包邮量子力学中的数学概念

1 Physical Background1.1 The Double-Slit Experiment1.2 Wave Functions1.3 State Space1.4 The Schr6dinger Equation1.5 Mathematical Supplement: Operators on Hilbert Spaces,2 Dynamics2.1 Conservation of Probability2.2 Existence of Dynamics2.3 The Free Propagator2.4 Mathematical Supplement: Operator Adjoints2.5 Mathematical Supplement: the Fourier Transform2.5.1 Definition of the Fourier Transform2.5.2 Properties of the Fourier Transform2.5.3 Functions of the Derivative3 Observables3,1 Mean Values and the Momentum Operator3.2 Observables3.3 The Heisenberg Representation3.4 Quantization3.5 Pseudodifferential Operators4 The Uncertainty Principle4.1 The Heisenberg Uncertainty Principle4.2 A Refined Uncertainty Principle4.3 Application: Stability of Hydrogen5 Spectral Theory5.1 The Spectrum of an Operator5.2 Functions of Operators and the Spectral Mapping Theorem..5.3 Applications to Schrodinger Operators5.4 Spectrum and Evolution5.5 Variational Characterization of Eigenvalues5.6 Number of Bound States5.7 Mathematical Supplement: Integral Operators6 Scattering States6.1 Short-raage Interactions:μ> 16.2 Long-range Interactions: μ6.3 Wave Operators7 Special Cases7.1 The Infinite Well7.2 The Torns7.3 A Potential Step7.4 The Square Well7.5 The Harmonic Oscillator7.6 A Particle on a Sphere7.7 The Hydrogen Atom7.8 A Particle in an External EM Field8 Many-particle Systems8.1 Quantization of a Many-particle System8.2 Separation of the Centre-of-mass Motion8.3 Break-ups8.4 The HVZ Theorem8.5 Intra- vs. Inter-cluster Motion8.6 Existence of Bound States for Atoms and Molecules8.7 Scattering States8.8 Mathematical Supplement: Tensor Products8.9 Appendix: Hartree and Gross-Pitaevski Equations9 Density Matrices9.1 Introduction9.2 States and Dynamics9.3 Open Systems9.4 The Thermodynamic Limit9.5 Equilibrium States9.6 The T →O Limit9.7 Example: a System of Harmonic Oscillators9.8 A Particle Coupled to a Reservoir9.9 Quantum Systems9.10 Problems9.11 Hilbert Space Approach9.12 BEC at T=O9.13 Appendix: the Ideal Bose Gas9.14 Appendix: Beee-Einstein Condensation9.15 Mathematical Supplement: the Trace, and Trace Class Operators9.16 Mathematical Supplement: Projections10 Perturbation Theory: Feshbach Method10.1 The Feshbach Method10.2 Example: The Zeeman Effect10.3 Example: Time-dependent Perturbations10.4 Appendix: Proof of Theorem 10.111 The Eeynman Path Integral ....11.1 The Feynman Path Integral11.2 Generalizations of the Path Integral11.3 Mathematical Supplement: The Trotter Product Formula12 Quasi-classical Analysis12.1 Quasi-classical Asymptoties of the Propagator12.2 Quasi-classical Asymptotics of Green's Function12.2.1 Appendix12.3 Bohr-Sommerfeld Semi-classical Quantization12.4 Quasi-classical Asymptotics for the Ground State Energy12.5 Mathematical Supplement: Operator Determinants13 Mathematical Supplement: The Calculus of Variations.13.1 Functionals13.2 The First Variation and Critical Points13.3 Constrained Variational Problems13.4 The Second Variation13.5 Conjugate Points and Jacobi Fields13.6 The Action of the Critical Path13.7 Appendix: Connection to Geodesics14 Resonances14.1 Tunneling and Resonances14.2 The Free Resonance Energy14.3 Instantons14.4 Positive Temperatures14.5 Pre-exponential Factor for the Bounce14.6 Contribution of the Zero-mode14.7 Bohr-Sommerfeld Quantization for Resonances15 Introduction to Quantum Field Theory15.1 The Place of QFT15.1.1 Physical Theories15.1.2 The Principle of Minimal Action15.2 Klein-Gordon Theory as a Hamiltoulan System15.2.1 The Legendre Transform15.2.2 Hamiltoninns15.2.3 Poison Brackets15.2.4 Hamilton's Equations15.3 Maxwell's Equations as a Hamiltonian System15.4 Quantization of the Klein-Gordon and Maxwell Equations..15.4.1 The Quantization Procedure15.4.2 Creation and Annihilation Operators15.4.3 Wick Ordering15.4.4 Quantizing Maxwall's Equations15.5 Fock Space15.6 Generalized Free Theory15.7 Interactions15.8 Quadratic Approximation15.8.1 Further Discussion15.8.2 A Brief Remark on Many-body Hamiltonians in Second Quantization and the Hartree Approximation .16 Quantum Electrodynamics of Non-relatlvistic Particles:The Theory of Radiation16.1 The Hamiltonian16.2 Perturbation Set-up16.3 Results16.4 Mathematical Supplements16.4.1 Spectral Projections16.4.2 Projecting-out Procedure17 Supplement: Renormalization Group17.1 The Decimation Map17.2 Relative Bounds17.3 Elimination of Particle and High Photon Energy Degrees of Freedom17.4 Generalized Normal Form of Operators on Fock Space17.5 The Hamiltonian H0(ε, z)17.6 A Banach Space of Operators17.7 Rescaling17.8 The Renormalization Map17.9 Linearized Flow17.10 Central-stable Manifold for RG and Spectra of Hamiltonians17.11 Appendix18 Comments on Missing Topics, Literatnre,and Further ReadingReferencesIndex