尽管量子物理思想在现代数学的许多领域发挥着重要的作用，但是针对数学家的量子力学书却几乎没有。该书用数学家熟悉的语言介绍了量子力学的主要思想。接触物理少的读者在会比较喜欢该书用会话的语调来讲述诸如用Hibert空间法研究量子理论、一维空间的薛定谔方程、有界无界自伴算子的谱定理、Ston-von Neumann定理、Wentzel-Kramers-Brillouin逼近、李群和李代数量子力学中的作用等

1 The Experimental Origins of Quantum Mechanics
1．1 Is Light a Wave or a Particle?
1．2 Is aElectroa Wave or a Particle?
1．3 SchrSdinger and Heisenberg
1．4 A Matter of Interpretation
1．5 Exercises

2 A First Approach to Classical Mechanics
2．1 MotioiR1
2．2 MotioiRn
2．3 Systems of Particles
2．4 Angular Momentum
2．5 PoissoBrackets and HamiltoniaMechanics
2．6 The Kepler Problem and the Runge-Lenz Vector
2．7 Exercises

3 A First Approach to Quantum Mechanics
3．1 Waves， Particles， and Probabilities
3．2 A Few Words About Operators and Their Adjoints
3．3 Positioand the PositioOperator
3．4 Momentum and the Momentum Operator
3．5 The Positioand Momentum Operators
3．6 Aoms of Quantum Mechanics： Operators and Measurements
3．7 Time-EvolutioiQuantum Theory
3．8 The Heisenberg Picture
3．9 Example： A Particle ia Box
3．10 Quantum Mechanics for a Particle iRn
3．11 Systems of Multiple Particles
3．12 Physics Notation
3．13 Exercises

4 The Free Schrodinger Equation
4．1 Solutioby Means of the Fourier Transform
4．2 Solutioas a Convolution
4．3 Propagatioof the Wave Packet： First Approach
4．4 Propagatioof the Wave Packet： Second Approach
4．5 Spread of the Wave Packet
4．6 Exercises

5 A Particle ia Square Well
5．1 The Time-Independent SchrSdinger Equation
5．2 DomaiQuestions and the Matching Conditions
5．3 Finding Square-Integrable Solutions
5．4 Tunneling and the Classically ForbiddeRegion
5．5 Discrete and Continuous Spectrum
5．6 Exercises

6 Perspectives othe Spectral Theorem
6．1 The Difficulties with the Infinite-Dimensional Case
6．2 The Goals of Spectral Theory
6．3 A Guide to Reading
6．4 The PositioOperator
6．5 MultiplicatioOperators
6．6 The Momentum Operator

7 The Spectral Theorem for Bounded Self-Adjoint Operators： Statements
7．1 Elementary Properties of Bounded Operators
7．2 Spectral Theorem for Bounded Self-Adjoint Operators， I
7．3 Spectral Theorem for Bounded Self-Adjoint Operators， II
7．4 Exercises

8 The Spectral Theorem for Bounded Self-Adjoint Operators： Proofs
8．1 Proof of the Spectral Theorem， First Version
8．2 Proof of the Spectral Theorem， Second Version
8．3 Exercises

9 Unbounded Self-Adjoint Operators
9．1 Introduction
9．2． Adjoint and Closure of aUnbounded Operator
9．3 Elementary Properties of Adjoints and Closed Operators
9．4 The Spectrum of aUnbounded Operator
9．5 Conditions for Self-Adjointness and Essential Self-Adjointness
9．6 A Counterexample
9．7 AExample
9．8 The Basic Operators of Quantum Mechanics
9．9 Sums of Self-Adjoint Operators
9．10 Another Counterexample
9．11 Exercises

10 The Spectral Theorem for Unbounded Self-Adjoint Operators
10．1 Statements of the Spectral Theorem
10．2 Stone's Theorem and One-Parameter Unitary Groups
10．3 The Spectral Theorem for Bounded Normal Operators
10．4 Proof of the Spectral Theorem for Unbounded Self-Adjoint Operators
10．5 Exercises

11 The Harmonic Oscillator
11．1 The Role of the Harmonic Oscillator
11．2 The Algebraic Approach
11．3 The Analytic Approach
11．4 DomaiConditions and Completeness
11．5 Exercises

12 The Uncertainty Principle
12．1 Uncertainty Principle， First Version
12．2 A Counterexample
12．3 Uncertainty Principle， Second Version
12．4 Minimum Uncertainty States
12．5 Exercises

13 QuantizatioSchemes for EuclideaSpace
13．1 Ordering Ambiguities
13．2 Some CommoQuantizatioSchemes
13．3 The Weyl Quantizatiofor R2n
13．4 The "No Go" Theorem of Groenewold
13．5 Exercises

14 The Stone-yoNeumanTheorem
14．1 A Heuristic Argument
14．2 The Exponentiated CommutatioRelations
14．3 The Theorem
14．4 The Segal-BargmanSpace
14．5 Exercises

15 The WKB Appromation
15．1 Introduction
15．2 The Old Quantum Theory and the Bohr-Sommerfeld Condition
15．3 Classical and Semiclassical Appromations
15．4 The WKB AppromatioAway from the Turning Points
15．5 The Airy Functioand the ConnectioFormulas
15．6 A Rigorous Error Estimate
15．7 Other Approaches
15．8 Exercises

16 Lie Groups， Lie Algebras， and Representations
16．1 Summary
16．2 Matrix Lie Groups
16．3 Lie Algebras
16．4 The Matrix Exponential
16．5 The Lie Algebra of a Matrix Lie Group
16．6 Relationships BetweeLie Groups and Lie Algebras
16．7 Finite-Dimensional Representations of Lie Groups and Lie Algebras
16．8 New Representations from Old
16．9 Infinite-Dimensional Unitary Representations
16．10 Exercises

17 Angular Momentum and Spin
17．1 The Role of Angular Momentum iQuantum Mechanics
17．2 TheAngular Momentum Operators iR3
17．3 Angular Momentum from the Lie Algebra Point of View
17．4 The Irreducible Representations of so(3)
17．5 The Irreducible Representations of S0(3)
17．6 Realizing the Representations Inside L2(S2)
17．7 Realizing the Representations Inside L2(~3)
17．8 Spin
17．9 Tensor Products of Representations： "Additioof Angular Momentum"
17．10 Vectors and Vector Operators
17．11 Exercises

18 Radial Potentials and the HydrogeAtom
18．1 Radial Potentials
18．2 The HydrogeAtom： Preliminaries
18．3 The Bound States of the HydrogeAtom
18．4 The Runge-Lenz Vector ithe Quantum Kepler Problem
18．5 The Role of Spin
18．6 Runge-Lenz Calculations
18．7 Exercises

19 Systems and Subsystems， Multiple Particles
19．1 Introduction
19．2 Trace-Class and Hilbert Schmidt Operators
19．3 Density Matrices： The General Notioof the State of a Quantum System
19．4 Modified Aoms for Quantum Mechanics
19．5 Composite Systems and the Tensor Product
19．6 Multiple Particles： Bosons and Fermions
19．7 "Statistics" and the Pauli ExclusioPrinciple
19．8 Exercises

20 The Path Integral Formulatioof Quantum Mechanics
20．1 Trotter Product Formula
20．2 Formal Derivatioof the FeynmaPath Integral
20．3 The Imaginary-Time Calculation
20．4 The Wiener Measure
20．5 The Feynman-Kac Formula
20．6 Path Integrals iQuantum Field Theory
20．7 Exercises

21 HamiltoniaMechanics oManifolds
21．1 Calculus oManifolds
21．2 Mechanics oSymplectic Manifolds
21．3 Exercises

22 Geometric QuantizatiooEuclideaSpace
22．1 Introduction
22．2 Prequantization
22．3 Problems with Prequantization
22．4 Quantization
22．5 Quantizatioof Observables
22．6 Exercises

23 Geometric QuantizatiooManifolds
23．1 Introduction
23．2 Line Bundles and Connections
23．3 Prequantization
23．4 Polarizations
23．5 QuantizatioWithout Half-Forms
23．6 Quantizatiowith Half-Forms： The Real Case
23．7 Quantizatiowith Half-Forms： The Complex Case
23．8 Pairing Maps
23．9 Exercises

A Review of Basic Material
A．1 Tensor Products of Vector Spaces
A．2 Measure Theory
A．3 Elementary Fumctional Analysis
A．4 Hilbert Spaces and Operators oThem
References
Index

Brian C. Hall(B.C. 霍尔，美国）是国际知名学者，在数学界享有盛誉。本书凝聚了作者多年科研和教学成果，适用于科研工作者、高校教师和研究生。