Aimed at graduate physics and chemistry students, this is the first comprechenslve 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 environm

1 Introduction
2 Quantal Phase Factors for Adiabatic Changes
2.1 Introduction
2.2 Adiabatic Approximation
2.3 Berry's 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
11 Experimental Detection of Geomentric PhasesII: Quantum Systems in Quantum Environments
12 Geometric Phase in Condensed Matter I: Bloch Bands
13 Geometric Phase in Condensed Matter II: The Quantum Hall Effect
14 Geometric Phase in Condensed Matter III: many-Body Systems
A. An Elementary Introduction to Manifolds and Lie Groups
References
Index