Quantum Mechanics, Volume 2

Claude Cohen-Tannoudji is a researcher at the Kastler-Brossel laboratory of the Ecole Normale Supérieure in Paris where he also studied and received his PhD in 1962. In 1973 he became Professor of atomic and molecular physics at the Collège des France. His main research interests were optical pumping, quantum optics and atom-photon interactions. In 1997, Claude Cohen-Tannoudji, together with Steven Chu and William D. Phillips, was awarded the Nobel Prize in Physics for his research on laser cooling and trapping of neutral atoms. Bernard Diu (1935-2025) was Professor at the Denis Diderot University (Paris VII). He was engaged in research at the Laboratory of Theoretical Physics and High Energy where his focus was on strong interactions physics and statistical mechanics. Franck Laloë is a researcher at the Kastler-Brossel laboratory of the Ecole Normale Supérieure in Paris. His first assignment was with the University of Paris VI before he was appointed to the CNRS, the French National Research Center. His research was focused on optical pumping, statistical mechanics of quantum gases, musical acoustics and the foundations of quantum mechanics.

• Volume II WAVES AND PARTICLES. INTRODUCTION TO THE BASIC IDEAS OF QUANTUM MECHANICS 1READER’S GUIDE FOR COMPLEMENTS 33AI Order of magnitude of the wavelengths associated with material particles 35BI Constraints imposed by the uncertainty relations 39CI Heisenberg relation and atomic parameters 41DI An experiment illustrating the Heisenberg relations 45EI A simple treatment of a two-dimensional wave packet 49FI The relationship between one- and three-dimensional problems 53GI One-dimensional Gaussian wave packet: spreading of the wave packet 57HI Stationary states of a particle in one-dimensional square potentials 63JI Behavior of a wave packet at a potential step 75II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS 87READER’S GUIDE FOR COMPLEMENTS 159AII The Schwarz inequality 161BII Review of some useful properties of linear operators 163CII Unitary operators 173DII A more detailed study of the r and p representations 181EII Some general properties of two observables, Q and P, whose commutator is equal to iℏ 187FII The parity operator 193GII An application of the properties of the tensor product: the twodimensional in;nite well 201HII Exercises 205III THE POSTULATES OF QUANTUM MECHANICS 213READER’S GUIDE FOR COMPLEMENTS 267AIII Particle in an in;nite one-dimensional potential well 271BIII Study of the probability current in some special cases 283CIII Root mean square deviations of two conjugate observables 289DIII Measurements bearing on only one part of a physical system 293EIII The density operator 299FIII The evolution operator 313GIII The Schrödinger and Heisenberg pictures 317HIII Gauge invariance 321JIII Propagator for the Schrödinger equation 335KIII Unstable states. Lifetime 343LIII Exercises 347MIII Bound states in a “potential well” of arbitrary shape 359NIII Unbound states of a particle in the presence of a potential well or barrier 367OIII Quantum properties of a particle in a one-dimensional periodic structure 375IV APPLICATIONS OF THE POSTULATES TO SIMPLE CASES: SPIN 1/2 AND TWO-LEVEL SYSTEMS 393READER’S GUIDE FOR COMPLEMENTS 423AIV The Pauli matrices 425BIV Diagonalization of a 2 × 2 Hermitian matrix 429CIV Fictitious spin 1/2 associated with a two-level system 435DIV System of two spin 1/2 particles 441EIV Spin density matrix 449FIV Spin 1/2 particle in a static and a rotating magnetic ;elds: magnetic resonance 455GIV A simple model of the ammonia molecule 469HIV E;ects of a coupling between a stable state and an unstable state 485JIV Exercises 491V THE ONE-DIMENSIONAL HARMONIC OSCILLATOR 497READER’S GUIDE FOR COMPLEMENTS 525AV Some examples of harmonic oscillators 527BV Study of the stationary states in the x representation. Hermite polynomials 547CV Solving the eigenvalue equation of the harmonic oscillator by the polynomial method 555DV Study of the stationary states in the momentum representation 563EV The isotropic three-dimensional harmonic oscillator 569FV A charged harmonic oscillator in a uniform electric ;eld 575GV Coherent “quasi-classical” states of the harmonic oscillator 583HV Normal vibrational modes of two coupled harmonic oscillators 599JV Vibrational modes of an in;nite linear chain of coupled harmonic oscillators; phonons 611KV Vibrational modes of a continuous physical system. Photons 631LV One-dimensional harmonic oscillator in thermodynamic equilibrium at a temperature T 647MV Exercises 661VI GENERAL PROPERTIES OF ANGULARMOMENTUM IN QUANTUM MECHANICS 667READER’S GUIDE FOR COMPLEMENTS 703AVI Spherical harmonics 705BVI Angular momentum and rotations 717CVI Rotation of diatomic molecules 739DVI Angular momentum of stationary states of a two-dimensional harmonic oscillator 755EVI A charged particle in a magnetic ;eld: Landau levels 771FVI Exercises 795VII PARTICLE IN A CENTRAL POTENTIAL, HYDROGEN ATOM 803READER’S GUIDE FOR COMPLEMENTS 831AVII Hydrogen-like systems 833BVII A soluble example of a central potential: The isotropic three-dimensional harmonic oscillator 841CVII Probability currents associated with the stationary states of the hydrogen atom 851DVII The hydrogen atom placed in a uniform magnetic ;eld. Paramagnetism and diamagnetism. The Zeeman e;ect 855EVII Some atomic orbitals. Hybrid orbitals 869FVII Vibrational-rotational levels of diatomic molecules 885GVII Exercises 899Index 901