量子力学数学概念

1 Physical Background

1.1 The Double-Slit Experiment

1.2 Wave Functions

1.3 State Space

1.4 The Schr6dinger Equation

1.5 Mathematical Supplement： Operators on Hilbert Spaces，

2 Dynamics

2.1 Conservation of Probability

2.2 Existence of Dynamics

2.3 The Free Propagator

2.4 Mathematical Supplement： Operator Adjoints

2.5 Mathematical Supplement： the Fourier Transform

2.5.1 Definition of the Fourier Transform

2.5.2 Properties of the Fourier Transform

2.5.3 Functions of the Derivative

3 Observables

3，1 Mean Values and the Momentum Operator

3.2 Observables

3.3 The Heisenberg Representation

3.4 Quantization

3.5 Pseudodifferential Operators

4 The Uncertainty Principle

4.1 The Heisenberg Uncertainty Principle

4.2 A Refined Uncertainty Principle

4.3 Application： Stability of Hydrogen

5 Spectral Theory

5.1 The Spectrum of an Operator

5.2 Functions of Operators and the Spectral Mapping Theorem..

5.3 Applications to Schrodinger Operators

5.4 Spectrum and Evolution

5.5 Variational Characterization of Eigenvalues

5.6 Number of Bound States

5.7 Mathematical Supplement： Integral Operators

6 Scattering States

6.1 Short-raage Interactions：μ> 1

6.2 Long-range Interactions： μ< 1

6.3 Wave Operators

7 Special Cases

7.1 The Infinite Well

7.2 The Torns

7.3 A Potential Step

7.4 The Square Well

7.5 The Harmonic Oscillator

7.6 A Particle on a Sphere

7.7 The Hydrogen Atom

7.8 A Particle in an External EM Field

8 Many-particle Systems

8.1 Quantization of a Many-particle System

8.2 Separation of the Centre-of-mass Motion

8.3 Break-ups

8.4 The HVZ Theorem

8.5 Intra- vs. Inter-cluster Motion

8.6 Existence of Bound States for Atoms and Molecules

8.7 Scattering States

8.8 Mathematical Supplement： Tensor Products

8.9 Appendix： Hartree and Gross-Pitaevski Equations

9 Density Matrices

9.1 Introduction

9.2 States and Dynamics

9.3 Open Systems

9.4 The Thermodynamic Limit

9.5 Equilibrium States

9.6 The T →O Limit

9.7 Example： a System of Harmonic Oscillators

9.8 A Particle Coupled to a Reservoir

9.9 Quantum Systems

9.10 Problems

9.11 Hilbert Space Approach

9.12 BEC at T=O

9.13 Appendix： the Ideal Bose Gas

9.14 Appendix： Beee-Einstein Condensation

9.15 Mathematical Supplement： the Trace， and Trace Class Operators

9.16 Mathematical Supplement： Projections

10 Perturbation Theory： Feshbach Method

10.1 The Feshbach Method

10.2 Example： The Zeeman Effect

10.3 Example： Time-dependent Perturbations

10.4 Appendix： Proof of Theorem 10.1

11 The Eeynman Path Integral ....

11.1 The Feynman Path Integral

11.2 Generalizations of the Path Integral

11.3 Mathematical Supplement： The Trotter Product Formula

12 Quasi-classical Analysis

12.1 Quasi-classical Asymptoties of the Propagator

12.2 Quasi-classical Asymptotics of Greens Function

12.2.1 Appendix

12.3 Bohr-Sommerfeld Semi-classical Quantization

12.4 Quasi-classical Asymptotics for the Ground State Energy

12.5 Mathematical Supplement： Operator Determinants

13 Mathematical Supplement： The Calculus of Variations.

13.1 Functionals

13.2 The First Variation and Critical Points

13.3 Constrained Variational Problems

13.4 The Second Variation

13.5 Conjugate Points and Jacobi Fields

13.6 The Action of the Critical Path

13.7 Appendix： Connection to Geodesics

14 Resonances

14.1 Tunneling and Resonances

14.2 The Free Resonance Energy

14.3 Instantons

14.4 Positive Temperatures

14.5 Pre-exponential Factor for the Bounce

14.6 Contribution of the Zero-mode

14.7 Bohr-Sommerfeld Quantization for Resonances

15 Introduction to Quantum Field Theory

15.1 The Place of QFT

15.1.1 Physical Theories

15.1.2 The Principle of Minimal Action

15.2 Klein-Gordon Theory as a Hamiltoulan System

15.2.1 The Legendre Transform

15.2.2 Hamiltoninns

15.2.3 Poison Brackets

15.2.4 Hamiltons Equations

15.3 Maxwells Equations as a Hamiltonian System

15.4 Quantization of the Klein-Gordon and Maxwell Equations..

15.4.1 The Quantization Procedure

15.4.2 Creation and Annihilation Operators

15.4.3 Wick Ordering

15.4.4 Quantizing Maxwalls Equations

15.5 Fock Space

15.6 Generalized Free Theory

15.7 Interactions

15.8 Quadratic Approximation

15.8.1 Further Discussion

15.8.2 A Brief Remark on Many-body Hamiltonians in Second Quantization and the Hartree Approximation .

16 Quantum Electrodynamics of Non-relatlvistic Particles：The Theory of Radiation

16.1 The Hamiltonian

16.2 Perturbation Set-up

16.3 Results

16.4 Mathematical Supplements

16.4.1 Spectral Projections

16.4.2 Projecting-out Procedure

17 Supplement： Renormalization Group

17.1 The Decimation Map

17.2 Relative Bounds

17.3 Elimination of Particle and High Photon Energy Degrees of Freedom

17.4 Generalized Normal Form of Operators on Fock Space

17.5 The Hamiltonian H0(ε， z)

17.6 A Banach Space of Operators

17.7 Rescaling

17.8 The Renormalization Map

17.9 Linearized Flow

17.10 Central-stable Manifold for RG and Spectra of Hamiltonians

17.11 Appendix

18 Comments on Missing Topics， Literatnre，and Further Reading

References

Index

The first fifteen chapters of these lectures （omitting four to six chapters each year） cover a one term course taken by a mixed group of senior undergraduate and junior graduate students specializing either in mathematics or physics. Typically， the mathematics students have some background in advanced analysis， while the physics students have had introductory quantum mechanics. To satisfy such a disparate audience， we decided to select material which is interesting from the viewpoint of modern theoretical physics， and which illustrates an interplay of ideas from various fields of mathematics such as operator theory， probability， differential equations， and differential geometry. Given our time constraint， we have often pursued mathematical content at the expense of rigor. However， wherever we have sacrificed the latter， we have tried to explain whether the result is an established fact， or， mathematically speaking， a conjecture， and in the former case， how a given argument can be made rigorous. The present book retains these features.