The Stability of Matter in Quantum Mechanics

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ISBN: 9780521191180 | 0521191181
Cover: Hardcover
Copyright: 12/14/2009

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Research into the stability of matter has been one of the most successful chapters in mathematical physics, and is a prime example of how modern mathematics can be applied to problems in physics. A unique account of the subject, this book provides a complete, self-contained description of research on the stability of matter problem. It introduces the necessary quantum mechanics to mathematicians, and aspects of functional analysis to physicists. The topics covered include electrodynamics of classical and quantized fields, Lieb-Thirring and other inequalities in spectral theory, inequalities in electrostatics, stability of large Coulomb systems, gravitational stability of stars, basics of equilibrium statistical mechanics, and the existence of the thermodynamic limit. The book is an up-to-date account for researchers, and its pedagogical style makes it suitable for advanced undergraduate and graduate courses in mathematical physics.

Preface	p. xiii
Prologue	p. 1
Introduction	p. 1
Brief Outline of the Book	p. 5
Introduction to Elementary Quantum Mechanics and Stability of the First Kind	p. 8
A Brief Review of the Connection Between Classical and Quantum Mechanics	p. 8
Hamiltonian Formulation	p. 10
Magnetic Fields	p. 10
Relativistic Mechanics	p. 12
Many-Body Systems	p. 13
Introduction to Quantum Mechanics	p. 14
Spin	p. 18
Units	p. 21
The Idea of Stability	p. 24
Uncertainty Principles: Domination of the Potential Energy by the Kinetic Energy	p. 26
The Hydrogenic Atom	p. 29
Many-Particle Systems and Stability of the Second Kind	p. 31
Many-Body Wave Functions	p. 31
The Space of Wave Functions	p. 31
Spin	p. 33
Bosons and Fermions (The Pauli Exclusion Principle)	p. 35
Density Matrices	p. 38
Reduced Density Matrices	p. 41
Many-Body Hamiltonians	p. 50
Many-Body Hamiltonians and Stability: Models with Static Nuclei	p. 50
Many-Body Hamiltonians: Models without Static Particles	p. 54
Monotonicity in the Nuclear Charges	p. 57
Unrestricted Minimizers are Bosonic	p. 58
Lieb-Thirring and Related Inequalities	p. 62
LT Inequalities: Formulation	p. 62
The Semiclassical Approximation	p. 63
The LT Inequalities; Non-Relativistic Case	p. 66
The LT Inequalities; Relativistic Case	p. 68
Kinetic Energy Inequalities	p. 70
The Birman-Schwinger Principle and LT Inequalities	p. 75
The Birman-Schwinger Formulation of the Schrödinger Equation	p. 75
Derivation of the LT Inequalities	p. 77
Useful Corollaries	p. 80
Diamagnetic Inequalities	p. 82
Appendix: An Operator Trace Inequality	p. 85
Electrostatic Inequalities	p. 89
General Properties of the Coulomb Potential	p. 89
Basic Electrostatic Inequality	p. 92
Application: Baxter's Electrostatic Inequality	p. 98
Refined Electrostatic Inequality	p. 100
An Estimation of the Indirect Part of the Coulomb Energy	p. 105
Introduction	p. 105
Examples	p. 107
Exchange Estimate	p. 110
Smearing Out Charges	p. 112
Proof of Theorem 6.1, a First Bound	p. 114
An Improved Bound	p. 118
Stability of Non-Relativistic Matter	p. 121
Proof of Stability of Matter	p. 122
An Alternative Proof of Stability	p. 125
Stability of Matter via Thomas-Fermi Theory	p. 127
Other Routes to a Proof of Stability	p. 129
Dyson-Lenard, 1967	p. 130
Federbush, 1975	p. 130
Some Later Work	p. 130
Extensivity of Matter	p. 131
Instability for Bosons	p. 133
The N^5/3 Law	p. 133
The N^7/5 Law	p. 135
Stability of Relativistic Matter	p. 139
Introduction	p. 139
Heuristic Reason for a Bound on ¿ Itself	p. 140
The Relativistic One-Body Problem	p. 141
A Localized Relativistic Kinetic Energy	p. 145
A Simple Kinetic Energy Bound	p. 146
Proof of Relativistic Stability	p. 148
Alternative Proof of Relativistic Stability	p. 154
Further Results on Relativistic Stability	p. 156
Instability for Large ¿, Large q or Bosons	p. 158
Magnetic Fields and the Pauli Operator	p. 164
Introduction	p. 164
The Pauli Operator and the Magnetic Field Energy	p. 165
Zero-Modes of the Pauli Operator	p. 166
A Hydrogenic Atom in a Magnetic Field	p. 168
The Many-Body Problem with a Magnetic Field	p. 171
Appendix: BKS Inequalities	p. 178
The Dirac Operator and the Brown-RavenhaU Model	p. 181
The Dirac Operator	p. 181
Gauge Invariance	p. 184
Three Alternative Hilbert Spaces	p. 185
The Brown-Ravenhall Model	p. 186
A Modified Brown-Ravenhall Model	p. 187
The Furry Picture	p. 188
The One-Particle Problem	p. 189
The Lonely Dirac Particle in a Magnetic Field	p. 189
The Hydrogenic Atom in a Magnetic Field	p. 190
Stability of the Modified Brown-Ravenhall Model	p. 193
Instability of the Original Brown-Ravenhall Model	p. 196
The Non-Relativistic Limit and the Pauli Operator	p. 198
Quantized Electromagnetic Fields and Stability of Matter	p. 200
Review of Classical Electrodynamics and its Quantization	p. 200
Maxwell's Equations	p. 200
Lagrangian and Hamiltonian of the Electromagnetic Field	p. 204
Quantization of the Electromagnetic Field	p. 207
Pauli Operator with Quantized Electromagnetic Field	p. 210
Dirac Operator with Quantized Electromagnetic Field	p. 217
The Ionization Problem, and the Dependence of the Energy on N and M Separately	p. 221
Introduction	p. 221
Bound on the Maximum Ionization	p. 222
How Many Electrons Can an Atom or Molecule Bind?	p. 228
Gravitational Stability of White Dwarfs and Neutron Stars	p. 233
Introduction and Astrophysical Background	p. 233
Stability and Instability Bounds	p. 235
A More Complete Picture	p. 240
Relativistic Gravitating Fermions	p. 240
Relativistic Gravitating Bosons	p. 242
Inclusion of Coulomb Forces	p. 243
The Thermodynamic Limit for Coulomb Systems	p. 247
Introduction	p. 247
Thermodynamic Limit of the Ground State Energy	p. 249
Introduction to Quantum Statistical Mechanics and the Thermodynamic Limit	p. 252
A Brief Discussion of Classical Statistical Mechanics	p. 258
The Cheese Theorem	p. 260
Proof of Theorem 14.2	p. 263
Proof for Special Sequences	p. 263
Proof for General Domains	p. 268
Convexity	p. 270
General Sequences of Particle Numbers	p. 271
The Jellium Model	p. 271
List of Symbols	p. 276
Bibliography	p. 279
Index	p. 290
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The Stability of Matter in Quantum Mechanics

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