Mathematical Concepts of Quantum Mechanics

Preface
Preface to the second edition
Preface to the enlarged second printing
From the preface to the first edition
Contents
1 Physical Background
1.1 The Double-Slit Experiment
1.2 Wave Functions
1.3 State Space
1.4 The Schrӧdinger Equation
1.5 Classical Limit
2 Dynamics
2.1 Conservation of Probability
2.2 Self-adjointness
2.3 Existence of Dynamics
2.4 The Free Propagator
2.5 Semi-classical Approximation
3 Observables
3.1 The Position and Momentum Operators
3.2 General Observables
3.3 The Heisenberg Representation
3.4 Conservation Laws
3.5 Conserved Currents
4 Quantization
4.1 Quantization
4.2 Quantization and Correspondence Principle
4.3 A Particle in an External Electro-magnetic Field
4.4 Spin
4.5 Many-particle Systems
4.6 Identical Particles
4.7 Supplement: Hamiltonian Formulation of Classical Mechanics
5 Uncertainty Principle and Stability of Atoms
and Molecules
5.1 The Heisenberg Uncertainty Principle
5.2 A Refined Uncertainty Principle
5.3 Application: Stability of Atoms and Molecules
6 Spectrum and Dynamics
6.1 The Spectrum of an Operator
6.2 Spectrum and Measurement Outcomes
6.3 Classification of Spectra
6.4 Bound and Decaying States
6.5 Spectra of Schrӧdinger Operators
6.6 Particle in a Periodic Potential
6.7 Angular Momentum
7
Special Cases and Extensions
7.1 The Square Well and Torus
7.2 Motion in a Spherically Symmetric Potential
7.3 The Hydrogen Atom
7.4 The Harmonic Oscillator
7.5 A Particle in a Constant Magnetic Field
7.6 Aharonov-Bohm Effect
7.7 Linearized Ginzburg-Landau Equations of Superconductivity
7.8 Ideal Quantum Gas and Ground States of Atoms
7.9 Supplement: L−equivariant functions
8 Bound States and Variational Principle
8.1 Variational Characterization of Eigenvalues
8.2 Exponential Decay of Bound States
8.3 Number of Bound States
9 Scattering States
9.1 Short-range Interactions: μ > 1
9.2 Long-range Interactions: μ ≤ 1
9.3 Wave Operators
9.4 Appendix: The Potential Step and Square Well
10 Existence of Atoms and Molecules
10.1 Essential Spectra of Atoms and Molecules
10.2 Bound States of Atoms and BO Molecules
10.3 Open Problems
11 Perturbation Theory: Feshbach-Schur Method
11.1 The Feshbach-Schur Method
11.2 The Zeeman Effect
11.3 Time-Dependent Perturbations
11.4 Appendix: Proof of Theorem 11.1
12 Born-Oppenheimer Approximation and Adiabatic Dynamics
12.1 Problem and Heuristics
12.2 Stationary Born-Oppenheimer Approximation
12.3 Complex ψy and Gauge Fields
12.4 Time-dependent Born-Oppenheimer Approximation
12.5 Adiabatic Motion
12.6 Geometrical Phases
12.7 Appendix: Projecting-out Procedure
12.8 Appendix: Proof of Theorem 12.11
13
General Theory of Many-particle Systems
13.1 Many-particle Schrӧdinger Operators
13.2 Separation of the Centre-of-mass Motion
13.3 Break-ups
13.4 The HVZ Theorem
13.5 Intra- vs. Inter-cluster Motion
13.6 Exponential Decay of Bound States
13.7 Remarks on Discrete Spectrum
13.8 Scattering States
14 Self-consistent Approximations
14.1 Hartree, Hartree-Fock and Gross-Pitaevski equations
14.2 Appendix: BEC at T=0
15 The Feynman Path Integral
15.1 The Feynman Path Integral
15.2 Generalizations of the Path Integral
15.3 Mathematical Supplement: the Trotter Product Formula
16 Semi-classical Analysis
16.1 Semi-classical Asymptotics of the Propagator
16.2 Semi-classical Asymptotics of Green’s Function
16.2.1 Appendix
16.3 Bohr-Sommerfeld Semi-classical Quantization
16.4 Semi-classical Asymptotics for the Ground State Energy
16.5 Mathematical Supplement: The Action of the Critical Path
16.6 Appendix: Connection to Geodesics
17 Resonances
17.1 Complex Deformation and Resonances
17.2 Tunneling and Resonances
17.3 The Free Resonance Energy
17.4 Instantons
17.5 Positive Temperatures
17.6 Pre-exponential Factor for the Bounce
17.7 Contribution of the Zero-mode
17.8 Bohr-Sommerfeld Quantization for Resonances
18 Quantum Statistics
18.1 Density Matrices
18.2 Quantum Statistics: General Framework
18.3 Stationary States
18.4 Hilbert Space Approach
18.5 Semi-classical Limit
18.6 Generalized Hartree-Fock and Kohn-Sham Equations
19 Open Quantum Systems
19.1 Information Reduction
19.2 Reduced dynamics
19.3 Some Proofs
19.4 Communication Channels
19.5 Quantum Dynamical Semigroups
19.6 Irreversibility
19.7 Decoherence and Thermalization
20 The Second Quantization
20.1 Fock Space and Creation and Annihilation Operators
20.2 Many-body Hamiltonian
20.3 Evolution of Quantum Fields
20.4 Relation to Quantum Harmonic Oscillator
20.5 Scalar Fermions
20.6 Mean Field Regime
20.7 Appendix: the Ideal Bose Gas
20.7.1 Bose-Einstein Condensation
21 Quantum Electro-Magnetic Field - Photons
21.1 Klein-Gordon Classical Field Theory
21.1.1 Principle of minimum action
21.1.2 Hamiltonians
21.1.3 Hamiltonian System
21.1.4 Complexification of the Klein-Gordon Equation
21.2 Quantization of the Klein-Gordon Equation
21.3 The Gaussian Spaces
21.4 Wick Quantization
21.5 Fock Space
21.6 Quantization of Maxwell’s Equations
22 Standard Model of Non-relativistic Matter and Radiation
22.1 Classical Particle System Interacting with an Electro-magnetic Field
22.2 Quantum Hamiltonian of Non-relativistic QED
22.2.1 Translation invariance
22.2.2 Fiber decomposition with respect to total momentum
22.3 Rescaling and decoupling scalar and vector
potentials
22.3.1 Self-adjointness of H(ε)
22.4 Mass Renormalization
22.5 Appendix: Relative bound on I(ε) and Pull-through Formulae
23
Theory of Radiation
23.1 Spectrum of the Uncoupled System
23.2 Complex Deformations and Resonances
23.3 Results
23.4 Idea of the proof of Theorem 23.1
23.5 Generalized Pauli-Fierz Transformation
23.6 Elimination of Particle and High Photon Energy Degrees of Freedom
23.7 The Hamiltonian H0(ε, z)
23.8 Estimates on the operator H0(ε, z)
23.9 Ground state of H(ε)
23.10 Appendix: Estimates on Iε and HPρˆ(ε)
23.11 Appendix: Key Bound
24 Renormalization Group
24.1 Main Result
24.2 A Banach Space of Operators
24.3 The Decimation Map
24.4 The Renormalization Map
24.5 Dynamics of RG and Spectra of Hamiltonians
24.6 Supplement: Group Property of Rρ
25
Mathematical Supplement: Elements of Operator Theory
25.1 Spaces
25.2 Operators on Hilbert Spaces
25.3 Integral Operators
25.4 Inverses and their Estimates
25.5 Self-adjointness
25.6 Exponential of an Operator
25.7 Projections
25.8 The Spectrum of an Operator
25.9 Functions of Operators and the Spectral Mapping Theorem
25.10 Weyl Sequences and Weyl Spectrum
25.11 The Trace, and Trace Class Operators
25.12 Operator Determinants
25.13 Tensor Products
25.14 The Fourier Transform
26
Mathematical Supplement: The Calculus of Variations
26.1 Functionals
26.2 The First Variation and Critical Points
26.3 The Second Variation
26.4 Conjugate Points and Jacobi Fields
26.5 Constrained Variational Problems
26.6 Legendre Transform and Poisson Bracket
26.7 Complex Hamiltonian Systems
26.8 Conservation Laws
27
Comments on Literature, and Further Reading
References
Index