Lectures on Quantum Mechanics and Attractors

Contents
Preface
Introduction
Notation
I Nonrelativistic Quantum Mechanics
I.1 Photons and Wave-Particle Duality
I.1.1 Planck's law and Einstein's photons
I.1.2 De Broglie's wave-particle duality
I.2 The Schrödinger Equation
I.2.1 Canonical quantization
I.2.2 Quasiclassical asymptotics and geometrical optics
I.3 Quantum Observables
I.3.1 Hamiltonian structure
I.3.2 Charge and current densities
I.3.3 Quantum momentum and angular momentum
I.3.4 Correspondence principle
I.3.5 Conservation laws
I.3.6 Proof of conservation laws
I.3.7 The Heisenberg picture
I.3.8 Plane waves as electron beams
I.3.9 The Heisenberg uncertainty principle
I.4 Bohr's postulates
I.4.1 Schrödinger's identification of stationary orbits
I.4.2 Perturbation theory
I.5 Coupled Maxwell–Schrödinger Equations
I.6 Hydrogen Spectrum
I.6.1 Spherical symmetry and separation of variables
I.6.2 Spherical coordinates
I.6.3 Radial equation
I.6.4 Eigenfunctions
I.7 Spherical Eigenvalue Problem
I.7.1 The Hilbert–Schmidt argument
I.7.2 The Lie algebra of quantum angular momenta
I.7.3 Irreducible representations
I.7.4 Spherical harmonics. Proof of Theorem I.6.6
I.7.5 Angular momentum in spherical coordinates
II Scattering of Light and Particles
II.1 Classical Scattering of Light
II.1.1 Incident wave
II.1.2 The Thomson scattering
II.1.3 Neglecting the selfaction
II.1.4 Dipole approximation
II.2 Quantum Scattering of Light
II.2.1 Scattering problem
II.2.2 Atomic form factor
II.2.3 Energy flux
II.3 Polarization and Dispersion
II.3.1 First-order approximation
II.3.2 Limiting amplitudes
II.3.3 The Kramers–Kronig formula
II.4 Photoelectric Effect
II.4.1 Resonance with the continuous spectrum
II.4.2 Limiting amplitude
II.4.3 Angular distribution: the Wentzel formula
II.4.4 Derivation of Einstein's rules
II.4.5 Further improvements
II.5 Classical Scattering of Charged Particles
II.5.1 The Kepler problem
II.5.2 Angle of scattering
II.5.3 The Rutherford scattering
II.6 Quantum Scattering of Electrons
II.6.1 Radiated outgoing wave
II.6.2 Differential cross section
II.7 Electron Diffraction
II.7.1 Introduction
II.7.2 Electron diffraction
II.7.3 Limiting absorption principle
II.7.4 The Fraunhofer asymptotics
II.7.5 Comparison with experiment
III Atom in Magnetic Field
III.1 Normal Zeeman Effect
III.1.1 Magnetic Schrödinger equation
III.1.2 Selection rules
III.2 Intrinsic Magnetic Moment of Electrons
III.2.1 The Einstein–de Haas experiment
III.2.2 The Landé vector model
III.2.3 The Stern–Gerlach experiment
III.2.4 The Goudsmit–Uhlenbeck hypothesis
III.3 Spin and the Pauli Equation
III.3.1 Uniform magnetic field
III.3.2 General Maxwell field
III.3.3 The Maxwell–Pauli equations
III.3.4 Rotation group and angular momenta
III.3.5 Rotational covariance
III.3.6 Conservation laws
III.4 Anomalous Zeeman Effect
III.4.1 Spin-orbital coupling
III.4.2 Gyroscopic ratio
III.4.3 Quantum numbers
III.4.4 The Landé formula
III.4.5 Applications of the Landé formula
IV Relativistic Quantum Mechanics
IV.1 Free Dirac Equation
IV.2 The Pauli Theorem
IV.3 The Lorentz Covariance
IV.4 Angular Momentum
IV.5 Negative Energies
IV.6 Coupling to the Maxwell Field
IV.6.1 Gauge invariance
IV.6.2 Antiparticles
IV.7 Charge and Current. Continuity Equation
IV.8 Nonrelativistic Limits
IV.8.1 Order 1/c
IV.8.2 Order 1/c2
IV.9 The Hydrogen Spectrum
IV.9.1 Spinor spherical functions
IV.9.2 Separation of variables
IV.9.3 Factorization method
V Quantum Postulates and Attractors
V.1 Quantum Jumps
V.1.1 Quantum jumps as global attraction
V.1.2 Einstein–Ehrenfest's paradox. Bifurcation of attractors
V.2 Conjecture on Attractors
V.2.1 Trivial symmetry group G = {e}
V.2.2 Symmetry group of translations G = Rn
V.2.3 Unitary symmetry group G = U(1)
V.2.4 Orthogonal symmetry group G = SO(3)
V.2.5 Generic equations
V.2.6 Empirical evidence
V.3 Wave-Particle Duality
V.3.1 Reduction of wave packets
V.3.2 Diffraction of electrons
V.3.3 Quasiclassical asymptotics for the electron gun
V.4 Probabilistic Interpretation
V.4.1 Diffraction current
V.4.2 Discrete registration of electrons
V.4.3 Superposition principle as a linear approximation
VI Attractors of Hamiltonian PDEs
VI.1 Global Attractors of Nonlinear PDEs
VI.2 Global Attraction to Stationary States
VI.2.1 The d'Alembert equation
VI.2.2 String coupled to a nonlinear oscillator
VI.2.3 String coupled to several nonlinear oscillators
VI.2.4 Nonlinear string
VI.2.5 Wave-particle system
VI.2.6 Coupled Maxwell–Lorentz equations
VI.3 Soliton-Like Asymptotics
VI.3.1 Global attraction to solitons
VI.3.2 Adiabatic effective dynamics
VI.3.3 Mass-energy equivalence
VI.4 Global Attraction to Stationary Orbits
VI.4.1 Nonlinear Klein–Gordon equation
VI.4.2 Generalizations and open questions
VI.4.3 Omega-limit trajectories
VI.4.4 Limiting absorption principle
VI.4.5 Nonlinear analog of the Kato theorem
VI.4.6 Dispersive and bound components
VI.4.7 Omega-compactness
VI.4.8 Reduction of spectrum to spectral gap
VI.4.9 Reduction of spectrum to a single point
VI.4.10 Nonlinear radiative mechanism
VI.5 Numerical Simulation of Soliton Asymptotics
VI.5.1 Kinks of relativistic Ginzburg–Landau equations
VI.5.2 Numerical simulation
VI.5.3 Soliton asymptotics
VI.5.4 Adiabatic effective dynamics
Appendix A Old Quantum Theory
A.1 Black-body radiation law
A.2 The Thomson electron and the Lorentz theory
A.3 The Zeeman effect
A.4 Debye's quantization rule
A.5 Correspondence principle and selection rules
A.6 The Bohr–Sommerfeld quantization
A.7 Atom in magnetic field
A.7.1 Rotating frame of reference. Larmor's theorem
A.7.2 Spatial quantization
A.8 Normal Zeeman effect
A.9 The Bohr–Pauli theory of periodic table
A.10 The Hamilton–Jacobi and eikonal equations
Appendix B The Noether Theory of Invariants
Appendix C Perturbation Theory
Bibliography
Index