Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- Acknowledgments -- Part I From classical to wave mechanics -- 1 Experimental foundations of quantum theory -- 1.1 The need for a quantum theory -- 1.2 Our path towards quantum theory -- 1.3 Photoelectric effect -- 1.4 Compton effect -- 1.4.1 Thomson scattering -- 1.5 Interference experiments -- 1.6 Atomic spectra and the Bohr hypotheses -- 1.7 The experiment of Franck and Hertz -- 1.8 Wave-like behaviour and the Bragg experiment -- 1.9 The experiment of Davisson and Germer -- 1.10 Position and velocity of an electron -- 1.11 Problems -- Appendix 1.A The phase 1-form -- 2 Classical dynamics -- 2.1 Poisson brackets -- 2.2 Symplectic geometry -- 2.3 Generating functions of canonical transformations -- 2.3.1 Time-dependent Hamiltonian formalism -- 2.3.2 Dynamical time -- 2.3.3 Various generating functions -- 2.3.4 An example: particle in a repulsive potential -- 2.3.5 The harmonic oscillator -- 2.4 Hamilton and Hamilton-Jacobi equations -- 2.5 The Hamilton principal function -- 2.5.1 Free particle on a line -- 2.5.2 One-dimensional harmonic oscillator -- 2.5.3 Time-dependent Hamiltonian -- 2.6 The characteristic function -- 2.6.1 Principal versus characteristic function -- 2.7 Hamilton equations associated with metric tensors -- 2.8 Introduction to geometrical optics -- 2.8.1 Variational principles -- 2.9 Problems -- Appendix 2.A Vector fields -- Appendix 2.B Lie algebras and basic group theory -- Groups and sub-groups -- The general linear group GL(n,R) -- Euclidean and rotation group -- Galilei group -- Lorentz and Poincaré groups -- Unitary group -- Appendix 2.C Some basic geometrical operations -- Appendix 2.D Space-time -- Appendix 2.E From Newton to Euler-Lagrange -- 3 Wave equations -- 3.1 The wave equation -- 3.2 Cauchy problem for the wave equation.

3.3 Fundamental solutions -- 3.3.1 Wave equations in spherical polar coordinates -- 3.4 Symmetries of wave equations -- 3.5 Wave packets -- 3.6 Fourier analysis and dispersion relations -- 3.6.1 The symbol of difierential operators -- 3.6.2 Dispersion relations -- 3.7 Geometrical optics from the wave equation -- 3.8 Phase and group velocity -- 3.9 The Helmholtz equation -- 3.10 Eikonal approximation for the scalar wave equation -- 3.11 Problems -- 4 Wave mechanics -- 4.1 From classical to wave mechanics -- 4.1.1 Continuity equation -- 4.1.2 Physical interpretation of the wave function -- 4.1.3 Mean values -- 4.1.4 Eigenstates and eigenvalues -- 4.2 Uncertainty relations for position and momentum -- 4.2.1 Uncertainty relations in relativistic systems -- 4.3 Transformation properties of wave functions -- 4.3.1 Direct approach to the transformation properties of the Schrödinger equation -- 4.3.2 Width of the wave packet -- 4.4 Green kernel of the Schrödinger equation -- 4.4.1 Free particle -- 4.5 Example of isometric non-unitary operator -- 4.6 Boundary conditions -- 4.6.1 Particle confined by a potential -- 4.6.2 Improper eigenfunctions -- 4.7 Harmonic oscillator -- 4.7.1 One-dimensional oscillator -- 4.7.2 Hermite polynomials -- 4.8 JWKB solutions of the Schrödinger equation -- 4.8.1 On the meaning of semi-classical -- 4.8.2 Example: alpha-decay -- 4.9 From wave mechanics to Bohr-Sommerfeld -- 4.9.1 Quantization of Keplerian motion -- 4.9.2 Harmonic oscillator -- 4.9.3 Rotator in a plane -- 4.10 Problems -- Appendix 4.A Glossary of functional analysis -- Appendix 4.B JWKB approximation -- Appendix 4.C Asymptotic expansions -- 5 Applications of wave mechanics -- 5.1 Reflection and transmission -- 5.2 Step-like potential: tunnelling effect -- 5.2.1 Step-like potential -- 5.2.2 Tunnelling effect -- 5.3 Linear potential.

5.4 The Schrödinger equation in a central potential -- 5.5 Hydrogen atom -- 5.5.1 A simpler derivation of the Balmer formula -- 5.6 Introduction to angular momentum -- 5.6.1 Lie algebra of O(3) and associated vector fields -- 5.6.2 Quantum definition of angular momentum -- 5.6.3 Harmonic polynomials and spherical harmonics -- 5.6.4 Back to central potentials in R -- 5.7 Homomorphism between SU(2) and SO(3) -- 5.8 Energy bands with periodic potentials -- 5.9 Problems -- Appendix 5.A Stationary phase method -- Appendix 5.B Bessel functions -- 6 Introduction to spin -- 6.1 Stern-Gerlach experiment and electron spin -- 6.2 Wave functions with spin -- 6.2.1 Addition of orbital and spin angular momentum -- 6.3 The Pauli equation -- 6.4 Solutions of the Pauli equation -- 6.5 Landau levels -- 6.6 Problems -- Appendix 6.A Lagrangian of a charged particle -- Appendix 6.B Charged particle in a monopole field -- 7 Perturbation theory -- 7.1 Approximate methods for stationary states -- 7.1.1 Rayleigh-Schrödinger expansion -- 7.1.2 Brillouin-Wigner expansion -- 7.1.3 Remark on quasi-stationary states -- 7.2 Very close levels -- 7.3 Anharmonic oscillator -- 7.4 Occurrence of degeneracy -- 7.5 Stark effect -- 7.6 Zeeman effect -- 7.7 Variational method -- 7.8 Time-dependent formalism -- 7.8.1 Harmonic perturbations -- 7.8.2 Fermi golden rule -- 7.9 Limiting cases of time-dependent theory -- 7.9.1 Adiabatic switch on and off of the perturbation -- 7.9.2 Perturbation suddenly switched on -- 7.10 The nature of perturbative series -- 7.10.1 Regular perturbation theory -- 7.10.2 Asymptotic perturbation theory -- 7.10.3 Spectral concentration -- 7.10.4 Singular perturbation theory -- 7.11 More about singular perturbations -- 7.11.1 The Harrell method -- 7.11.2 Extension to other singular potentials -- 7.11.3 Concluding remarks -- 7.12 Problems.

Appendix 7.A Convergence in the strong resolvent sense -- 8 Scattering theory -- 8.1 Aims and problems of scattering theory -- 8.2 Integral equation for scattering problems -- 8.3 The Born series and potentials of the Rollnik class -- 8.4 Partial wave expansion -- 8.5 The Levinson theorem -- 8.6 Scattering from singular potentials -- 8.7 Resonances -- 8.8 Separable potential model -- 8.9 Bound states in the completeness relationship -- 8.10 Excitable potential model -- 8.11 Unitarity of the Möller operator -- 8.12 Quantum decay and survival amplitude -- 8.12.1 Law of radioactive decay: Poisson distribution -- 8.12.2 Quantum decay transitions. The survival amplitude -- 8.12.3 Decay amplitude under a Lorentz transformation -- 8.12.4 Quantum mechanics in dual spaces -- 8.13 Problems -- Part II Weyl quantization and algebraic methods -- 9 Weyl quantization -- 9.1 The commutator in wave mechanics -- 9.2 Abstract version of the commutator -- 9.3 Canonical operators and the Wintner theorem -- 9.4 Canonical quantization of commutation relations -- 9.5 Weyl quantization and Weyl systems -- 9.5.1 Representations -- 9.5.2 Unitary equivalence -- 9.5.3 Weyl quantization -- 9.6 The Schrödinger picture -- 9.7 From Weyl systems to commutation relations -- 9.8 Heisenberg representation for temporal evolution -- 9.9 Generalized uncertainty relations -- 9.9.1 Time-energy uncertainty relation -- 9.10 Unitary operators and symplectic linear maps -- 9.10.1 Translations -- 9.10.2 Rotations -- 9.10.3 Harmonic oscillator -- 9.11 On the meaning of Weyl quantization -- 9.12 The basic postulates of quantum theory -- 9.12.1 Rigged Hilbert spaces -- Position and momentum operators -- 9.13 Problems -- 10 Harmonic oscillators and quantum optics -- 10.1 Algebraic formalism for harmonic oscillators -- 10.2 A thorough understanding of Landau levels -- 10.3 Coherent states.

10.4 Weyl systems for coherent states -- 10.5 Two-photon coherent states -- 10.6 Problems -- 11 Angular momentum operators -- 11.1 Angular momentum: general formalism -- 11.1.1 Algebraic method for the spectrum -- 11.1.2 Representations -- 11.1.3 Hilbert space -- 11.2 Two-dimensional harmonic oscillator -- 11.2.1 Introduction of different bases -- 11.3 Rotations of angular momentum operators -- 11.4 Clebsch-Gordan coefficients and the Regge map -- 11.5 Postulates of quantum mechanics with spin -- 11.6 Spin and Weyl systems -- 11.7 Monopole harmonics -- 11.8 Problems -- 12 Algebraic methods for eigenvalue problems -- 12.1 Quasi-exactly solvable operators -- 12.2 Transformation operators for the hydrogen atom -- 12.3 Darboux maps: general framework -- 12.4 SU(1, 1) structures in a central potential -- 12.5 The Runge-Lenz vector -- 12.6 Problems -- 13 From densitymatrix to geometrical phases -- 13.1 The density matrix -- 13.2 Applications of the density matrix -- 13.3 Quantum entanglement -- 13.4 Hidden variables and the Bell inequalities -- 13.5 Entangled pairs of photons -- 13.6 Production of statistical mixtures -- 13.7 Pancharatnam and Berry phases -- 13.7.1 More concerning non-integrable phases -- 13.8 The Wigner theorem and symmetries -- 13.9 A modern perspective on the Wigner theorem -- 13.10 Problems -- Part III Selected topics -- 14 From classical to quantum statistical mechanics -- 14.1 Aims and main assumptions -- 14.2 Canonical ensemble -- 14.3 Microcanonical ensemble -- 14.4 Partition function -- 14.5 Equipartition of energy -- 14.6 Specific heats of gases and solids -- 14.7 Black-body radiation -- 14.7.1 The Kirchhoff laws -- 14.7.2 Stefan and displacement laws -- 14.7.3 The Planck model -- 14.7.4 The contributions of Einstein -- 14.7.5 Dynamic equilibrium of the radiation field -- 14.8 Quantum models of specific heats.

14.9 Identical particles in quantum mechanics.