Aimedatgraduatephysicsandchemistry,students,thisisthefirstComprechensivemonographcoveringtheconceptofthegeometricphaseinquantumphysicsfromitsmathematicalfoundationstoitsphysicalapplicationsandexperimentalmanifestations,ItcontainsallthepremisesoftheadiabaticBerryphaseaswellastheexactAnandan-Aharonovphase.Itdiscussesquantumsystemsinaclassicaltime-independentenvironment(timedependentHamiltonians)andquantumsystemsinachangingenvironment(gaugetheoryofmolecularphysics).ThemathematicalmethodsusedareacombinationofdifferentialgeometryandthetheoryofIinearoperatorsinHilbertSpace,Asaresult,themonographdemonstrateshownon-trivialgaugetheoriesnaturallyariseandhowtheconsequencescanbeexperimentallyobserved.Readersbenefitbygainingadeepunderstandingofthelong-ignoredgaugetheoreticeffectsofquantummethanicsandhowtomeasurethem.

The Geometric Phase in Quantum Systems : Foundations, mathematical concepts, and applications in molecular and condensed matter physics

量子系统中的几何相位 : 基本原理、数学概念及其在分子物理和凝聚态物理中的应用(影印版)

XBWL.s10

Edited by A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, J. Zwanziger

A. Bohm. [et al]著; Hm Bo

A.Bohm[等著]

lyl

Science Press

国外物理名著系列, Reprinted ed, Beijing, 2009

China, People's Republic, China

Reprinted ed., China, 2009

1, 国外物理名著系列, 2009-03

producers:
AFPL Ghostscript 8.50

Includes bibliographical references and index.
Reprint. Originally published : Springer-Verlag, c2003.

目录 10
1. Introduction 15
2. Quantal Phase Factors for Adiabatic Changes 19
2.1 Introduction 19
2.2 Adiabatic Approximation 24
2.3 Berry's Adiabatic Phase 28
2.4 Topological Phases and the Aharonov-Bohm Effect 36
Problems 43
3. Spinning Quantum System in an External Magnetic Field 45
3.1 Introduction 45
3.2 The Parameterization of the Basis Vectors 45
3.3 Mead-Berry Connection and Berry Phase for Adiabatic Evolutions-Magnetic Monopole Potentials 50
3.4 The Exact Solution of the SchrSdinger Equation 56
3.5 Dynamical and Geometrical Phase Factors for Non-Adiabatic Evolution 62
Problems 66
4. Quantal Phases for General Cyclic Evolution 67
4.1 Introduction 67
4.2 Aharonov-Anandan Phase 67
4.3 Exact Cyclic Evolution for Periodic Hamiltonians 74
Problems 78
5. Fiber Bundles and Gauge Theories 79
5.1 Introduction 79
5.2 From Quantal Phases to Fiber Bundles 79
5.3 An Elementary Introduction to Fiber Bundles 81
5.4 Geometry of Principal Bundles and the Concept of Holonomy 90
5.5 Gauge Theories 101
5.6 Mathematical Foundations of Gauge Theories and Geometry of Vector Bundles 109
Problems 116
6. Mathematical Structure of the Geometric Phase I: The Abelian Phase 121
6.1 Introduction 121
6.2 Holonomy Interpretations of the Geometric Phase 121
6.3 Classification of U(1) Principal Bundles and the Relation Between the Berry-Simon and Aharonov-Anandan Interpretations of the Adiabatic Phase 127
6.4 Holonomy Interpretation of the Non-Adiabatic Phase Using a Bundle over the Parameter Space 132
6.5 Spinning Quantum System and Topological Aspects of the Geometric Phase 137
Problems 140
7. Mathematical Structure of the Geometric Phase II: The Non-Abelian Phase 143
7.1 Introduction 143
7.2 The Non-Abelian Adiabatic Phase 143
7.3 The Non-Abelian Geometric Phase 150
7.4 Holonomy Interpretations of the Non-Abelian Phase 153
7.5 Classification of U(N) Principal Bundles and the Relation Between the Berry-Simon and Aharonov-Anandan Interpretations of Non-Abelian Phase 155
Problems 159
8. A Quantum Physical System in a Quantum Environment-The Gauge Theory of Molecular Physics 161
8.1 Introduction 161
8.2 The Hamiltonian of Molecular Systems 162
8.3 The Born-Oppenheimer Method 171
8.4 The Gauge Theory of Molecular Physics 180
8.5 The Electronic States of Diatomic Molecule 188
8.6 The Monopole of the Diatomic Molecule 190
Problems 205
9. Crossing of Potential Energy Surfaces and the Molecular Aharonov-Bohm Effect 209
9.1 Introduction 209
9.2 Crossing of Potential Energy Surfaces 210
9.3 Conical Intersections and Sign-Change of Wave Functions 212
9.4 Conical Intersections in Jahn-Teller Systems 223
9.5 Symmetry of the Ground State in Jahn Teller Systems 227
9.6 Geometric Phase in Two Kramers Doublet Systems 233
9.7 Adiabatic Diabatic Transformation 236
10. Experimental Detection of Geometric Phases I: Quantum Systems in Classical Environments 239
10.1 Introduction 239
10.2 The Spin Berry Phase Controlled by Magnetic Fields 239
10.2.1 Spins in Magnetic Fields: The Laboratory Frame 239
10.2.2 Spins in Magnetic Fields: The Rotating Frame 245
10.2.3 Adiabatic Reorientation in Zero Field 251
10.3 Observation of the Aharonov-Anandan Phase Through the Cyclic Evolution of Quantum States 262
Problems 266
11. Experimental Detection of Geometric Phases II: Quantum Systems in Quantum Environments 269
11.1 Introduction 269
11.2 Internal Rotors Coupled to External Rotors 270
11.3 Electronic-Rotational Coupling 273
11.4 Vibronic Problems in Jahn-Teller Systems 274
11.4.1 Transition Metal Ions in Crystals 275
11.4.2 Hydrocarbon Radicals 278
11.4.3 Alkali Metal Trimers 279
11.5 The Geometric Phase in Chemical Reactions 284
12. Geometric Phase in Condensed Matter I: Bloch Bands 291
12.1 Introduction 291
12.2 Bloch Theory 292
12.2.1 One-Dimensional Case 292
12.2.2 Three-Dimensional Case 294
12.2.3 Band Structure Calculation 295
12.3 Semiclassical Dynamics 297
12.3.1 Equations of Motion 297
12.3.2 Symmetry Analysis 299
12.3.3 Derivation of the Semiclassical Formulas 300
12.3.4 Tiine-Dependent Bands 301
12.4 Applications of Semiclassical Dynamics 302
12.4.1 Uniform DC Electric Field 302
12.4.2 Uniform and Constant Magnetic Field 303
12.4.3 Perpendicular Electric and Magnetic Fields 304
12.4.4 Transport 304
12.5 Wannier Functions 306
12.5.1 General Properties 306
12.5.2 Localization Properties 307
12.6 Some Issues on Band Insulators 309
12.6.1 Quantized Adiabatic Particle Transport 309
12.6.2 Polarization 311
Problems 313
13. Geometric Phase in Condensed Matter II: The Quantum Hall Effect 315
13.1 Introduction 315
13.2 Basics of the Quantum Hall Effect 316
13.2.1 The Hall Effect 316
13.2.2 The Quantum Hall Effect 316
13.2.3 The Ideal Model 318
13.2.4 Corrections to Quantization 319
13.3 Magnetic Bands in Periodic Potentials 321
13.3.1 Single-Band Approximation in a Weak Magnetic Field 321
13.3.2 Harper's Equation and Hofstadter's Butterfly 323
13.3.3 Magnetic Translations 325
13.3.4 Quantized Hall Conductivity 328
13.3.5 Evaluation of the Chern Number 330
13.3.6 Semiclassical Dynamics and Quantization 332
13.3.7 Structure of Magnetic Bands and Hyperorbit Levels 335
13.3.8 Hierarchical Structure of the Butterfly 339
13.3.9 Quantization of Hyperorbits and Rule of Band Splitting 341
13.4 Quantization of Hall Conductance in Disordered Systems 343
13.4.1 Spectrum and Wave Functions 343
13.4.2 Perturbation and Scattering Theory 345
13.4.3 Laughlin's Gauge Argument 346
13.4.4 Hall Conductance as a Topological Invariant 347
14. Geometric Phase in Condensed Matter III: Many-Body Systems 351
14.1 Introduction 351
14.2 Fractional Quantum Hall Systems 351
14.2.1 Laughlin Wave Function 351
14.2.2 Fractional Charged Excitations 354
14.2.3 Fractional Statistics 355
14.2.4 Degeneracy and Fractional Quantization 358
14.3 Spin-Wave Dynamics in Itinerant Magnets 360
14.3.1 General Formulation 360
14.3.2 Tight-Binding Limit and Beyond 362
14.3.3 Spin Wave Spectrum 364
14.4 Geometric Phase in Doubly-Degenerate Electronic Bands 367
Problem 373
A. An Elementary Introduction to Manifolds and Lie Groups 375
A.1 Introduction 375
A.2 Differentiable Manifolds 385
A.3 Lie Groups 402
B. A Brief Review of Point Groups of Molecules with Application to Jahn-Teller Systems 421
References 443
Index 451
参考文献 375

2025-10-27