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.

1 Physical Background

 1.1 The Double-Slit Experiment

 1.2 Wave Functions

 1.3 State Space

 1.4 The SchrSdinger 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 Schr6dinger 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-range Interactions: ■

 6.2 Long-range Interactions: ■

 6.3 Wave Operators

7 Special Cases

 7.1 The Infinite Well

 7.2 The Torus

 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 ~ 0 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=0

 9.13 Appendix: the Ideal Bose Gas

 9.14 Appendix: Bose-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 Feynman 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 Asymptotics of the Propagator

 12.2 Quasi-classical Asymptotics of Green's 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 Hamiltonian System

 15.2.1 The Legendre Transform

 15.2.2 Hamiltonians

 15.2.3 Poisson Brackets

 15.2.4 Hamilton's Equations

 15.3 Maxwelrs 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 Maxwelrs 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-relativistic 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(E, 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, Literature,and Further Reading

References

Index