THE PHYSICS OF QUANTUM MECHANICS The Physics of Quantum Mechanics JAMES BINNEY RudolfPeierlsCentreforTheoreticalPhysicsandMertonCollege, UniversityofOxford DAVID SKINNER DepartmentofAppliedMathematicsandTheoreticalPhysics, UniversityofCambridge 3 3 GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries (cid:2)c JamesBinneyandDavidSkinner2014 FirstpublishedbyCappellaArchivein2008 Themoralrightsoftheauthorshavebeenasserted FirstEditionpublishedin2014 Impression:1 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressControlNumber:2013940021 ISBN978–0–19–968856–2(hbk.) ISBN978–0–19–968857–9(pbk.) Printedandboundby CPIGroup(UK)Ltd,Croydon,CR04YY This book is a consequence of the vision and munificence of Walter of Merton, who in 1264 launched something good Contents Preface xii 1 Introduction 1 1.1 Origins 2 1.2 Measurements 6 • Measurement involves disturbance 7 ⊲ Heisenberg microscope 7 • Ideal measurements 8 • Summary 10 1.3 Probability amplitudes 11 • Two-slit interference 12 1.4 Quantum states 14 • Observables 14 ⊲ Complete sets of amplitudes 14 • Vector spaces and their duals 16 • The energy represen- tation 19 • Polarisation of photons 20 1.5 Summary 23 Problems 24 2 Operators, measurement and time evolution 25 2.1 Operators 25 ⊲ Functions of operators 29 ⊲ Commutators 29 2.2 Evolution in time 31 • Evolution of expectation values 32 2.3 The position representation 34 • Hamiltonian of a particle 37 • Wavefunction for well- defined momentum 38 ⊲ The uncertainty principle 39 • Dynamics of a free particle 41 • Back to two-slit in- terference 43 • Generalisation to three dimensions 44 ⊲ Probability current 45 ⊲ The virial theorem 46 2.4 Summary 47 Problems 48 viii Contents 3 Oscillators 52 3.1 Stationary states of a harmonic oscillator 52 3.2 Dynamics of oscillators 56 • Anharmonic oscillators 58 Problems 61 4 Transformations and observables 66 4.1 Transforming kets 66 • Translating kets 67 • Continuous transformations and generators 69 • The rotation operator 71 • Discrete transformations 72 ⊲ The parity operator 72 ⊲ Mirror operators 73 4.2 Transformations of operators 74 ⊲ The parity operator 76 ⊲ Mirror operators 78 4.3 Symmetries and conservation laws 79 4.4 The Heisenberg picture 81 4.5 What is the essence of quantum mechanics? 83 Problems 85 5 Motion in step potentials 88 5.1 Square potential well 88 • Limiting cases 91 ⊲ Infinitely deep well 91 ⊲ Infinitely narrow well 92 5.2 A pair of square wells 93 • Ammonia 96 ⊲ The ammonia maser 97 5.3 Scattering of free particles 99 ⊲ The scattering cross-section 102 • Tunnelling through a potential barrier 103 • Scattering by a classically allowed region 104 • Resonant scattering 106 ⊲ The Breit–Wigner cross-section 109 5.4 How applicable are our results? 113 5.5 Summary 115 Problems 117 6 Composite systems 123 6.1 Composite systems 124 • Collapse of the wavefunction 128 • Operators for com- posite systems 129 • Development of entanglement 131 • Einstein–Podolski–Rosen experiment 132 ⊲ Bell’s inequality 134 6.2 Quantum computing 137 Contents ix 6.3 The density operator 144 • Reduced density operators 148 • Shannon entropy 151 6.4 Thermodynamics 154 6.5 Measurement 158 Problems 162 7 Angular momentum 167 2 7.1 Eigenvalues of Jz and J 168 • Rotation spectra of diatomic molecules 171 7.2 Spin and orbital angular momentum 174 • Orbital angular momentum 174 ⊲ L as the generator of 2 circular translations 176 ⊲ Spectra of L and Lz 177 • Spin angular momentum 177 7.3 Physics of spin 180 • Spin-half matrices 181 • Spin-one matrices 182 • The Stern–Gerlach experiment 183 ⊲ Stern– Gerlach experiment with spin-one atoms 186 • Precession in a magnetic field 187 • The classical limit 188 7.4 Orbital angular-momentum eigenfunctions 191 • Orbital angular momentum and parity 196 • Orbital angular momentum and kinetic energy 196 • Legendre polynomials 198 7.5 Three-dimensional harmonic oscillator 199 7.6 Addition of angular momenta 205 • Case of two spin-half systems 209 • Case of spin-one and spin-half 211 • The classical limit 212 Problems 213 8 Hydrogen 219 8.1 Gross structure of hydrogen 220 • Emission-line spectra 226 • Radial eigenfunctions 226 • Shielding 231 • Expectation values for r k 234 − 8.2 Fine structure and beyond 236 • Spin–orbit coupling 236 • Hyperfine structure 241 Problems 243 9 Motion in a magnetic field 247 9.1 Hamiltonian for motion in a magnetic field 248 • Classical equations of motion 248 9.2 Gauge transformations 250 • Probability current 251