商品簡介
Aimed at graduate physics and chemistry students, this is the first comprechensive 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 Iinear 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 methanics and how to measure them.
目次
1. Introduction
2. Quantal Phase Factors for Adiabatic Changes
2.1 Introduction
2.2 Adiabatic Approximation
2.3 Berrys Adiabatic Phase
2.4 Topological Phases and the Aharonov-Bohm Effect
Problems
3. Spinning Quantum System in an External Magnetic Field
3.1 Introduction
3.2 The Parameterization of the Basis Vectors
3.3 Mead-Berry Connection and Berry Phase for Adiabatic Evolutions - Magnetic Monopole Potentials
3.4 The Exact Solution of the SchrSdinger Equation
3.5 Dynamical and Geometrical Phase Factors for Non-Adiabatic Evolution
Problems
4. Quantal Phases for General Cyclic Evolution
4.1 Introduction
4.2 Aharonov-Anandan Phase
4.3 Exact Cyclic Evolution for Periodic Hamiltonians
Problems
5. Fiber Bundles and Gauge Theories
5.1 Introduction
5.2 From Quantal Phases to Fiber Bundles
5.3 An Elementary Introduction to Fiber Bundles
5.4 Geometry of Principal Bundles and the Concept of Holonomy
5.5 Gauge Theories
5.6 Mathematical Foundations of Gauge Theories and Geometry of Vector Bundles
Problems
6. Mathematical Structure of the Geometric Phase I:The Abelian Phase
6.1 Introduction
6.2 Holonomy Interpretations of the Geometric Phase
6.3 Classification of U(1) Principal Bundles and the Relation
Between the Berry-Simon and Aharonov-Anandan
Interpretations of the Adiabatic Phase
6.4 Holonomy Interpretation of the Non-Adiabatic Phase
Using a Bundle over the Parameter Space
6.5 Spinning Quantum System and Topological Aspects of the Geometric Phase
Problems
7. Mathematical Structure of the Geometric Phase II:The Non-Abelian Phase
7.1 Introduction
7.2 The Non-Abelian Adiabatic Phase
7.3 The Non-Abelian Geometric Phase
7.4 Holonomy Interpretations of the Non-Abelian Phase
7.5 Classification of U(N) Principal Bundles and the Relation
Between the Berry-Simon and Aharonov-Anandan
Interpretations of Non-Abelian Phase
Problems
8. A Quantum Physical System in a Quantum Environment - The Gauge Theory of Molecular Physics
8.1 Introduction
8.2 The Hamiltonian of Molecular Systems
8.3 The Born-Oppenheimer Method
8.4 The Gauge Theory of Molecular Physics
8.5 The Electronic States of Diatomic Molecule
8.6 The Monopole of the Diatomic Molecule
Problems
9. Crossing of Potential Energy Surfaces
and the Molecular Aharonov-Bohm Effect
9.1 Introduction
9.2 Crossing of Potential Energy Surfaces
9.3 Conical Intersections and Sign-Change of Wave Functions
9.4 Conical Intersections in Jahn-Teller Systems
9.5 Symmetry of the Ground State in Jahn-Teller Systems
9.6 Geometric Phase in Two Kramers Doublet Systems
9.7 Adiabatic-Diabatic Transformation
10. Experimental Detection of Geometric Phases I
Quantum Systems in Classical Environments
10.1 Introduction
10.2 The Spin Berry Phase Controlled by Magnetic Fields .
10.2.1 Spins in Magnetic Fields: The Laboratory Frame
10.2.2 Spins in Magnetic Fields: The Rotating Frame
10.2.3 Adiabatic Reorientation in Zero Field
10.3 Observation of the Aharonov-Anandan Phase
Through the Cyclic Evolution of Quantum States
Problems
11. Experimental Detection of Geometric Phases II:
Quantum Systems in Quantum Environments
11.1 Introduction
11.2 Internal Rotors Coupled to External Rotors
11,3 Electronic-Rotational Coupling
11.4 Vibronic Problems in Jahn-Teller Systems
11.4.1 Transition Metal Ions in Crystals
11.4.2 Hydrocarbon Radicals
11.4.3 Alkali Metal Trimers
11,5 The Geometric Phase in Chemical Reactions
12. Geometric Phase in Condensed Matter I: Bloch Bands
12.1 Introduction
12.2 Bloch Theory
12.2.1 One-Dimensional Case
12.2.2 Three-Dimensional Case
12.2.3 Band Structure Calculation
12.3 Semiclassical Dynamics
12.3.1 Equations of Motion
12.3.2 Symmetry Analysis
12.3.3 Derivation of the Semiclassical Formulas
12.3.4 Time-Dependent Bands
12.4 Applications of Semiclassical Dynamics
12.4.1 Uniform DC Electric Field
12,4.2 Uniform and Constant Magnetic Field
12.4.3 Per
12.5.1 General Properties
12.5.2 Localization Properties
12.6 Some Issues on Band Insulators
12.6.1 Quantized Adiabatic Particle Transport
12.6.2 Polarization
Problems
13. Geometric Phase in Condensed Matter II:The Quantum Hall Effect
13.1 Introduction
13.2 Basics of the Quantum Hall Effect
13.2.1 The Halt Effect
13.2.2 The Quantum Hall Effect
13.2.3 The Ideal Model
13.2.4 Corrections to Quantization
13.3 Magnetic Bands in Periodic Potentials
13.3.1 Single-Band Approximation in a Weak Magnetic Field
13.3.2 Harpers Equation and Hofstadters Butterfly
13.3.3 Magnetic Translations
13.3.4 Quantized Hall Conductivity
13.3.5 Evaluation of the Chern Number
13.3.6 Semiclassical Dynamics and Quantization
13.3.7 Structure of Magnetic Bands and Hyperorbit Levels
13.3.8 Hierarchical Structure of the Butterfly
13.3.9 Quantization of Hyperorbits and Rule of Band Splitting
13.4 Quantization of Hall Conductance in Disordered Systems
13.4.1 Spectrum and Wave Functions
13.4.2 Perturbation and Scattering Theory
13.4.3 Laughlins Gauge Argument
13.4.4 Hall Conductance as a Topological Invariant
14. Geometric Phase in Condensed Matter III:Many-Body Systems
14.1 Introduction
14.2 Fractional Quantum Hall Systems
14.2.1 Laughlin Wave Function
14.2.2 Fractional Charged Excitations
14.2.3 Fractional Statistics
14.2.4 Degeneracy and Fractional Quantization
14.3 Spin-Wave Dynamics in Itinerant Magnets
14.3.1 General Formulation
14.3.2 Tight-Binding Limit and Beyond
14.3.3 Spin Wave Spectrum
14.4 Geometric Phase in Doubly-Degenerate Electronic Bands
Problem
A. An Elementary Introduction to Manifolds and Lie Groups
A.1 Introduction
A.2 Differentiable Manifolds
A.3 Lie Groups
B. A Brief Review of Point Groups of Molecules with A
