Quantum Theory: A Mathematical Approach

Introductory remarks.- Historical background.- Physics in The Twentieth Century and Beyond.- Methodological Remarks.- References.- Classical Mechanics.- Introduction.-- Historical Remarks.- Newtonian Classical Mechanics.- The Lagrangian Formulation of Classical Mechanics.- The Hamiltonian Formulation of Classical Mechanics.- An Intrinsic Formulation.- An Algebraic Reformulation.- References.- Quantum Theory: General Principles.- Historical Background.- The Beginning of Quantum Mechanics.- Quantum Theory. General Remarks.- The basic concepts of quantum theory.- A preview.- States and Observables.- Time Evolution.- Symmetries.- References.- Quantum Mechanics of a Single Particle I.- Introduction.- `Diagonalizing' the Pj . The Fourier Transform.- A General Uncertainty Relation.- The Heisenberg Uncertainty Relation.- Minimal Uncertainty States.- The Heisenberg Inequality. Examples.- The 3-dimensional Case.5.- Quantum Mechanics of a Single Particle II.- Time Evolution of Wave Functions.- Pseudo-Classical Behavior of Expectation Values.- The Free Particle.- A Particle in a Box.- The Tunnel Effect.- The Harmonic Oscillator.- Introduction.- The Classical Harmonic Oscillator.- The Quantum Oscillator.- Lowering and Raising Operators.- Time Evolution.- Coherent States.- Time Evolution of Coherent States.- The 3-Dimensional Harmonic Oscillator.- The Hydrogen Atom.- Spin.- Many-Particle Systems.- Introduction.- Combining Quantum Systems - Systems of N Particles.- System of Identical Particles.- An Example: The Helium Atom.- Historical Remarks.- The Fock Space Formulation for Many-Particle Systems.- A Heuristic Formulation. `Second Quantization'.- References.- Review of Classical Statistical Physics.- Introduction.- Thermodynamics.- Classical Statistical Physics (Continued).- The Three main Ensembles.- The Microcanonical Ensemble.- The Canonical Ensemble.- The Grand Canonical Ensemble.- The Canonical Ensemble in the Approach of Gibbs.- From Statistical Mechanics to Thermodynamics.- Summary.- Kinetic Gas Theory.- General Statistical Physics.- References.- Quantum Statistical Physics.- Introduction.- What is an Ensemble in Quantum Statistical Physics?.- An Intermezzo - Is there a Quantum Phase Space?.- An Approach in Terms of Linear Functionals.- An Extended System of Axioms for Quantum Theory.- The Explicit Form of the Main Quantum Ensembles.- Planck's Formula for Black Body Radiation.- Bose-Einstein Condensation.- References.- Physical Theories as Algebraic Systems.- Introduction.- `Spaces'. Commutative and Noncommutative.- An Explicit Description of Physical Systems I.- An Explicit Description of Physical Systems II.- Quantum Theory: Von Neumann Versus Birkhoff.- References.- Quantization. The Classical Limit.- Towards Relativistic Quantum Theory.- Introduction.- Einstein's Special Theory of Relativity.- The Klein-Gordon Equation.- The Dirac Equation.- References.- An Introduction to Quantum Field Theory.- Introductory Remarks. Some History.- Quantum Field Theory as a Many Particle Theory.- Fock Space and its Operators.- The Scalar Quantum Field.- The Scalar Quantum Field.- The Field Operators.- The Scalar Field with Self-Interaction.- Towards a Rigorous Quantum Field Theory.- Concluding Remark.- References.