com This page intentionally left blank www.com ii T H E S TA B I L I T Y O F M AT T E R IN QUANTUM MECHANICS 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 physi- cists. 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. Lieb is a Professor of Mathematics and Higgins Professor of Physics at Princeton University. He has been a leader of research in mathematical physics for 45 years, and his achievements have earned him numerous prizes and awards, including the Heineman Prize in Mathematical Physics of the American Physical Society, the Max-Planck medal of the German Physical Society, the Boltzmann medal in statistical mechanics of the International Union of Pure and Applied Physics, the Schock prize in mathematics by the Swedish Academy of Sciences, the Birkhoff prize in applied mathematics of the American Mathematical Society, the Austrian Medal of Honor for Science and Art, and the Poincaré prize of the International Association of Mathematical Physics. Robert Seiringer is an Assistant Professor of Physics at Princeton University.
His research is centered largely on the quantum-mechanical many-body problem, and has been recognized by a Fellowship of the Sloan Foundation, by a U. National Science Foundation Early Career award, and by the 2009 Poincaré prize of the International Association of Mathematical Physics.com ii THE STABILITY OF MATTER IN QUA N T U M ME CH A N I CS ELLIOTT H. LIEB AND ROBERT SEIRINGER Princeton University www.com iii CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.org Information on this title: www. Seiringer 2010 This publication is in copyright.
Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2009 ISBN-13 978-0-511-65818-1 eBook (NetLibrary) ISBN-13 978-0-521-19118-0 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.com To Christiane, Letizzia and Laura www.com vi Contents Preface xiii 1 Prologue 1 1.2 Brief Outline of the Book 5 2 Introduction to Elementary Quantum Mechanics and Stability of the First Kind 8 2.1 A Brief Review of the Connection Between Classical and Quantum Mechanics 8 2.4 Many-Body Systems 13 2.5 Introduction to Quantum Mechanics 14 2.2 The Idea of Stability 24 2.1 Uncertainty Principles: Domination of the Potential Energy by the Kinetic Energy 26 2.2 The Hydrogenic Atom 29 3 Many-Particle Systems and Stability of the Second Kind 31 3.1 Many-Body Wave Functions 31 3.1 The Space of Wave Functions 31 3.3 Bosons and Fermions (The Pauli Exclusion Principle) 35 vii www.com viii Contents 3.5 Reduced Density Matrices 41 3.2 Many-Body Hamiltonians 50 3.1 Many-Body Hamiltonians and Stability: Models with Static Nuclei 50 3.2 Many-Body Hamiltonians: Models without Static Particles 54 3.3 Monotonicity in the Nuclear Charges 57 3.4 Unrestricted Minimizers are Bosonic 58 4 Lieb--Thirring and Related Inequalities 62 4.1 LT Inequalities: Formulation 62 4.1 The Semiclassical Approximation 63 4.2 The LT Inequalities; Non-Relativistic Case 66 4.3 The LT Inequalities; Relativistic Case 68 4.2 Kinetic Energy Inequalities 70 4.3 The Birman–Schwinger Principle and LT Inequalities 75 4.1 The Birman–Schwinger Formulation of the Schrödinger Equation 75 4.2 Derivation of the LT Inequalities 77 4.5 Appendix: An Operator Trace Inequality 85 5 Electrostatic Inequalities 89 5.1 General Properties of the Coulomb Potential 89 5.2 Basic Electrostatic Inequality 92 5.3 Application: Baxter’s Electrostatic Inequality 98 5.4 Refined Electrostatic Inequality 100 6 An Estimation of the Indirect Part of the Coulomb Energy 105 6.4 Smearing Out Charges 112 6.5 Proof of Theorem 6.6 An Improved Bound 118 www.com Contents ix 7 Stability of Non-Relativistic Matter 121 7.1 Proof of Stability of Matter 122 7.2 An Alternative Proof of Stability 125 7.3 Stability of Matter via Thomas–Fermi Theory 127 7.4 Other Routes to a Proof of Stability 129 7.3 Some Later Work 130 7.5 Extensivity of Matter 131 7.6 Instability for Bosons 133 7.2 The N 7/5 Law 135 8 Stability of Relativistic Matter 139 8.1 Heuristic Reason for a Bound on α Itself 140 8.2 The Relativistic One-Body Problem 141 8.3 A Localized Relativistic Kinetic Energy 145 8.4 A Simple Kinetic Energy Bound 146 8.5 Proof of Relativistic Stability 148 8.6 Alternative Proof of Relativistic Stability 154 8.7 Further Results on Relativistic Stability 156 8.8 Instability for Large α, Large q or Bosons 158 9 Magnetic Fields and the Pauli Operator 164 9.2 The Pauli Operator and the Magnetic Field Energy 165 9.3 Zero-Modes of the Pauli Operator 166 9.4 A Hydrogenic Atom in a Magnetic Field 168 9.5 The Many-Body Problem with a Magnetic Field 171 9.6 Appendix: BKS Inequalities 178 10 The Dirac Operator and the Brown--Ravenhall Model 181 10.1 The Dirac Operator 181 10.2 Three Alternative Hilbert Spaces 185 10.1 The Brown–Ravenhall Model 186 www.2 A Modified Brown–Ravenhall Model 187 10.3 The Furry Picture 188 10.3 The One-Particle Problem 189 10.1 The Lonely Dirac Particle in a Magnetic Field 189 10.2 The Hydrogenic Atom in a Magnetic Field 190 10.4 Stability of the Modified Brown–Ravenhall Model 193 10.5 Instability of the Original Brown–Ravenhall Model 196 10.6 The Non-Relativistic Limit and the Pauli Operator 198 11 Quantized Electromagnetic Fields and Stability of Matter 200 11.1 Review of Classical Electrodynamics and its Quantization 200 11.2 Lagrangian and Hamiltonian of the Electromagnetic Field 204 11.3 Quantization of the Electromagnetic Field 207 11.2 Pauli Operator with Quantized Electromagnetic Field 210 11.3 Dirac Operator with Quantized Electromagnetic Field 217 12 The Ionization Problem, and the Dependence of the Energy on N and M Separately 221 12.2 Bound on the Maximum Ionization 222 12.3 How Many Electrons Can an Atom or Molecule Bind? 228 13 Gravitational Stability of White Dwarfs and Neutron Stars 233 13.1 Introduction and Astrophysical Background 233 13.2 Stability and Instability Bounds 235 13.3 A More Complete Picture 240 13.1 Relativistic Gravitating Fermions 240 13.2 Relativistic Gravitating Bosons 242 13.3 Inclusion of Coulomb Forces 243 14 The Thermodynamic Limit for Coulomb Systems 247 14.2 Thermodynamic Limit of the Ground State Energy 249 14.3 Introduction to Quantum Statistical Mechanics and the Thermodynamic Limit 252 www.com Contents xi 14.4 A Brief Discussion of Classical Statistical Mechanics 258 14.5 The Cheese Theorem 260 14.6 Proof of Theorem 14.1 Proof for Special Sequences 263 14.2 Proof for General Domains 268 14.4 General Sequences of Particle Numbers 271 14.7 The Jellium Model 271 List of Symbols 276 Bibliography 279 Index 290 www.com xii Preface The fundamental theory that underlies the physicist’s description of the material world is quantum mechanics – specifically Erwin Schrödinger’s 1926 formula- tion of the theory. This theory also brought with it an emphasis on certain fields of mathematical analysis, e., Hilbert space theory, spectral analysis, differen- tial equations, etc., which, in turn, encouraged the development of parts of pure mathematics. Despite the great success of quantum mechanics in explaining details of the structure of atoms, molecules (including the complicated molecules beloved of organic chemists and the pharmaceutical industry, and so essential to life) and macroscopic objects like transistors, it took 41 years before the most fundamental question of all was resolved: Why doesn’t the collection of negatively charged electrons and positively charged nuclei, which are the basic constituents of the theory, implode into a minuscule mass of amorphous matter thousands of times denser than the material normally seen in our world? Even today hardly any physics textbook discusses, or even raises this question, even though the basic conclusion of stability is subtle and not easily derived using the elementary means available to the usual physics student.
There is a tendency among many physicists to regard this type of question as uninteresting because it is not easily reducible to a quantitative one. Matter is either stable or it is not; since nature tells us that it is so, there is no question to be answered. Nevertheless, physicists firmly believe that quantum mechanics is a ‘theory of everything’ at the level of atoms and molecules, so the question whether quantum mechanics predicts stability cannot be ignored. The depth of the question is further revealed when it is realized that a world made of bosonic particles would be unstable.
It is also revealed by the fact that the seemingly innocuous interaction of matter and electromagnetic radiation at ordinary, every-day energies – quantum electrodynamics – should be a settled, closed subject, but it is not and it can be understood only in the context xiii www.com xiv Preface of perturbation theory. Given these observations, it is clearly important to know that at least the quantum-mechanical part of the story is well understood. It is this stability question that will occupy us in this book. After four decades of development of this subject, during which most of the basic questions have gradually been answered, it seems appropriate to present a thorough review of the material at this time.
Schrödinger’s equation is not simple, so it is not surprising that some inter- esting mathematics had to be developed to understand the various aspects of the stability of matter. In particular, aspects of the spectral theory of Schrödinger operators and some new twists on classical potential theory resulted from this quest. Some of these theorems, which play an important role here, have proved useful in other areas of mathematics. The book is directed towards researchers on various aspects of quantum mechanics, as well as towards students of mathematics and students of physics.
We have tried to be pedagogical, recognizing that students with diverse back- grounds may not have all the basic facts at their finger tips. Physics students will come equipped with a basic course in quantum mechanics but perhaps will lack familiarity with modern mathematical techniques. These techniques will be introduced and explained as needed, and there are many mathematics texts which can be consulted for further information; among them is [118], which we will refer to often. Students of mathematics will have had a course in real anal- ysis and probably even some basic functional analysis, although they might still benefit from glancing at [118].
They will find the necessary quantum-mechanical background self-contained here in chapters two and three, but if they need more help they can refer to a huge number of elementary quantum mechanics texts, some of which, like [77, 22], present the subject in a way that is congenial to mathematicians. While we aim for a relaxed, leisurely style, the proofs of theorems are either completely rigorous or can easily be made so by the interested reader. It is our hope that this book, which illustrates the interplay between mathematical and physical ideas, will not only be useful to researchers but can also be a basis for a course in mathematical physics. To keep things within bounds, we have purposely limited ourselves to the subject of stability of matter in its various aspects (non-relativistic and relativis- tic mechanics, inclusion of magnetic fields, Chandrasekhar’s theory of stellar collapse and other topics).