| 词条 | 量子力学中的数学概念 |
| 释义 | 图书信息出版社: 世界图书出版公司; 第1版 (2009年8月1日) 外文书名: Mathematical Concepts of Quantum Mechanics 平装: 286页 正文语种: 英语 开本: 24 ISBN: 7510005027, 9787510005022 条形码: 9787510005022 尺寸: 22.2 x 14.8 x 1.4 cm 重量: 399 g 作者简介作者：(加拿大)格斯特松 内容简介《量子力学中的数学概念(英文版)》介绍了：The first fifteen chapters of these lectures （omitting four to six chapters each year） cover a one term course taken by a mixed group of senior undergraduate and junior graduate students specializing either in mathematics or physics. Typically, the mathematics students have some background in advanced analysis, while the physics students have had introductory quantum mechanics. To satisfy such a disparate audience, we decided to select material which is interesting from the viewpoint of modern theoretical physics, and which illustrates an interplay of ideas from various fields of mathematics such as operator theory, probability, differential equations, and differential geometry. Given our time constraint, we have often pursued mathematical content at the expense of rigor. However, wherever we have sacrificed the latter, we have tried to explain whether the result is an established fact, or, mathematically speaking, a conjecture, and in the former case, how a given argument can be made rigorous. The present book retains these features. 目录1 Physical Background 1.1 The Double-Slit Experiment 1.2 Wave Functions 1.3 State Space 1.4 The Schr6dinger 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 2.5.1 Definition of the Fourier Transform 2.5.2 Properties of the Fourier Transform 2.5.3 Functions of the Derivative 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 Schrodinger 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-raage Interactions:μ> 1 6.2 Long-range Interactions: μ< 1 6.3 Wave Operators 7 Special Cases 7.1 The Infinite Well 7.2 The Torns 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 8.9 Appendix: Hartree and Gross-Pitaevski Equations 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 →O 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 BEC at T=O 9.13 Appendix: the Ideal Bose Gas 9.14 Appendix: Beee-Einstein Condensation 9.15 Mathematical Supplement: the Trace, and Trace Class Operators 9.16 Mathematical Supplement: Projections 10 Perturbation Theory: Feshbach Method 10.1 The Feshbach Method 10.2 Example: The Zeeman Effect 10.3 Example: Time-dependent Perturbations 10.4 Appendix: Proof of Theorem 10.1 11 The Eeynman Path Integral .... 11.1 The Feynman Path Integral 11.2 Generalizations of the Path Integral 11.3 Mathematical Supplement: The Trotter Product Formula 12 Quasi-classical Analysis 12.1 Quasi-classical Asymptoties of the Propagator 12.2 Quasi-classical Asymptotics of Green's Function 12.2.1 Appendix 12.3 Bohr-Sommerfeld Semi-classical Quantization 12.4 Quasi-classical Asymptotics for the Ground State Energy 12.5 Mathematical Supplement: Operator Determinants 13 Mathematical Supplement: The Calculus of Variations. 13.1 Functionals 13.2 The First Variation and Critical Points 13.3 Constrained Variational Problems 13.4 The Second Variation 13.5 Conjugate Points and Jacobi Fields 13.6 The Action of the Critical Path 13.7 Appendix: Connection to Geodesics 14 Resonances 14.1 Tunneling and Resonances 14.2 The Free Resonance Energy 14.3 Instantons 14.4 Positive Temperatures 14.5 Pre-exponential Factor for the Bounce 14.6 Contribution of the Zero-mode 14.7 Bohr-Sommerfeld Quantization for Resonances 15 Introduction to Quantum Field Theory 15.1 The Place of QFT 15.1.1 Physical Theories 15.1.2 The Principle of Minimal Action 15.2 Klein-Gordon Theory as a Hamiltoulan System 15.2.1 The Legendre Transform 15.2.2 Hamiltoninns 15.2.3 Poison Brackets 15.2.4 Hamilton's Equations 15.3 Maxwell's Equations as a Hamiltonian System 15.4 Quantization of the Klein-Gordon and Maxwell Equations.. 15.4.1 The Quantization Procedure 15.4.2 Creation and Annihilation Operators 15.4.3 Wick Ordering 15.4.4 Quantizing Maxwall's Equations 15.5 Fock Space 15.6 Generalized Free Theory 15.7 Interactions 15.8 Quadratic Approximation 15.8.1 Further Discussion 15.8.2 A Brief Remark on Many-body Hamiltonians in Second Quantization and the Hartree Approximation . 16 Quantum Electrodynamics of Non-relatlvistic Particles:The Theory of Radiation 16.1 The Hamiltonian 16.2 Perturbation Set-up 16.3 Results 16.4 Mathematical Supplements 16.4.1 Spectral Projections 16.4.2 Projecting-out Procedure 17 Supplement: Renormalization Group 17.1 The Decimation Map 17.2 Relative Bounds 17.3 Elimination of Particle and High Photon Energy Degrees of Freedom 17.4 Generalized Normal Form of Operators on Fock Space 17.5 The Hamiltonian H0(ε, z) 17.6 A Banach Space of Operators 17.7 Rescaling 17.8 The Renormalization Map 17.9 Linearized Flow 17.10 Central-stable Manifold for RG and Spectra of Hamiltonians 17.11 Appendix 18 Comments on Missing Topics, Literatnre,and Further Reading References Index |
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