1 Physical Background -- 1.1 The Double-Slit Experiment -- 1.2 Wave Functions -- 1.3 State Space -- 1.4 The Schrödinger Equation -- 1.5 Mathematical Supplement: Operators on Hilbert Spaces -- 2 Dynamics -- 2.1 Conservation of Probability -- 2.2 Existence of Dynamics -- 2.3 The Free Propagator -- 2.4 Mathematical Supplement: Operator Adjoints -- 2.5 Mathematical Supplement: the Fourier Transform -- 3 Observables -- 3.1 Mean Values and the Momentum Operator -- 3.2 Observables -- 3.3 The Heisenberg Representation -- 3.4 Quantization -- 3.5 Pseudodifferential Operators -- 4 The Uncertainty Principle -- 4.1 The Heisenberg Uncertainty Principle -- 4.2 A Refined Uncertainty Principle -- 4.3 Application: Stability of Hydrogen -- 5 Spectral Theory -- 5.1 The Spectrum of an Operator -- 5.2 Functions of Operators and the Spectral Mapping Theorem -- 5.3 Applications to Schrödinger Operators -- 5.4 Spectrum and Evolution -- 5.5 Variational Characterization of Eigenvalues -- 5.6 Number of Bound States -- 5.7 Mathematical Supplement: Integral Operators -- 6 Scattering States -- 6.1 Short-range Interactions: µ > 1 -- 6.2 Long-range Interactions: µ ? 1 -- 6.3 Existence of Wave Operators -- 7 Special Cases -- 7.1 The Infinite Well -- 7.2 The Torus -- 7.3 A Potential Step -- 7.4 The Square Well -- 7.5 The Harmonic Oscillator -- 7.6 A Particle on a Sphere -- 7.7 The Hydrogen Atom -- 7.8 A Particle in an External EM Field -- 8 Many-particle Systems -- 8.1 Quantization of a Many-particle System -- 8.2 Separation of the Centre-of-mass Motion -- 8.3 Break-ups -- 8.4 The HVZ Theorem -- 8.5 Intra- vs. Inter-cluster Motion -- 8.6 Existence of Bound States for Atoms and Molecules -- 8.7 Scattering States -- 8.8 Mathematical Supplement: Tensor Products -- 9 Density Matrices -- 9.1 Introduction -- 9.2 States and Dynamics -- 9.3 Open Systems -- 9.4 The Thermodynamic Limit -- 9.5 Equilibrium States -- 9.6 The T ? 0 Limit -- 9.7 Example: a System of Harmonic Oscillators -- 9.8 A Particle Coupled to a Reservoir -- 9.9 Quantum Systems -- 9.10 Problems -- 9.11 Hilbert Space Approach -- 9.12 Appendix: the Ideal Bose Gas -- 9.13 Appendix: Bose-Einst ein Condensation -- 9.14 Mathematical Supplement: the Trace, and Trace Class Operators -- 10 The Feynman Path Integral -- 10.1 The Feynman Path Integral -- 10.2 Generalizations of the Path Integral -- 10.3 Mathematical Supplement: the Trotter Product Formula -- 11 Quasi-classical Analysis -- 11.1 Quasi-classical Asymptotics of the Propagator -- 11.2 Quasi-classical Asymptotics of Green's Function -- 11.3 Bohr-Sommerfeld Semi-classical Quantization -- 11.4 Quasi-classical Asymptotics for the Ground State Energy -- 11.5 Mathematical Supplement: Operator Determinants -- 12 Mathematical Supplement: the Calculus of Variations -- 12.1 Functionals -- 12.2 The First Variation and Critical Points -- 12.3 Constrain ed Variational Problems -- 12.4 The Second Variation -- 12.5 Conjugate Points and Jacobi Fields -- 12.6 The Action of the Critical Path -- 12.7 Appendix: Connection to Geodesics -- 13 Resonances -- 13.1 Tunneling and Resonances -- 13.2 The Free Resonance Energy -- 13.3 Instantons -- 13.4 Positive Temperatures -- 13.5 Pre-exponential Factor for the Bounce -- 13.6 Contribution of the Zero-mode -- 13.7 Bohr-Sommerfeld Quantization for Resonances -- 14 Introduction to Quantum Field Theory -- 14.1 The Place of QFT -- 14.2 Klein-Gordon Theory as a Hamiltonian System -- 14.3 Maxwell's Equations as a Hamiltonian System -- 14.4 Quantization of the Klein-Gordon and Maxwell Equations -- 14.5 Fock Space -- 14.6 Generalized Free Theory -- 14.7 Interactions -- 14.8 Quadratic Approximation -- 15 Quantum Electrodynamics of Non-relativistic Particles: the Theory of Radiation -- 15.1 The Hamiltonian -- 15.2 Perturbation Set-up -- 15.3 Results -- 15.4 Mathematical Supplements -- 16 Supplement: Renormalization Group -- 16.1 The Decimation Map -- 16.2 Relative Bounds -- 16.3 Elimination of Particle and High Photon Energy Degrees of Freedom -- 16.4 Generalized Normal Form of Operators on Fock Space -- 16.5 The Hamiltonian H0(?, z) -- 16.6 A Banach Space of Operators -- 16.7 Rescaling -- 16.8 The Renormalization Map -- 16.9 Linearized Flow -- 16.10 Central-stable Manifold for RG and Spectra of Hamiltonians -- 16.11 Appendix -- 17 Comments on Missing Topics, Literature, and Further Reading -- References.