Foundations of Quantum Mechanics

Cover......Page 1 Title: FOUNDATIONS OFQUANTUM MECHANICS......Page 3 LCCN: 67-23978.......Page 4 PREFACE......Page 6 CONTENTS......Page 10 PART 1: Mathematical Foundations......Page 14 1-1. SOME NOTIONS AND NOTATIONS FROM SET THEORY......Page 16 PROBLEMS......Page 19 1-2. THE MEASURE SPACE......Page 20 1-3. MEASURABLE AND INTEGRABLE FUNCTIONS......Page 22 PROBLEMS......Page 26 1-4. THE THEOREM OF RADON-NIKODYM......Page 27 PROBLEMS......Page 28 1-5. FUNCTION SPACES......Page 29 REFERENCES......Page 30 I. H is a linear vector space with complex coefficients......Page 31 2-2. COMMENTS ON THE AXIOMS......Page 32 PROBLEMS......Page 35 2-3. REALIZATIONS OF HILBERT SPACE......Page 36 2-4. LINEAR MANIFOLDS AND SUBSPACES......Page 37 2-5. THE LATTICE OF SUI3SPACES......Page 39 PROBLEMS......Page 41 REFERENCES......Page 42 3.I. BOUNDED LINEAR FUNCTIONALS......Page 43 3.2. SESQUILINEAR FUNCTIONALS AND QUADRATIC FORMS......Page 45 3.3. BOUNDED LINEAR OPERATORS......Page 46 3.4. PROJECTIONS......Page 50 PROBLEMS......Page 52 3.5. UNBOUNDED OPERATORS......Page 53 1. The position operator Q......Page 55 2. The momentum operator......Page 56 PROBLEMS......Page 57 REFERENCES......Page 58 CHAPTER 4: SPECTRAL THEOREM AND SPECTRAL REPRESENTATION......Page 59 4-1. SELF-ADJOINT OPERATORS IN FINITE-DIMENSIONAL SPACES......Page 60 PROBLEMS......Page 64 4-2. THE RESOLVENT AND THE SPECTRUM......Page 65 4-3. THE SPECTRAL THEOREM......Page 66 PROBLEMS......Page 67 4-4. THE FUNCTIONAL CALCULUS......Page 68 PROBLEMS......Page 70 4-5. SPECTRAL DENSITIES AND GENERATING VECTORS......Page 71 4-6. THE SPECTRAL REPRESENTATION......Page 73 4-7. EIGENFUNCTION EXPANSIONS......Page 75 REFERENCES......Page 77 PART 2: Physical Foundations......Page 78 CHAPTER 5: THE PROPOSITIONAL CALCULUS......Page 80 5-1. HISTORIC-PHILOSOPHIC PRELUDE......Page 81 5-2. YES-NO EXPERIMENTS......Page 85 5-3. THE PROPOSITIONAL CALCULUS......Page 87 5-4. CLASSICAL SYSTEMS AND BOOLEAN LATTICES......Page 91 5-5. COMPATIBLE AND INCOMPATIBLE PROPOSITIONS......Page 93 PROBLEMS......Page 95 5-6. MODULARITY......Page 96 5-7. THE LATTICE OF SURSPACES......Page 97 5-8. PROPOSITION SYSTEMS......Page 99 PROBLEMS......Page 100 REFERENCES......Page 101 6-1. THE NOTION OF STATE......Page 103 6-2. THE MEASUREMENT OF THE STATE......Page 106 6-3. DESCRIPTION OF STATES......Page 107 6-4. THE NOTION OF OBSERVABLES......Page 110 6-5. PROPERTIES OF OBSERVABLES......Page 112 PROBLEMS......Page 113 6-7. THE FUNCTIONAL CALCULUS FOR OBSERVABLES......Page 114 PROBLEMS......Page 117 6-8. THE SUPERPOSITION PRINCIPLE......Page 118 Principle of Superposition......Page 119 PROBLEMS......Page 121 6-9. SUPERSELECTION RULES......Page 122 REFERENCES......Page 123 CHAPTER 7: HIDDEN VARIABLES......Page 124 7-1. A THOUGHT EXPERIMENT......Page 125 7-2. DISPERSION-FREE STATES......Page 127 7-3. HIDDEN VARIABLES......Page 129 7-4. ALTERNATIVE WAYS OF INTRODUCING HIDDEN VARIABLES......Page 132 REFERENCES......Page 133 8-I. PROJECTIVE GEOMETRIES......Page 134 8-2. REDUCTION THEORY......Page 137 PROBLEMS......Page 139 8-3. THE STRUCTURE OF IRREDUCIBLE PROPOSITION SYSTEMS......Page 140 PROBLEMS......Page 141 8-4. ORTHOCOMPLEMENTATION AND THE METRIC OF THE VECTOR SPACE......Page 142 8-5. QUANTUM MECHAMCS IN HILBERT SPACE......Page 144 REFERENCES......Page 146 9-1. THE MEANING OF SYMMETRY......Page 148 9-2. ABSTRACT GROUPS......Page 150 PROBLEMS......Page 151 b) Local properties of Lie groups......Page 152 e) The global properties of Lie groups......Page 154 9-4. THE AUTOMORPHISMS OF A PROPOSITION SYSTEM......Page 155 PROBLEMS......Page 157 PROBLEMS......Page 158 9-6. PROJECTIVE REPRESENTATION OF GROUPS......Page 159 PROBLEMS......Page 162 REFERENCES......Page 163 10-1. THE TiME EVOLUTION OF A SYSTEM......Page 164 10-2. THE DYNAMTCAL GROUP......Page 166 10-3. DIFFERENT DESCRIPTIONS OF THE TIME EVOLUTION......Page 168 10-4. NONCONSERVATIVE SYSTEMS......Page 170 11-1. UNCERTAINTY RELATIONS......Page 173 11-2. GENERAL DESCRIPTION OF THE MEASURING PROCESS......Page 176 11-3. DESCRIPTION OF THE MEASURING PROCESS FOR QUANTUM-MECHANICAL SYSTEMS......Page 177 11-4. PROPERTIES OF THE MEASURING DEVICE......Page 181 11-5. EQUIVALENT STATES......Page 183 11-6. EVENTS AND DATA......Page 186 11-7. MATHEMATICAL INTERLUDE: THE TENSOR PRODUCT......Page 188 PROBLEMS......Page 191 11-8. THE UNION AND SEPARATION OF SYSTEMS......Page 192 PROBLEMS......Page 195 11-9. A MODEL OF THE MEASURING PROCESS......Page 196 b) Einstein's element of physical reality......Page 198 c) Wigner's friend.......Page 200 d) Discussion of the paradoxes......Page 201 REFERENCES......Page 204 PART 3: Elementary Particles......Page 206 12-1. LOCALIZARILITY......Page 208 12-2. HOMOGENEITY......Page 210 12-3. THE CANONICAL COMMUTATION RULES......Page 212 12-4. THE ELEMENTARY PARTICLE......Page 218 12-5. VELOCITY AND GALILEI INVARIANCE......Page 219 PROBLEM......Page 223 12-6. THE HARMONIC OSCILLATOR......Page 224 PROBLEMS......Page 227 12-7. A HILBERT SPACE OF ANALYTICAL FUNCTIONS [13]......Page 228 PROBLEMS......Page 231 12-8. LOCALIZABILITY AND MODULARITY......Page 232 REFERENCES......Page 234 13-1. LOCALIZABILITY......Page 235 13-2. HOMOGENEITY AND ISOTROPY......Page 237 13-3. ROTATIONS AS KINEMATICAL SYMMETRIES......Page 240 13-4. VELOCITY AND GALILEI INVARIANCE......Page 247 13-5. GAUGE TRANSFORMATIONS AND GAUGE INVARIANCE......Page 250 13-6. DENSITY AND CURRENT OF AN OBSERVABLE......Page 252 13-7. SPACE INVERSION......Page 254 PROBLEMS......Page 256 13-8. TIME REVERSAL......Page 257 PROBLEMS......Page 258 REFERENCES......Page 259 14-1. SPIN, A NONCLASSICAL DEGREE OF FREEDOM......Page 260 14-2. THE DESCRIPTION OF A PARTICLE WITH SPIN......Page 262 14-3. SPIN AND ROTATIONS......Page 266 PROBLEMS......Page 270 14-4. SPIN AND ORBITAL ANGULAR MOMENTUM......Page 271 PROBLEMS......Page 272 14-5. SPIN UNDER SPACE REFLECTION AND TIME INVERSION......Page 273 14-6. SPIN IN AN EXTERNAL FORCE FIELD......Page 274 PROBLEMS......Page 276 14-7. ELEMENTARY PARTICLE WITH ARBITRARY SPIN......Page 277 REFERENCES......Page 281 15-1. ASSEMBLY OF SEVERAL PARTICLES......Page 282 PROBLEMS......Page 285 15-2. MATHEMATICAL DIGRESSION:THE MULTIPLE TENSOR PRODUCT......Page 286 15-3. THE NOTION OF IDENTITY IN QUANTUM MECHAICS......Page 288 15-4. SYSTEMS OF SEVERAL IDENTICAL PARTICLES......Page 291 15-5. THE BOSE GAS......Page 293 PROBLEMS......Page 297 15-6. THE FERMI GAS......Page 298 REFERENCES......Page 300 INDEXES......Page 302 AUTHOR INDEX......Page 304 SUBJECT INDEX......Page 306 Contents: Chapter 1: Measure and Integral Chapter 2: The Axioms of Hilbert Space Chapter 3: Linear Functionals and Operators Chapter 4: Spectral Theorem and Spectral Representation Chapter 5: The Propositional Calculus Chapter 6: States and Observables Chapter 7: Hidden Variables Chapter 8: Proposition Systems and Projective Geometries Chapter 9: Symmetries and Groups Chapter 10: The Dynamical Structure Chapter 11: The Measuring Process Chapter 12: The Elementary Particle in One Dimension Chapter 13: The Elementary Particle Without Spin Chapter 14: Particles with Spin Chapter 15: Identical Particles