آنتروپی و کوانتوم: مدرسه تحلیلی آریزونا با کاربردها، 16-20 مارس 2009 دانشگاه آریزونا (ریاضیات معاصر)

These lecture notes provide a pedagogical introduction to quantum mechanics and to some of the mathematics that has been motivated by this field. They are a product of the school ""Entropy and the Quantum"", which took place in Tucson, Arizona, in 2009. They have been written primarily for young mathematicians, but they will also prove useful to more experienced analysts and mathematical physicists. In the first contribution, William Faris introduces the mathematics of quantum mechanics. Robert Seiringer and Eric Carlen review certain recent developments in stability of matter and analytic inequalities, respectively. Bruno Nachtergaele and Robert Sims review locality results for quantum systems, and Christopher King deals with additivity conjectures and quantum information theory. The final article, by Christian Hainzl, describes applications of analysis to the Shandrasekhar limit of stellar masses.|These lecture notes provide a pedagogical introduction to quantum mechanics and to some of the mathematics that has been motivated by this field. They are a product of the school ""Entropy and the Quantum"", which took place in Tucson, Arizona, in 2009. They have been written primarily for young mathematicians, but they will also prove useful to more experienced analysts and mathematical physicists. In the first contribution, William Faris introduces the mathematics of quantum mechanics. Robert Seiringer and Eric Carlen review certain recent developments in stability of matter and analytic inequalities, respectively. Bruno Nachtergaele and Robert Sims review locality results for quantum systems, and Christopher King deals with additivity conjectures and quantum information theory. The final article, by Christian Hainzl, describes applications of analysis to the Shandrasekhar limit of stellar masses Contents 6 Preface 8 List of Participants 10 Outline of Quantum Mechanics 12 1. The setting for quantum mechanics 13 1.1. Introduction 13 1.2. Plan of the exposition 15 1.3. Hilbert space 16 1.4. Unitary operators 17 2. The Schrödinger equation 19 2.1. Diffusion and the free motion Schrödinger equation 19 2.2. The free particle 20 2.3. Shift operators 23 2.4. The Schrödinger equation with a potential energy function 23 2.5. Motion with constant force 24 2.6. Spectral and propagator solutions 25 2.7. Particle in a box 26 2.8. Particle on the half-line 28 2.9. The diffusion with drift representation 29 2.10. The harmonic oscillator 30 3. Self-adjoint operators 34 3.1. Stone's theorem 34 3.2. Multiplication operators 35 3.3. The spectral theorem 36 3.4. Spectral measures 37 3.5. Generalized vectors 39 3.6. Dirac notation 40 3.7. Spectral representation for multiplicity one 42 3.8. Form sums and Schrödinger operators 44 4. The role of Planck's constant 46 4.1. The uncertainty principle 46 4.2. Classical mechanics 48 5. Spin and statistics 49 5.1. Spin 1⁄2 49 5.2. Composite systems 51 5.3. Statistics 53 6. Fundamental structures of quantum mechanics 53 6.1. Self-adjoint operators as dynamics 53 6.2. Self-adjoint operators as states 54 6.3. Self-adjoint operators as observables? 57 6.4. Self-adjoint operators with measurement as observables 59 Acknowledgments 63 References 63 Inequalities for Schrödinger Operators and Applications to the Stability of Matter Problem 64 Trace Inequalities and Quantum Entropy: An Introductory Course 84 1. Introduction 85 1.1. Basic definitions and notation 85 1.2. Trace inequalities and entropy 87 2. Operator convexity and monotonicity 92 2.1. Some examples and the Löwner-Heinz Theorem 92 2.2. Convexity and monotonicity for trace functions 96 2.3. Klein’s Inequality and the Peierls-Bogoliubov Inequality 98 3. The joint convexity of certain operator functions 101 3.1. The joint convexity of the map (A, B) → B*A−1B* on H+n × Mn 101 3.2. Joint concavity of the harmonic mean 102 3.3. Joint concavity of the geometric mean 104 3.4. The arithmetic-geometric-harmonic mean inequality 105 4. Projections onto *-subalgebras and convexity inequalities 105 4.1. A simple example 105 4.2. The von Neumann Double Commutant Theorem 107 4.3. Properties of the conditional expectation 111 4.4. Pinching, conditional expectations, and the Operator Jensen Inequality 117 5. Tensor products 119 5.1. Basic definitions and elementary properties of tensor products 119 5.2. Tensor products and inner products 122 5.3. Tensor products of matrices 124 5.4. The partial trace 125 5.5. Ando's identity 129 6. Lieb's Concavity Theorem and related results 129 6.1. Lieb's Concavity Theorem 129 6.2. Ando's Convexity Theorem 130 6.3. Lieb's Concavity Theorem and joint convexity of the relative entropy 131 6.4. Monotonicity of the relative entropy 132 6.5. Subadditivity and strong subadditivity of the entropy 133 7. Lp norms for matrices and entropy inequalities 134 7.1. The matricial analogs of the Lp norms 134 7.2. Convexity of A → Tr [(B*ApB)q/p] and certain of its applications 136 7.3. Proof of the convexity of A → Tr [(B*ApB)q/p] 140 8. Brascamp-Lieb type inequalities for traces 142 8.1. A generalized Young's inequality in the context of non-commutative integration 143 8.2. A generalized Young's inequality for tensor products 144 8.3. Subadditivity of Entropy and Generalized Young's Inequalities 146 Acknoledgements 149 References 149 Lieb-Robinson Bounds in Quantum Many-Body Physics 152 Remarks on the Additivity Conjectures for Quantum Channels 188 On the Static and Dynamical Collapse of White Dwarfs 200 Providing a pedagogical introduction to quantum mechanics, these lecture notes from ""Entropy and the Quantum"", include contributions on recent developments in stability of matter and analytic inequalities, a review of locality results for quantum systems, additivity conjectures and quantum information theory, and applications of analysis to the Shandrasekhar limit of stellar masses.