量子系統(tǒng)中的幾何相位

前言

　　對(duì)于國內(nèi)的物理學(xué)工作者和青年學(xué)生來講，研讀國外優(yōu)秀的物理學(xué)著作是系統(tǒng)掌握物理學(xué)知識(shí)的一個(gè)重要手段。但是，在國內(nèi)并不能及時(shí)、方便地買到國外的圖書，且國外圖書不菲的價(jià)格往往令國內(nèi)的讀者卻步，因此，把國外的優(yōu)秀物理原著引進(jìn)到國內(nèi)，讓國內(nèi)的讀者能夠方便地以較低的價(jià)格購買是一項(xiàng)意義深遠(yuǎn)的工作，將有助于國內(nèi)物理學(xué)工作者和青年學(xué)生掌握國際物理學(xué)的前沿知識(shí)，進(jìn)而推動(dòng)我國物理學(xué)科研和教學(xué)的發(fā)展?！　榱藵M足國內(nèi)讀者對(duì)國外優(yōu)秀物理學(xué)著作的需求，科學(xué)出版社啟動(dòng)了引進(jìn)國外優(yōu)秀著作的工作，出版社的這一舉措得到了國內(nèi)物理學(xué)界的積極響應(yīng)和支持，很快成立了專家委員會(huì)，開展了選題的推薦和篩選工作，在出版社初選的書單基礎(chǔ)上確定了第一批引進(jìn)的項(xiàng)目，這些圖書幾乎涉及了近代物理學(xué)的所有領(lǐng)域，既有闡述學(xué)科基本理論的經(jīng)典名著，也有反映某一學(xué)科專題前沿的專著。在選擇圖書時(shí)，專家委員會(huì)遵循了以下原則：基礎(chǔ)理論方面的圖書強(qiáng)調(diào)“經(jīng)典”，選擇了那些經(jīng)得起時(shí)間檢驗(yàn)、對(duì)物理學(xué)的發(fā)展產(chǎn)生重要影響、現(xiàn)在還不“過時(shí)”的著作（如狄拉克的《量子力學(xué)原理》）。反映物理學(xué)某一領(lǐng)域進(jìn)展的著作強(qiáng)調(diào)“前沿”和“熱點(diǎn)”，根據(jù)國內(nèi)物理學(xué)研究發(fā)展的實(shí)際情況，選擇了能夠體現(xiàn)相關(guān)學(xué)科最新進(jìn)展，對(duì)有關(guān)方向的科研人員和研究生有重要參考價(jià)值的圖書。這些圖書都是最新版的，多數(shù)圖書都是2000年以后出版的，還有相當(dāng)一部分是當(dāng)年出版的新書。因此，這套叢書具有權(quán)威性、前瞻性和應(yīng)用性強(qiáng)的特點(diǎn)。由于國外出版社的要求，科學(xué)出版社對(duì)部分圖書進(jìn)行了少量的翻譯和注釋（主要是目錄標(biāo)題和練習(xí)題），但這并不會(huì)影響圖書“原汁原味”的感覺，可能還會(huì)方便國內(nèi)讀者的閱讀和理解?！　　八街梢怨ビ瘛?，希望這套叢書的出版能夠?yàn)閲鴥?nèi)物理學(xué)工作者和青年學(xué)生的工作和學(xué)習(xí)提供參考，也希望國內(nèi)更多專家參與到這一工作中來，推薦更多的好書。

內(nèi)容概要

　　Aimed at graduate physics and chemistry students, this is the first comprechenslve monograph covering the concept of thegeometric 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 theoryofmolecular 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.

書籍目錄

1 Introduction2 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　Problems3 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　Problems4 Quantal Phases for General Cyclic Evolution　4.1  Introduction　4.2  Aharonov-Anandan Phase　4.3  Exact Cyclic Evolution for Periodic Hamiltonians　Problems5 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　Problems6 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　Problems7 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　Problems8 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　Problems9 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 Transformation10 Experimental Detection of Geometric Phases I：Quantum Systems in Classical Environments11 Experimental Detection of Geomentric PhasesII: Quantum Systems in Quantum Environments12 Geometric Phase in Condensed Matter I: Bloch Bands 13 Geometric Phase in Condensed Matter II: The Quantum Hall Effect14 Geometric Phase in Condensed Matter III: many-Body SystemsA. An Elementary Introduction to Manifolds and Lie GroupsReferencesIndex

章節(jié)摘錄

　　6.1 Introduction　　In the preceding chapter, we have developed the parts of the theory of fiber bundles which are relevant to our study of geometric phases and briefly described gauge theories. We introduced abstract gauge theories as generalizations of the Abelian gauge theory of electromagnetism. There is also another Abelian gauge theory which we encountered in Chap. 4. We call the latter the Abelian gauge theory of quantum mechanics. The parameter space of this gauge theory is the projective Hilbert space P（H） associated with a Hilbert space H, the matter fields are the pure state vectors which belong to H, the gauge or symmetry group is the group U（1） of the phases of the state vectors,and the gauge potential is the Aharonov-Anandan （A-A） connection.　　The defining PFB associated with this gauge theory is the A-A bundle n whose structure is determined by the Hilbert space H. The A-A connection defines a natural geometric structure on n. The associated vector bundle to　　that yields the state vectors as its global sections is the one defined by the standard representation of U（1）. Thus it is a complex line bundle over 9（H）.　　In this chapter, we shall present a detailed description of the mathematical structure of the Abelian gauge theory of quantum mechanics. In particular we offer different holonomy interpretations of the Abelian geometric phase and reveal their relationship.　　6.2 Holonomy Interpretations of the Geometric Phase　　In Chap. 4, we outlined a holonomy interpretation of the geometric phase."This interpretation used the U（1） PFB n （4.33） and identified the phase with the holonomy of a particular connection which we called the AharonovAnandan （A-A） connection. We shall devote this section to a more systematic discussion of the holonomy interpretations of the geometric phase. We shall start our analysis by first describing an alternative interpretation of the adiabatic phase.

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