| Introduction | p. 1 |
| Quantal Phase Factors for Adiabatic Changes | p. 5 |
| Introduction | p. 5 |
| Adiabatic Approximation | p. 10 |
| Berry's Adiabatic Phase | p. 14 |
| Topological Phases and the Aharonov-Bohm Effect | p. 22 |
| Problems | p. 29 |
| Spinning Quantum System in an External Magnetic Field | p. 31 |
| Introduction | p. 31 |
| The Parameterization of the Basis Vectors | p. 31 |
| Mead-Berry Connection and Berry Phase for Adiabatic Evolutions - Magnetic Monopole Potentials | p. 36 |
| The Exact Solution of the Schrödinger Equation | p. 42 |
| Dynamical and Geometrical Phase Factors for Non-Adiabatic Evolution | p. 48 |
| Problems | p. 52 |
| Quantal Phases for General Cyclic Evolution | p. 53 |
| Introduction | p. 53 |
| Aharonov-Anandan Phase | p. 53 |
| Exact Cyclic Evolution for Periodic Hamiltonians | p. 60 |
| Problems | p. 64 |
| Fiber Bundles and Gauge Theories | p. 65 |
| Introduction | p. 65 |
| From Quantal Phases to Fiber Bundles | p. 65 |
| An Elementary Introduction to Fiber Bundles | p. 67 |
| Geometry of Principal Bundles and the Concept of Holonomy | p. 76 |
| Gauge Theories | p. 87 |
| Mathematical Foundations of Gauge Theories and Geometry of Vector Bundles | p. 95 |
| Problems | p. 102 |
| Mathematical Structure of the Geometric Phase I: The Abelian Phase | p. 107 |
| Introduction | p. 107 |
| Holonomy Interpretations of the Geometric Phase | p. 107 |
| Classification of U(1) Principal Bundles and the Relation Between the Berry-Simon and Aharonov-Anandan Interpretations of the Adiabatic Phase | p. 113 |
| Holonomy Interpretation of the Non-Adiabatic Phase Using a Bundle over the Parameter Space | p. 118 |
| Spinning Quantum System and Topological Aspects of the Geometric Phase | p. 123 |
| Problems | p. 126 |
| Mathematical Structure of the Geometric Phase II: The Non-Abelian Phase | p. 129 |
| Introduction | p. 129 |
| The Non-Abelian Adiabatic Phase | p. 129 |
| The Non-Abelian Geometric Phase | p. 136 |
| Holonomy Interpretations of the Non-Abelian Phase | p. 139 |
| Classification of <$>U({\cal N})<$> Principal Bundles and the Relation Between the Berry-Simon and Aharonov-Anandan Interpretations of Non-Abelian Phase | p. 141 |
| Problems | p. 145 |
| A Quantum Physical System in a Quantum Environment - The Gauge Theory of Molecular Physics | p. 147 |
| Introduction | p. 147 |
| The Hamiltonian of Molecular Systems | p. 148 |
| The Born-Oppenheimer Method | p. 157 |
| The Gauge Theory of Molecular Physics | p. 166 |
| The Electronic States of Diatomic Molecule | p. 174 |
| The Monopole of the Diatomic Molecule | p. 176 |
| Problems | p. 191 |
| Crossing of Potential Energy Surfaces and the Molecular Aharonov-Bohm Effect | p. 195 |
| Introduction | p. 195 |
| Crossing of Potential Energy Surfaces | p. 196 |
| Conical Intersections and Sign-Change of Wave Functions | p. 198 |
| Conical Intersections in Jahn-Teller Systems | p. 209 |
| Symmetry of the Ground State in Jahn-Teller Systems | p. 213 |
| Geometric Phase in Two Kramers Doublet Systems | p. 219 |
| Adiabatic-Diabatic Transformation | p. 222 |
| Experimental Detection of Geometric Phases I: Quantum Systems in Classical Environments | p. 225 |
| Introduction | p. 225 |
| The Spin Berry Phase Controlled by Magnetic Fields | p. 225 |
| Spins in Magnetic Fields: The Laboratory Frame | p. 225 |
| Spins in Magnetic Fields: The Rotating Frame | p. 231 |
| Adiabatic Reorientation in Zero Field | p. 237 |
| Observation of the Aharonov-Anandan Phase Through the Cyclic Evolution of Quantum States | p. 248 |
| Problems | p. 252 |
| Experimental Detection of Geometric Phases II: Quantum Systems in Quantum Environments | p. 255 |
| Introduction | p. 255 |
| Internal Rotors Coupled to External Rotors | p. 256 |
| Electronic-Rotational Coupling | p. 259 |
| Vibronic Problems in Jahn-Teller Systems | p. 260 |
| Transition Metal Ions in Crystals | p. 261 |
| Hydrocarbon Radicals | p. 264 |
| Alkali Metal Trimers | p. 265 |
| The Geometric Phase in Chemical Reactions | p. 270 |
| Geometric Phase in Condensed Matter I: Bloch Bands | p. 277 |
| Introduction | p. 277 |
| Bloch Theory | p. 278 |
| One-Dimensional Case | p. 278 |
| Three Dimensional Case | p. 280 |
| Band Structure Calculation | p. 281 |
| Semiclassical Dynamics | p. 283 |
| Equations of Motion | p. 283 |
| Symmetry Analysis | p. 285 |
| Derivation of the Semiclassical Formulas | p. 286 |
| Time-Dependent Bands | p. 287 |
| Applications of Semiclassical Dynamics | p. 288 |
| Uniform DC Electric Field | p. 288 |
| Uniform and Constant Magnetic Field | p. 289 |
| Perpendicular Electric and Magnetic Fields | p. 290 |
| Transport | p. 290 |
| Wannier Functions | p. 292 |
| General Properties | p. 292 |
| Localization Properties | p. 293 |
| Some Issues on Band Insulators | p. 295 |
| Quantized Adiabatic Particle Transport | p. 295 |
| Polarization | p. 297 |
| Problems | p. 299 |
| Geometric Phase in Condensed Matter II: The Quantum Hall Effect | p. 301 |
| Introduction | p. 301 |
| Basics of the Quantum Hall Effect | p. 302 |
| The Hall Effect | p. 302 |
| The Quantum Hall Effect | p. 302 |
| The Ideal Model | p. 304 |
| Corrections to Quantization | p. 305 |
| Magnetic Bands in Periodic Potentials | p. 307 |
| Single-Band Approximation in a Weak Magnetic Field | p. 307 |
| Harper's Equation and Hofstadter's Butterfly | p. 309 |
| Magnetic Translations | p. 311 |
| Quantized Hall Conductivity | p. 314 |
| Evaluation of the Chern Number | p. 316 |
| Semiclassical Dynamics and Quantization | p. 318 |
| Structure of Magnetic Bands and Hyperorbit Levels | p. 321 |
| Hierarchical Structure of the Butterfly | p. 325 |
| Quantization of Hyperorbits and Rule of Band Splitting | p. 327 |
| Quantization of Hall Conductance in Disordered Systems | p. 329 |
| Spectrum and Wave Functions | p. 329 |
| Perturbation and Scattering Theory | p. 331 |
| Laughlin's Gauge Argument | p. 332 |
| Hall Conductance as a Topological Invariant | p. 333 |
| Geometric Phase in Condensed Matter III: Many-Body Systems | p. 337 |
| Introduction | p. 337 |
| Fractional Quantum Hall Systems | p. 337 |
| Laughlin Wave Function | p. 337 |
| Fractional Charged Excitations | p. 340 |
| Fractional Statistics | p. 341 |
| Degeneracy and Fractional Quantization | p. 344 |
| Spin-Wave Dynamics in Itinerant Magnets | p. 346 |
| General Formulation | p. 346 |
| Tight-Binding Limit and Beyond | p. 348 |
| Spin Wave Spectrum | p. 350 |
| Geometric Phase in Doubly-Degenerate Electronic Bands | p. 353 |
| Problem | p. 359 |
| An Elementary Introduction to Manifolds and Lie Groups | p. 361 |
| Introduction | p. 361 |
| Differentiable Manifolds | p. 371 |
| Lie Groups | p. 388 |
| A Brief Review of Point Groups of Molecules with Application to Jahn-Teller Systems | p. 407 |
| References | p. 429 |
| Index | p. 437 |
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