
The Geometric Phase in Quantum Systems
Foundations, Mathematical Concepts, and Applications in Molecular and Condensed Matter Physics
By: Arno Bohm, Ali Mostafazadeh, Hiroyasu Koizumi
Hardcover | 12 June 2003
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460 Pages
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Intended for graduate physics and chemistry students, this is the first comprehensive monograph covering the concept of the geometric phase in quantum physics, from its mathematical foundations to its physical applications and experimental manifestations. It contains all the premises of the adiabatic Berry phase as well as the exact Anandan-Aharonov phase. It discusses quantum systems in a classical time-independent environment (time dependent Hamiltonians) and quantum systems in a changing environment (gauge theory of molecular physics). The mathematical methods used are a combination of differential geometry and the theory of linear operators in Hilbert Space. As a result, the monograph demonstrates how non-trivial gauge theories naturally arise and how the consequences can be experimentally observed. Readers benefit by gaining a deep understanding of the long-ignored gauge theoretic effects of quantum mechanics and how to measure them.
Industry Reviews
From the reviews:
"[...] the present book is a modern presentation of the subject that takes the rich developments over recent decades, to which the authors contributed significantly, into account. [...] Overall it is a well written book that enjoys its uniqueness as a textbook in its field." (J.Pachos, Contemporary Physics 2004, vol. 45, page 273)
"The book is oriented to advanced undergraduate and graduate students in physics and chemistry but the clarity of exposition makes it accessible to much larger audience." (Zentralblatt MATH 2004, vol. 1039, page 54)
"The aim of this book is to systematically present the geometric phase, which can be regarded as one of the major discoveries of 20th century physics, in order to help readers to better understand the richness of the quantum world. ... this is a very interesting book, which deserves to be found in any physical library. This is the first book to be dedicated to the geometric phase ... ." (Daniela Dragoman, Optics & Photonics News, Vol. 16 (7-8), July/August, 2005)
"The book covers a very wide territory, showing in a convincing way the huge range of applicability of geometric phases. The mathematical treatment is clear and essentially correct ... and thus highlights the effectiveness of geometrical methods in quantum physics. Altogether, we have here a book that fills a conspicuous gap in the literature, and it does it rather well. Thus it can be recommended to physicists and chemists with a taste for new tools, and, of course, for physics departments libraries." (Jean-Pierre Antoine, Physicalia, Vol. 57 (3), 2005)
"I am pleased to introduce The Geometrical Phase in Quantum Systems ... which is the first book on this popular research subject in theoretical and experimental physics. ... Undoubtedly, the present book is a modern presentation of the subject that takes the rich developments over recent decades, to which the authors contributed significantly, intoaccount. ... Overall it is a well written book that enjoys its uniqueness as a textbook in its field." (J. Pachos, Contemporary Physics, Vol. 45 (3), 2004)
"The exposition is uniform in depth and level and well balanced regarding exposition of mathematical and physical ideas and concepts. ... The book is oriented to advanced undergraduate and graduate students in physics and chemistry but the clarity of exposition makes it accessible to much larger audience. ... being the first comprehensive book on the subject it should be recognized as a quite valuable addition to the Springer Text and Monographs in Physics series in particular and mathematical physics literature in general." (Ivailo Mladenov, Zentralblatt MATH, Vol. 1039 (8), 2004)
| 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 |
| Table of Contents provided by Publisher. All Rights Reserved. |
ISBN: 9783540000310
ISBN-10: 3540000313
Series: TEXTS AND MONOGRAPHS IN PHYSICS
Published: 12th June 2003
Format: Hardcover
Language: English
Number of Pages: 460
Audience: Professional and Scholarly
Publisher: Springer Nature B.V.
Country of Publication: DE
Dimensions (cm): 24.77 x 16.51 x 1.91
Weight (kg): 0.79
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