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
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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
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Comments on Literature, and Further Reading
References
Index