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The Geometric Phase in Quantum Systems

Foundations, Mathematical Concepts, and Applications in Molecular and Condensed Matter Physics

  • Book
  • © 2003

Overview

Authors:
  1. Arno Bohm

    1. Physics Department, University of Texas at Austin, Austin, USA

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  2. Ali Mostafazadeh

    1. Department of Mathematics, Koç University, Istanbul, Turkey

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  3. Hiroyasu Koizumi

    1. Faculty of Science, Himeji Institute of Technology, Hyogo, Japan

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  4. Qian Niu

    1. Physics Department, University of Texas at Austin, Austin, USA

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  5. Joseph Zwanziger

    1. Department of Chemistry, Indiana University, Bloomington, USA

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  • This is the first textbook devoted to the phenomenon of Berry phases
  • It also includes experimental findings otherwise not to be found in publications on geometric phases
  • Includes supplementary material: sn.pub/extras

Part of the book series: Theoretical and Mathematical Physics (TMP)

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Table of contents (16 chapters)

  1. Front Matter

    Pages I-XVII
    Download chapter PDF

  2. Introduction

    • Arno Bohm, Ali Mostafazadeh, Hiroyasu Koizumi, Qian Niu, Joseph Zwanziger
    Pages 1-4

  3. Quantal Phase Factors for Adiabatic Changes

    • Arno Bohm, Ali Mostafazadeh, Hiroyasu Koizumi, Qian Niu, Joseph Zwanziger
    Pages 5-30

  4. Spinning Quantum System in an External Magnetic Field

    • Arno Bohm, Ali Mostafazadeh, Hiroyasu Koizumi, Qian Niu, Joseph Zwanziger
    Pages 31-52

  5. Quantal Phases for General Cyclic Evolution

    • Arno Bohm, Ali Mostafazadeh, Hiroyasu Koizumi, Qian Niu, Joseph Zwanziger
    Pages 53-64

  6. Fiber Bundles and Gauge Theories

    • Arno Bohm, Ali Mostafazadeh, Hiroyasu Koizumi, Qian Niu, Joseph Zwanziger
    Pages 65-105

  7. Mathematical Structure of the Geometric Phase I: The Abelian Phase

    • Arno Bohm, Ali Mostafazadeh, Hiroyasu Koizumi, Qian Niu, Joseph Zwanziger
    Pages 107-127

  8. Mathematical Structure of the Geometric Phase II: The Non-Abelian Phase

    • Arno Bohm, Ali Mostafazadeh, Hiroyasu Koizumi, Qian Niu, Joseph Zwanziger
    Pages 129-145

  9. A Quantum Physical System in a Quantum Environment — The Gauge Theory of Molecular Physics

    • Arno Bohm, Ali Mostafazadeh, Hiroyasu Koizumi, Qian Niu, Joseph Zwanziger
    Pages 147-193

  10. Crossing of Potential Energy Surfaces and the Molecular Aharonov-Bohm Effect

    • Arno Bohm, Ali Mostafazadeh, Hiroyasu Koizumi, Qian Niu, Joseph Zwanziger
    Pages 195-224

  11. Experimental Detection of Geometric Phases I: Quantum Systems in Classical Environments

    • Arno Bohm, Ali Mostafazadeh, Hiroyasu Koizumi, Qian Niu, Joseph Zwanziger
    Pages 225-253

  12. Experimental Detection of Geometric Phases II: Quantum Systems in Quantum Environments

    • Arno Bohm, Ali Mostafazadeh, Hiroyasu Koizumi, Qian Niu, Joseph Zwanziger
    Pages 255-275

  13. Geometric Phase in Condensed Matter I: Bloch Bands

    • Arno Bohm, Ali Mostafazadeh, Hiroyasu Koizumi, Qian Niu, Joseph Zwanziger
    Pages 277-299

  14. Geometric Phase in Condensed Matter II: The Quantum Hall Effect

    • Arno Bohm, Ali Mostafazadeh, Hiroyasu Koizumi, Qian Niu, Joseph Zwanziger
    Pages 301-336

  15. Geometric Phase in Condensed Matter III: Many-Body Systems

    • Arno Bohm, Ali Mostafazadeh, Hiroyasu Koizumi, Qian Niu, Joseph Zwanziger
    Pages 337-359

  16. An Elementary Introduction to Manifolds and Lie Groups

    • Arno Bohm, Ali Mostafazadeh, Hiroyasu Koizumi, Qian Niu, Joseph Zwanziger
    Pages 361-405

  17. A Brief Review of Point Groups of Molecules with Application to Jahn—Teller Systems

    • Arno Bohm, Ali Mostafazadeh, Hiroyasu Koizumi, Qian Niu, Joseph Zwanziger
    Pages 407-427

  18. Back Matter

    Pages 429-443
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Keywords

About this book

Aimed 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.

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)

Authors and Affiliations

  • Physics Department, University of Texas at Austin, Austin, USA

    Arno Bohm, Qian Niu

  • Department of Mathematics, Koç University, Istanbul, Turkey

    Ali Mostafazadeh

  • Faculty of Science, Himeji Institute of Technology, Hyogo, Japan

    Hiroyasu Koizumi

  • Department of Chemistry, Indiana University, Bloomington, USA

    Joseph Zwanziger

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