The Geometric Phase in Quantum Systems: Foundations, Mathematical Concepts, and Applications in Molecular and Condensed Matter Physics

1. Introduction.- 2. Quantal Phase Factors for Adiabatic Changes.- 3. Spinning Quantum System in an External Magnetic Field.- 4. Quantal Phases for General Cyclic Evolution.- 5. Fiber Bundles and Gauge Theories.- 6. Mathematical Structure of the Geometric Phase I: The Abelian Phase.- 7. Mathematical Structure of the Geometric Phase II: The Non-Abelian Phase.- 8. A Quantum Physical System in a Quantum Environment - The Gauge Theory of Molecular Physics.- 9. Crossing of Potential Energy Surfaces and the Molecular Aharonov-Bohm Effect.- 10. Experimental Detection of Geometric Phases I: Quantum Systems in Classical Environments.- 11. Experimental Detection of Geometric Phases II: 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.- B. A Brief Review of Point Groups of Molecules with Application to Jahn-Teller Systems.- References.