## The Geometric Phase in Quantum Systems: Foundations, Mathematical Concepts, and Applications in Molecular and Condensed Matter PhysicsAimed at 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. |

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### Contents

1 | |

3 | |

Spinning Quantum System in an External Magnetic Field | 31 |

Quantal Phases for General Cyclic Evolution | 53 |

Fiber Bundles and Gauge Theories | 65 |

Mathematical Structure of the Geometric Phase | 107 |

Problems | 126 |

5 | 141 |

Experimental Detection of Geometric Phases | 225 |

Quantum Systems in Quantum Environments | 255 |

Bloch Bands 277 | 276 |

Problems | 299 |

A An Elementary Introduction to Manifolds and Lie Groups 361 | 360 |

B A Brief Review of Point Groups of Molecules | 407 |

428 | |

437 | |

### Other editions - View all

The Geometric Phase in Quantum Systems Arno Bohm,Ali Mostafazadeh,Hiroyasu Koizumi No preview available - 2014 |

### Common terms and phrases

Abelian adiabatic adiabatic approximation angular momentum axis basis vectors Berry curvature Berry phase Born–Oppenheimer calculated called Chap Chern number closed curve commutation components conical intersection connection one-form consider coordinate corresponding covariant curvature two-form cyclic evolution defined definition degeneracy degenerate denote density operator depends derivative differential discussion dynamical eigenstates eigenvalue eigenvectors electronic en(R equivalence example expression fiber bundles function f gauge potential gauge theory gauge transformation geometric phase given global Hamiltonian Hilbert space holonomy integral Lie algebra Lie group lift magnetic field matrix elements molecular molecule motion non-Abelian non-adiabatic nuclear obtained parameter space particle phase angle phase factor physical potential energy surfaces problem quantization quantum number quantum system relation representation result rotation Schrödinger equation semiclassical single-valued smooth manifold spin structure subspace symmetry tangent tensor tion transition functions unitary values vector bundle vector potential vibronic Wannier functions wave function