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Trang 1Quantum eld theory in
Henrik Bruus and Karsten Flensberg
rsted Laboratory
Niels Bohr Institute
Copenhagen,1 September 2001
Trang 31 First and second quantization 3
1.1 Firstquantization,single-particlesystems 4
1.2 Firstquantization,many-particlesystems 6
1.2.1 Permutationsymmetry and indistinguishability 6
1.2.2 Thesingle-particlestatesas basisstates 7
1.2.3 Operators in rstquantization 9
1.3 Second quantization, basicconcepts 11
1.3.1 Theoccupationnumberrepresentation 11
1.3.2 Thebosoncreation and annihilationoperators 12
1.3.3 Thefermion creation andannihilationoperators 14
1.3.4 Thegeneral formforsecond quantizationoperators 16
1.3.5 Changeofbasisinsecond quantization 17
1.3.6 Quantum eldoperators andtheir Fouriertransforms 19
1.4 Second quantization, speci c operators 20
1.4.1 Theharmonicoscillatorinsecond quantization 20
1.4.2 Theelectromagnetic eldinsecond quantization 21
1.4.3 Operators forkineticenergy,spin, density,and current 23
1.4.4 TheCoulomb interactioninsecond quantization 25
1.4.5 Basisstates forsystems withdi erentkinds ofparticles 26
1.5 Second quantizationand statistical mechanics 27
1.5.1 Thedistributionfunction fornon-interactingfermions 30
1.5.2 Distributionfunctions fornon-interactingbosons 30
1.6 Summaryand outlook 31
2 The electron gas 33 2.1 The non-interactingelectron gas 34
2.1.1 Bloch theoryof electronsina staticion lattice 35
2.1.2 Non-interactingelectronsinthejellium model 37
2.1.3 Non-interactingelectronsat nitetemperature 40
2.2 Electron interactions inperturbationtheory 41
2.2.1 Electroninteractionsin1 st order perturbationtheory 43
2.2.2 Electroninteractionsin2 nd order perturbation theory 45
2.3 Electron gases in3,2,1, and0 dimensions 46
Trang 42.3.1 3D electron gases: metals and semiconductors 47
2.3.2 2D electron gases: GaAs/Ga 1 x Al x Asheterostructures 48
2.3.3 1D electron gases: carbon nanotubes 50
2.3.4 0D electron gases: quantum dots 51
3 Phonons; coupling to electrons 53 3.1 JelliumoscillationsandEinsteinphonons 54
3.2 Electron-phononinteractionand thesound velocity 55
3.3 Lattice vibrationsand phononsin1D 55
3.4 Acoustical and opticalphonons in3D 58
3.5 The speci c heatof solidsintheDebye model 61
3.6 Electron-phononinteractioninthe lattice model 63
3.7 Electron-phononinteractioninthe jelliummodel 65
3.8 Summaryand outlook 66
4 Mean eld theory 67 4.1 The artof mean eld theory 70
4.2 Hartree{Fockapproximation 71
4.3 Broken symmetry 73
4.4 Ferromagnetism 75
4.4.1 The Heisenbergmodelof ionicferromagnets 75
4.4.2 The Stoner model ofmetallicferromagnets 77
4.5 Superconductivity 80
4.5.1 Breaking ofglobal gaugesymmetry andits consequences 80
4.5.2 Microscopic theory 83
4.6 Summaryand outlook 87
5 Time evolution pictures 89 5.1 The Schrodingerpicture 89
5.2 The Heisenbergpicture 90
5.3 The interactionpicture 90
5.4 Time-evolution inlinearresponse 93
5.5 Time dependent creation and annihilationoperators 93
5.6 Summaryand outlook 95
6 Linear response theory 97 6.1 The generalKuboformula 97
6.2 Kuboformulaforconductivity 100
6.3 Kuboformulaforconductance 102
6.4 Kuboformulaforthedielectric function 103
6.4.1 Dielectric functionfortranslation-invariant system 105
6.4.2 Relation betweendielectricfunctionand conductivity 105
Trang 57 Green's functions 107
7.1 \Classical" Green'sfunctions 107
7.2 Green's functionforthesingleparticleSchrodingerequation 108
7.3 The single-particleGreen'sfunctionof amany-bodysystem 111
7.3.1 Green'sfunctionof translation-invariantsystems 112
7.3.2 Green'sfunctionof freeelectrons 113
7.3.3 TheLehmannrepresentation 114
7.3.4 Thespectral function 115
7.3.5 Broadeningofthespectralfunction 117
7.4 Measuring thesingle-particlespectralfunction 117
7.4.1 Tunnelingspectroscopy 118
7.4.2 Opticalspectroscopy 121
7.5 The two-particle correlationfunctionofa many-bodysystem 122
7.6 Summaryand outlook 124
8 Equation of motiontheory 127 8.1 The single-particleGreen'sfunction 127
8.1.1 Non-interactingparticles 129
8.2 Anderson's modelformagnetic impurities 129
8.2.1 Theequation of motionfortheAnderson model 131
8.2.2 Mean- ... Tunnelinginto spinlessLuttinger liquid 292
18.6.1 Tunnelingintothe endofLuttingerliquid 292
18.7 What is aLuttingerliquid? 292
18.8 Experimentalrealizationsof Luttingerliquidphysics... class="page_container" data-page="9">
Thisintroductionto quantum eldtheoryincondensedmatterphysicshasemerged from
ourcoursesforgraduateandadvancedundergraduatestudentsattheNielsBohrInstitute,... resultinginthedevelopmentofquantum
electrodynam-ics(QED)andquantum eldtheory(QFT)ingeneral Byconvention,theoriginalformof
quantummechanicsisdenoted rstquantization,whilequantum eldtheoryisformulated