Quantum: The Quantum Theory of Particles, Fields and Cosmology

ISBN-13: 9783642643279 / Angielski / Miękka / 2011 / 602 str.

Edgard Elbaz

Opis

I consider teaching to have been one of my greatest privileges in life. I have al ways attemped to make a matter, sometimes difficult, sometimes in constant evolution, simpler and easier to understand. There comes a time then, when one feels the necessity to write and publish a book. For this reason, I have already published (in french) several lecture books on Classical Mechanics, Quantum Mechanics, Matrix Optics, Electroweak Interaction, General Rel ativity and Gravitation, Cosmology. Teaching quantum theory has been a particular delight as its constant evolution and enlargement, embraces all domains of physics. Even if the general layout remains relatively unchanged, the evolution of our understanding of the physical world imposes the in troduction of new approaches. It therefore seemed that a physics textbook, even one with a graduate readership in mind, had to introduce the Dirac Electron Theory and some rudimentary material on quantum field theory. This holds for also for explanations of the spontaneous symmetry breaking of Higgs scalar fields that gives mass to the bosons involved in short range interactions. We have deliberately cut down on material on some "classical" topics of quantum theory to make space for less known methods. Examples of the latter include the Feynman path integral "3rd quantization" and the interpretation of quantum mechanics in terms of phase focusing coherence."

Spis treści

1. Particles and Waves.- 1. Generalized Coordinates.- 2. Energy Conservation.- 3. Principle of Virtual Work.- 4. Lagrange Equations.- 5. Hamilton’s Principle of Least Action.- 6. Conservation Laws.- 7. The Center of Gravity Theorem.- 8. Symmetry and Invariance.- 9. Hamilton’s Equations.- 10. Maupertuis’ Least-Action Principle.- 11. Canonical Transformations.- 12. Poisson’s Equations.- 13. Hamilton-Jacobi Equations.- 14. Propagation Equation of Geometric Optics.- 15. Mechanics and Waves.- 16. Wave Mechanics.- 17. Analytical Mechanics - Schematic Summary.- 2. Wave Mechanics.- 1. Conservative Systems.- 2. Interpretations of the Wave Function.- 3. Mean Values of Observables.- 4. Heisenberg’s Uncertainty Relation.- 5. Interpretations of Wave Mechanics.- 6. Wave Mechanics - Schematic Summary.- 3. One-Dimensional Systems.- 1. Motion of the Center of Mass.- 2. General Properties of the Wave Function.- 3. Stationary Schrödinger Equation.- 4. Semi-classical (W.K.B.) Approximation.- 4. Dirac Formalism.- 1. Postulates of Quantum Mechanics.- 2. Dirac Formalism.- 3. Observables.- 4. Complete Set of Compatible Observables.- 5. Heisenberg’s Uncertainty Relations.- 6. Dirac Formalism - Schematic Summary.- 5. Representations.- 1. Wave Mechanics.- 2. Matrix Mechanics.- 3. Wave Mechanics - Schematic Summary.- 4. Matrix Mechanics - Schematic Summary.- 6. Quantum Dynamics.- 1. Time-Evolution Operator.- 2. Two Different Descriptions of Quantum Dynamics.- 3. The Correspondence Principle.- 4. Intermediate Description.- 5. Interpretation of Decoherence in Quantum Mechanics.- 6. Quantum Dynamics - Schematic Summary.- 7. Path Integrals.- 1. Evolution of the Wave Function.- 2. Path Integrals.- 3. Equivalence to the Schrödinger Equation.- 4. Action in Quantum Physics.- 5. Quantum Paths and Fractals.- 6. Path Integrals - Schematic Summary.- 8. Symmetries and Invariances.- 1. Unitary Transformations.- 2. Anti-unitary Transformations.- 3. Gauge Transformations.- 9. Angular Momentum.- 1. General Definition.- 2. Standard Representation.- 3. Examples of Angular Momenta.- 4. Magnetic Moments.- 10. Addition Theorems.- 1. Coupling of Two Angular Momenta.- 2. Clebsch-Gordan Coefficients.- 3. 3jm Coefficients.- 4. Coupling of n Angular Momenta.- 5. Change of Coupling Mode.- 6. Introduction to the Graphical Method.- Appendix 1: Table of Selected 3jm and 6j Coefficients.- Appendix 2: Diquark.- 11. Rotations.- 1. Introduction.- 2. Infinitesimal Rotations.- 3. Finite Rotations.- 4. Standard Representation.- 5. Fundamental Theorem of Pinch.- 6. Transformation Rules of Graphical Spin Algebra.- 12. Spherical Harmonics.- 1. Orbital Angular Momentum.- 2. Spherical Harmonic Y?o.- 3. General Statement.- 4. Parity.- 5. Orthonormalization and Closure.- 6. Graphical Representation.- 7. Integration of Three Spherical Harmonics.- 8. Contraction.- 9. Addition Theorem.- 10. Coupled Spherical Harmonics.- 11. Solid Spherical Harmonics.- 12.Schrödinger Equation in a Central Potential.- 13. Irreducible Tensor Operators.- 1. Definitions.- 2. Graphical Representation.- 3. Hermitian Conjugate.- 4. Scalar Product.- 5. Tensor Product.- 6. Standardization of a Vector Operator.- 7. Wigner-Eckart Theorem.- 8. Magnetic Moments.- 9. Density Operator.- 10. Differential Operators.- 11. Irreducible Tensor Operators - Schematic Summary.- Appendix 1: Reduced Matrix Elements.- Appendix 2: Graphical Vector Analysis.- 14. Perturbations.- 1. Stationary Perturbations.- 2. Variational Method.- 3. Time-Dependent Perturbations.- 15. Scattering of Particles.- 1. Scattering Operator.- 2. Transition Operator.- 3. Differential Cross Section.- 4. Scattering Amplitude and Interaction Potential.- 5. Lippmann-Schwinger Equations.- 6. Phase-Shift Method.- 7. Angular Distribution and Polarization.- 8. Scattering of Particles - Schematic Summary.- 16. Second Quantization.- 1. One-Dimensional Harmonic Oscillator.- 2. The p-Dimensional Harmonic Oscillator.- 3. Bosons and Fermions.- 4. Fermion Systems.- 5. Boson Systems.- 6. Simple Harmonic Oscillator - Schematic Summary.- 7. Second Quantization - Schematic Summary.- Appendix: The q-Commutators.- 17. Boson Fields.- 1. Classical Field Theory.- 2. Massive Scalar Fields.- 3. Higgs Fields.- 4. The Electromagnetic Field.- 5. Quantization of a Boson Field.- 18. Fermion Fields.- 1. Dirac Equation of a Free Particle.- 2. Dirac ? Matrices.- 3. The Adjoint Dirac Equation.- 4. Lagrangian Density and Field Equations.- 5. Probability Density.- 6. Gauge Invariance and Noether’s Theorem.- 7. Interaction with the Electromagnetic Field.- 8. Local Gauge Invariance.- 9. Electron Spin.- 10. Solutions for a Free Electron.- 11. Charge Conjugation.- 12. Hydrogen Atom.- 13. Coulomb Elastic Scattering.- 14. Quantization of a Fermion Field.- 15. Dirac Equation - Schematic Summary.- 19. Quantum X-Dynamics.- 1. Quantum Electrodynamics.- 2. Elementary Introduction to Quantum Chromodynamics.- 3. The GSW Standard Model.- 4. Grand Unification Theories.- Appendix: Introduction to Supersymmetry.- 20. Quantum Cosmology.- 1. Introduction.- 2. Essentials of General Relativity.- 3. The Correspondence Principle.- 4. Essentials of the Standard Model of Cosmology.- 5. Hamiltonian Statement of Einstein Equations.- 6. Fundamental Equation of Quantum Cosmology.- 7. Quantization in the Mini-Superspaces.- Exercises.- References.- Author Index.

Blurb

This textbook represents a new approach in the teaching of quantum physics. The first seven chapters present nonrelativistic quantum mechanics and its interpretation as well as perturbations and scattering theory. They include Dirac's and Feynman's formalisms. The chapter on symmetry treats gauge transformations, too. The quantum theory of angular momentum includes the isospin of leptons and quarks. It uses as a new tool the graphical spin algebra. The second part of the book is devoted to quantum fields: Boson fields including Higgs fields, Dirac's theory of Fermion fields, quantum electrodynamic and quantum chromodynamics. A brief review is appended to raise the students' interests in quantum cosmology.
The readers will also find many detailed worked examples and many problems to test their own understanding.