
This text focuses on the physics of symmetries, developing symmetries and transformations through concrete physical examples and contexts rather than presenting the information axiomatically, mathematically, and abstractly. Readers are introduced gradually to advanced mathematical procedures, including the Wigner and Racah algebras and their applications to various symmetry groups. The book also includes some of the latest research on the use of non-invariance and non-compact groups in the consideration of relativistic and many-particle problems of atoms and nuclei. This book is an updated replacement for the text Irreducible Tensorial Sets (Academic Press, 1959). Parts A and B of the present book grew out of occasional lectures in the intervening decades at the University of Chicago, where it became neccessary to update or elaborate upon certain points. Part C has been built more recently to deal with innovations and new information in the field of mathematical physics. The book as a whole develops the subject of symmetry from a physical point of view, allowing students and researchers to gain new insight on their subject. This book can be used both as a text and as a reference by students and scientists in the field. Adapts and extends the earlier Irreducible Tensor Sets (Academic Press, 1959) to classroom use Extends to multi-particle systems and relativity Includes problems in each chapter for homework assignments Embraces the latest research on non-invariance groups
Classic undergraduate text explores wave functions for the hydrogen atom, perturbation theory, the Pauli exclusion principle, and the structure of simple and complex molecules. Numerous tables and figures.
This text systematically presents the basics of quantum mechanics, emphasizing the role of Lie groups, Lie algebras, and their unitary representations. The mathematical structure of the subject is brought to the fore, intentionally avoiding significant overlap with material from standard physics courses in quantum mechanics and quantum field theory. The level of presentation is attractive to mathematics students looking to learn about both quantum mechanics and representation theory, while also appealing to physics students who would like to know more about the mathematics underlying the subject. This text showcases the numerous differences between typical mathematical and physical treatments of the subject. The latter portions of the book focus on central mathematical objects that occur in the Standard Model of particle physics, underlining the deep and intimate connections between mathematics and the physical world. While an elementary physics course of some kind would be helpful to the reader, no specific background in physics is assumed, making this book accessible to students with a grounding in multivariable calculus and linear algebra. Many exercises are provided to develop the reader's understanding of and facility in quantum-theoretical concepts and calculations.
This textbook gives a connected mathematical derivation of the important mathematical results, concentrating on the central ideas without including elaborate detail or unnecessary rigour, and explaining in the simplest terms the symbols and concepts which confront the researcher in solid state, nuclear or high-energy physics.
Relativistic Quantum Mechanics. Wave Equations concentrates mainly on the wave equations for spin-0 and spin-1/2 particles. Chapter 1 deals with the Klein-Gordon equation and its properties and applications. The chapters that follow introduce the Dirac equation, investigate its covariance properties and present various approaches to obtaining solutions. Numerous applications are discussed in detail, including the two-center Dirac equation, hole theory, CPT symmetry, Klein's paradox, and relativistic symmetry principles. Chapter 15 presents the relativistic wave equations for higher spin (Proca, Rarita-Schwinger, and Bargmann-Wigner). The extensive presentation of the mathematical tools and the 62 worked examples and problems make this a unique text for an advanced quantum mechanics course. This third edition has been slightly revised to bring the text up-to-date.
Author: Angelo Bassi
Publisher: Amer Inst of Physics
This conference brought together experts in different fields related to the foundations of quantum mechanics, ranging from mathematical physics to experimental physics, as well as the philosophy of science. The major topics discussed are: collapse models, Bohemian mechanics and their relativistic extensions, other alternative formulation of quantum mechanics, properties of entanglement, statistical physics and probability theory, new experimental results, as well as philosophical and epistemological issues.
Quantum Field Theory
Author: Mark Srednicki
Publisher: Cambridge University Press
Quantum field theory is the basic mathematical framework that is used to describe elementary particles. This textbook provides a complete and essential introduction to the subject. Assuming only an undergraduate knowledge of quantum mechanics and special relativity, this book is ideal for graduate students beginning the study of elementary particles. The step-by-step presentation begins with basic concepts illustrated by simple examples, and proceeds through historically important results to thorough treatments of modern topics such as the renormalization group, spinor-helicity methods for quark and gluon scattering, magnetic monopoles, instantons, supersymmetry, and the unification of forces. The book is written in a modular format, with each chapter as self-contained as possible, and with the necessary prerequisite material clearly identified. It is based on a year-long course given by the author and contains extensive problems, with password protected solutions available to lecturers at www.cambridge.org/9780521864497.
The solution of the Dirac equation for an electron in a Coulomb field is systematically treated here by utilizing new insights provided by supersymmetry. It is shown that each of the concepts has its analogue in the non-relativistic case. Indeed, the non-relativistic case is developed first, in order to introduce the new concepts in a familiar context. The symmetry of the non-relativistic model is already present in the classical limit, so the classical Kepler problem is first discussed in order to bring out the role played by the Laplace vector, one of the central concepts of the whole book. Analysis of the concept of eccentricity of the orbits turns out to be essential to understanding the relation of the classical and quantum mechanical models. The opportunity is taken to relive the great moments of physics: From Kepler's discovery of the laws of motion of the planets the development is traced through the Dirac equation up to modern advances, which bring the concepts of supersymmetry to bear on the derivation of the solutions.
Quantum Field Theory
Author: Lewis H. Ryder
Publisher: Cambridge University Press
This book is a modern introduction to the ideas and techniques of quantum field theory. After a brief overview of particle physics and a survey of relativistic wave equations and Lagrangian methods, the author develops the quantum theory of scalar and spinor fields, and then of gauge fields. The emphasis throughout is on functional methods, which have played a large part in modern field theory. The book concludes with a brief survey of "topological" objects in field theory and, new to this edition, a chapter devoted to supersymmetry. Graduate students in particle physics and high energy physics will benefit from this book.
This book continues the fundamental work of Arnold Sommerfeld and David Hestenes formulating theoretical physics in terms of Minkowski space-time geometry. We see how the standard matrix version of the Dirac equation can be reformulated in terms of a real space-time algebra, thus revealing a geometric meaning for the “number i” in quantum mechanics. Next, it is examined in some detail how electroweak theory can be integrated into the Dirac theory and this way interpreted in terms of space-time geometry. Finally, some implications for quantum electrodynamics are considered. The presentation of real quantum electromagnetism is expressed in an addendum. The book covers both the use of the complex and the real languages and allows the reader acquainted with the first language to make a step by step translation to the second one.
Author: Askold M Perelomov, Yakov B Zel’dovich
Publisher: World Scientific
It can serve as a good supplement to any quantum mechanics textbook, filling the gap between standard textbooks and higher-level books on the one hand and journal articles on the other. This book provides a detailed treatment of the scattering theory, multidimensional quasi-classical approximation, non-stationary problems for oscillators and the theory of unstable particles. It will be useful for postgraduate students and researchers who wish to find new, interesting information hidden in the depths of non-relativistic quantum mechanics. Contents: Discrete SpectrumContinuous SpectrumAnalytic Properties of Wave FunctionInverse Scattering ProblemThe Green Functions and Perturbation TheoryQuasi-classical ApproximationExact Solutions of Non-stationary Problems for OscillatorQuasi-stationary StatesAppendices:Specific Cases of the Schrödinger Equation SpectrumQuasi-classical Properties of Highly Excited Levels in the Coulomb Field Readership: Undergraduates, academics and researchers in physics. Keywords:Accidental Degeneracy;Bertrand Theorem;Coherent State;Coulomb Potential;Green Function;Inverse Scattering Problem;Isospectral Deformation;Perturbation Theory;Reflectionless Potential;Quantum Mechanics;Quasi-Classical Approximation;Quasi-Stationary State
Assuming a background in basic classical physics, multivariable calculus, and differential equations, A Concise Introduction to Quantum Mechanics provides a self-contained presentation of the mathematics and physics of quantum mechanics. The relevant aspects of classical mechanics and electrodynamics are reviewed, and the basic concepts of wave-particle duality are developed as a logical outgrowth of experiments involving blackbody radiation, the photoelectric effect, and electron diffraction. The Copenhagen interpretation of the wave function and its relation to the particle probability density is presented in conjunction with Fourier analysis and its generalization to function spaces. These concepts are combined to analyze the system consisting of a particle confi ned to a box, developing the probabilistic interpretation of observations and their associated expectation values. The Schrödinger equation is then derived by using these results and demanding both Galilean invariance of the probability density and Newtonian energy-momentum relations. The general properties of the Schrödinger equation and its solutions are analyzed, and the theory of observables is developed along with the associated Heisenberg uncertainty principle. Basic applications of wave mechanics are made to free wave packet spreading, barrier penetration, the simple harmonic oscillator, the Hydrogen atom, and an electric charge in a uniform magnetic fi eld. In addition, Dirac notation, elements of Hilbert space theory, operator techniques, and matrix algebra are presented and used to analyze coherent states, the linear potential, two state oscillations, and electron diffraction. Applications are made to photon and electron spin and the addition of angular momentum, and direct product multiparticle states are used to formulate both the Pauli exclusion principle and quantum decoherence. The book concludes with an introduction to the rotation group and the general properties of angular momentum.