北京大学：量子多体的理论与计算（讲义）Quantum Theory of Many Particle Systems

QuantumTheory of ManyParticleSystems施均仁(junrenshi@pku.edu.cn)March 28,2023

Quantum Theory of Many Particle Systems 施均仁 (junrenshi@pku.edu.cn) March 28, 2023

Contents11 Second quantization and coherent statesNO51.111.1Quantum mechanicsNO52.121.2QuantumstatisticalmechanicsNO51.231.3Identical particles41.4Creation and annihilation operatorsNO51.441.4.1Basics.......71.4.2Secondquantized Hamiltonians9NO51.51.5Coherent states.91.5.1Boson coherent states101.5.2Grassmann algebra1.5.3Fermion coherent states .111.5.413Gaussian integrals1.613Summary152Green'sfunctions152.1 Green'sfunctions and observables.152.1.1Time-ordered Green'sfunctionsNO55.1172.1.2Evaluation ofObservablesNO52.1182.1.3Response functions.HJ53.2202.1.4Species of Green's functions2.2:21Fluctuation-dissipationtheoremHJ53.32.2.121Real timeGreen'sfunctionsFW5312.2.2Thermal Green's function and analytic continuation22HJ542.322Non-equilibrium Green's function242.41 Summary25NO52.23Functional integrals253.1 Feynman path integrals.263.2Imaginary time path integrals and the partition function. .3.328Functional integrals.293.4PartitionfunctionandGreen'sfunctionsi

Contents 1 Second quantization and coherent states 1 1.1 Quantum mechanics NO§1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Quantum statistical mechanics NO§2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Identical particles NO§1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Creation and annihilation operators . . . . . . . . . . . . . . . . . . . . . . . 4 1.4.1 Basics NO§1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4.2 Second quantized Hamiltonians . . . . . . . . . . . . . . . . . . . . . 7 1.5 Coherent states NO§1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5.1 Boson coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5.2 Grassmann algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5.3 Fermion coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5.4 Gaussian integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Green’s functions 15 2.1 Green’s functions and observables . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.1 Time-ordered Green’s functions . . . . . . . . . . . . . . . . . . . . . 15 2.1.2 Evaluation of Observables NO§5.1 . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.3 Response functions NO§2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.4 Species of Green’s functions HJ§3.2 . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Fluctuation-dissipation theorem . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.1 Real time Green’s functions HJ§3.3 . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.2 Thermal Green’s function and analytic continuation FW§31 . . . . . . . . 22 2.3 Non-equilibrium Green’s function HJ§4 . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 Functional integrals NO§2.2 25 3.1 Feynman path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Imaginary time path integrals and the partition function . . . . . . . . . . 26 3.3 Functional integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.4 Partition function and Green’s functions . . . . . . . . . . . . . . . . . . . . 29 ii

32NO52.34 PerturbationtheoryNO52.3：324.1General strategy4.2 Finite temperature formalism34：344.2.1Labeled Feynman diagrams.364.2.2UnlabeledFeynmanDiagrams4.2.336Hugenholtz diagrams374.2.4Frequencyandmomentumrepresentation394.2.5Thelinked clustertheorem.394.2.6Green'sfunctionsNO53.14.3Zero temperatureformalism41414.3.1Ground state energy and Green's function434.3.2Diagram Rules4.3.344FreeFermion propagatorsHJ 54.3454.4Contourformalism454.5Summary495Effectiveactiontheoryandenergyfunctionals495.1Effectiveaction..NO52.4525.2Irreducible diagrams and integral equations525.2.1Self-energy and Dyson's equation.535.2.2Perturbative construction of the Luttinger-Ward functional545.2.3Second ordervertexfunction555.2.4Higher order equationsHJ55.25.2.556KeldyshFormulation5.2.656Other sourcesGV58575.3Landau Fermi-liquid theoryGV58.3575.3.1PhenomenologicalapproachGV$8.5595.3.2Microscopic underpinning5.461Generalizations636 TheoryofelectronliquidFW512636.1Energy....636.1.1Hartree-Fock approximation.：656.1.2High order contributions6.1.366General structure of the self-energy:676.2Densityresponsefunction.GV 53.3676.2.1Basic propertiesGV 55.36.2.269Random phase approximation (RPA)GV $5.4726.2.3Local field correction756.3Plasmon...GV 55.3.3756.3.1CollectiveexcitationAS 56.26.3.277Functional integralsofplasmonsAS 56.2796.3.3Collective excitationsii

4 Perturbation theory NO§2.3 32 4.1 General strategy NO§2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2 Finite temperature formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.2.1 Labeled Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . 34 4.2.2 Unlabeled Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . 36 4.2.3 Hugenholtz diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2.4 Frequency and momentum representation . . . . . . . . . . . . . . 37 4.2.5 The linked cluster theorem . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2.6 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.3 Zero temperature formalism NO§3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.3.1 Ground state energy and Green’s function . . . . . . . . . . . . . . 41 4.3.2 Diagram Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3.3 Free Fermion propagators . . . . . . . . . . . . . . . . . . . . . . . . 44 4.4 Contour formalism HJ §4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5 Effective action theory and energy functionals 49 5.1 Effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Irreducible diagrams and integral equations NO§2.4 . . . . . . . . . . . . . . . . . 52 5.2.1 Self-energy and Dyson’s equation . . . . . . . . . . . . . . . . . . . . 52 5.2.2 Perturbative construction of the Luttinger-Ward functional . . . . 53 5.2.3 Second order vertex function . . . . . . . . . . . . . . . . . . . . . . 54 5.2.4 Higher order equations . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.2.5 Keldysh Formulation HJ§5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.2.6 Other sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.3 Landau Fermi-liquid theory GV§8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.3.1 Phenomenological approach GV§8.3 . . . . . . . . . . . . . . . . . . . . . . . 57 5.3.2 Microscopic underpinning GV§8.5 . . . . . . . . . . . . . . . . . . . . . . . . 59 5.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6 Theory of electron liquid 63 6.1 Energy FW§12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.1.1 Hartree-Fock approximation . . . . . . . . . . . . . . . . . . . . . . . 63 6.1.2 High order contributions . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.1.3 General structure of the self-energy . . . . . . . . . . . . . . . . . . 66 6.2 Density response function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.2.1 Basic properties GV §3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.2.2 Random phase approximation (RPA) GV §5.3 . . . . . . . . . . . . . . . . . . 69 6.2.3 Local field correction GV §5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.3 Plasmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.3.1 Collective excitation GV §5.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.3.2 Functional integrals of plasmons AS §6.2 . . . . . . . . . . . . . . . . . . . . 77 6.3.3 Collective excitations AS §6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 iii

817Phasetransitions and spontaneous symmetrybreaking817.1Generaltheory.....NO 54.1817.1.1PhasetransitionsNO 54.1, 4.27.1.2Landautheory81NO 54.37.1.3Meanfieldtheory84NO 54.47.1.4Fluctuations8587AS 56.37.2Bose-Einstein condensation and superfluidity7.2.187Phasetransition.887.2.2Superfluidity897.2.3BogoliubovtransformationAS 56.4907.3Superconductivity7.3.190Introduction917.3.2Cooper instability.7.3.3Mean field theory93957.3.4EffectivefieldtheoryandAnderson-Higgsmechanismiv

7 Phase transitions and spontaneous symmetry breaking 81 7.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.1.1 Phase transitions NO §4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.1.2 Landau theory NO §4.1, 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.1.3 Mean field theory NO §4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 7.1.4 Fluctuations NO §4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.2 Bose-Einstein condensation and superfluidity AS §6.3 . . . . . . . . . . . . . . . . . 87 7.2.1 Phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.2.2 Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.2.3 Bogoliubov transformation . . . . . . . . . . . . . . . . . . . . . . . . 89 7.3 Superconductivity AS §6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.3.2 Cooper instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.3.3 Mean field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.3.4 Effective field theory and Anderson-Higgs mechanism . . . . . . . 95 iv

PrefaceOverview:Anintroductorycourseforthequantumtheoryofmany-bodysystems;·Asurvey of general principles and language;:Anoverviewonmanybodytechniques;.Bridging the gap between thequantum many-body theory and real material calculations;Contents.BasicTheory-Secondquantizationandcoherentstates-Green's functions-Functionalintegralformalism-Perturbationtheory-Effectiveactiontheory:Applicationstophysical systems-Theoryofelectronliquids-BrokensymmetryandphasetransitionsV

Preface Overview • An introductory course for the quantum theory of many-body systems; • A survey of general principles and language; • An overview on many body techniques; • Bridging the gap between the quantum many-body theory and real material calculations; Contents • Basic Theory – Second quantization and coherent states – Green’s functions – Functional integral formalism – Perturbation theory – Effective action theory • Applications to physical systems – Theory of electron liquids – Broken symmetry and phase transitions v

ResourcesTherearemanyexcellentbooksonthequantumtheoryofmany-particlesystems.Thefollowingisalistofthebooksuponwhichthislectureisbased.Theywill bereferredinmaintext/sidenotesbyusingthefollowingabbreviations:NOJ. W. Negele and H. Orland, Quantum Many-Particle Systems (Addison-Wesley, 1988)-Functional integral formalismASAlexanderAltlandandBenD.Simons,CondensedMatterFieldTheory(CambridgeUni-versityPress,2010)-Various applications in condensed matter physicsGVG. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge UniversityPress,2005).-Theory of the electron liquidHJHartmut Haug and Antti-Pekka Jauho, Quantum Kinetics in Transport and Optics of Semi-conductors (Springer,2008).-Non-equilibrium Green'sfunctionFWA. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw Hill2003).MHG.D.Mahan, Many-Particle Physics (Kluwer Academic, 2000)FRE.Fradkin,FieldTheoriesofCondensedMatterPhysics(CambridgeUniversityPress,2013)vi

Resources There are many excellent books on the quantum theory of many-particle systems. The following is a list of the books upon which this lecture is based. They will be referred in main text/side notes by using the following abbreviations: NO J. W. Negele and H. Orland, Quantum Many-Particle Systems (Addison-Wesley, 1988). –Functional integral formalism AS Alexander Altland and Ben D. Simons, Condensed Matter Field Theory (Cambridge Uni￾versity Press, 2010). –Various applications in condensed matter physics GV G. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University Press, 2005). –Theory of the electron liquid HJ Hartmut Haug and Antti-Pekka Jauho, Quantum Kinetics in Transport and Optics of Semi￾conductors (Springer, 2008). –Non-equilibrium Green’s function FW A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw Hill, 2003). MH G. D. Mahan, Many-Particle Physics (Kluwer Academic, 2000). FR E. Fradkin, Field Theories of Condensed Matter Physics (Cambridge University Press, 2013). vi

Acknowledgements谭奕typosinEq.(1.107)林益浩signerrorsinEq.(2.8)and(2.9);ambiguitiesintheenergyreferencesofEq.(5.24)and(5.26)尹超inconsistencyinthefluctuation-dissipationrelations;atypo inEq.(3.29)陆易einEq.(3.51)shouldbelessthanh刘震寰atypoinEq.(3.12)王法杰atypoinEq.(3.45李德馨typosinEq.(5.27),Eq.(7.45),andEq.(7.144)刘鹏飞atypoinEq.(5.60)刘怡然Eq.(6.17a)is independentofthedensityonlyforitson-shellvalue张凡atypoinEq.（6.58)黄鑫懿atypoinEq.(1.27)娄琴剑 typosin Eq.(4.10),Eq.(7.87),Eq.(2.66),Eq.(7.148),Eq.(2.32),and Problem 4of S4梁靖雲atypoinEq.（6.89)林织星typosinEq.(6.79)胡世昕typos inEq.(5.43),Eq.(7.19),and Eq.(7.31)王晨冰atypoinEq.(6.144)贡晓苟typosinEqs.(7.106,7.109),Eg.(7.142),Eg.(7.91),andelectromagneticunitsinS7.3.4陈天扬typosinEq.(4.47),Eq.(5.6),andEq.(5.92)薛泽洋typosinEq.(2.49)andEq.（6.99)杨天骅typosinEq(6.89)andEq(7.107)颜子涵typosinEq.(6.119)andEq.(6.131)洪源 typo in Eq. (3.37) and margin note in s1.5冯杰超typosinEqs.(4.38.5.80-5.82.5.86.6.37,6.144)andthemarginnoteforEq.6.15）vii

Acknowledgements 谭奕 typos in Eq. (1.107) 林益浩 sign errors in Eq. (2.8) and (2.9); ambiguities in the energy references of Eq. (5.24) and (5.26) 尹超 inconsistency in the fluctuation-dissipation relations; a typo in Eq. (3.29) 陆易 ϵ in Eq. (3.51) should be less than ℏβ 刘震寰 a typo in Eq. (3.12) 王法杰 a typo in Eq. (3.45) 李德馨 typos in Eq. (5.27), Eq. (7.45), and Eq. (7.144) 刘鹏飞 a typo in Eq. (5.60) 刘怡然 Eq. (6.17a) is independent of the density only for its on-shell value 张凡 a typo in Eq. (6.58) 黄鑫懿 a typo in Eq. (1.27) 娄琴剑 typos in Eq. (4.10), Eq. (7.87), Eq. (2.66), Eq. (7.148), Eq. (2.32), and Problem 4 of §4 梁靖雲 a typo in Eq. (6.89) 林织星 typos in Eq. (6.79) 胡世昕 typos in Eq. (5.43), Eq. (7.19), and Eq. (7.31) 王晨冰 a typo in Eq. (6.144) 贡晓荀 typos in Eqs. (7.106, 7.109), Eq. (7.142), Eq. (7.91), and electromagnetic units in §7.3.4 陈天扬 typos in Eq. (4.47), Eq. (5.6), and Eq. (5.92). 薛泽洋 typos in Eq. (2.49) and Eq. (6.99) 杨天骅 typos in Eq. (6.89) and Eq. (7.107) 颜子涵 typos in Eq. (6.119) and Eq. (6.131) 洪源 typo in Eq. (3.37) and margin note in §1.5 冯杰超 typos in Eqs. (4.38,5.80–5.82,5.86,6.37,6.144) and the margin note for Eq.6.15) vii

Chapter 1Second quantization and coherentstatesNO51.11.1QuantummechanicsBasicconcepts. States and observables: position eigenstates r), momentum eigenstates (p):(1.1)(r)=r(r),(1.2)p(p) = p [p) .The concept of sTATE can be generalized to eigenstates of any observables/operators, notlimited to the position/momentum.An example is the spin eigenstate:h(1.3)32 (±) =/±).Hilbertspace:allstateswithfinitenorms..Completeness (closure) relations:[ dr [r) (r| = 1,(1.4)(1.5)dp p) (p| = 1,[dr 1r) () = [dr[r) (r),(1.6)[) = -- Note that 1 here (with or without a subscript) denotes an identity operator. It is as-sociated with a particular Hilbert space. Identity operators associated with differentHilbertspacesarenotequal:(1.7)I+) (+/ + I-) (-| = 1s.(1.8)1+ 1s..Overlapsbetweenstates:(1.9)(rlr) =8(r -r'),(1.10)(plp") = 8(p-p"),1)3/2(rlp) = ("exp(1.11)（2元h）Schrodinger equation1

Chapter 1 Second quantization and coherent states 1.1 Quantum mechanics NO§1.1 Basic concepts • States and observables: position eigenstates |r⟩, momentum eigenstates |p⟩: rˆ |r⟩ = r |r⟩, (1.1) pˆ|p⟩ = p |p⟩. (1.2) The concept of STATE can be generalized to eigenstates of any observables/operators, not limited to the position/momentum. An example is the spin eigenstate: sˆz |±⟩ = ± ℏ 2 |±⟩. (1.3) • Hilbert space: all states with finite norms. • Completeness (closure) relations: Z dr |r⟩ ⟨r| = 1, (1.4) Z dp |p⟩ ⟨p| = 1, (1.5) |ψ⟩ = Z dr |r⟩ ⟨r| ψ⟩ ≡ Z dr |r⟩ ψ(r). (1.6) – Note that 1 here (with or without a subscript) denotes an identity operator. It is as￾sociated with a particular Hilbert space. Identity operators associated with different Hilbert spaces are not equal: |+⟩ ⟨+| + |−⟩ ⟨−| = 1S. (1.7) 1 ̸= 1S. (1.8) • Overlaps between states: ⟨r|r ′ ⟩ = δ(r − r ′ ), (1.9) ⟨p|p ′ ⟩ = δ(p − p ′ ), (1.10) ⟨r|p⟩ =  1 2πℏ 3/2 exp  ip · r ℏ  . (1.11) Schrödinger equation 1

.Wave function(1.12)(r)=(r [).:Momentumoperatorinthepositionbasis[ dpeip-r/p(p|b)/ dp (r ) (p[ b) = ((r [P[ ) =(1.13)(10) =ih0(n)7dp(r/p) (p/) =-i.(1.14)FOrIr. Schrodinger equation:d+V()13)(1.15)130)hinow(r) = (P-[ + () ) =+ V(r)(r)(1.16)-1.-ihotHeisenbergand Schrodingerrepresentations· In the Schrodinger representation, states evolve with time:b(t) = e-it/ [b(0),(1.17)·IntheHeisenbergrepresentation,operators (observables)evolvewithtime:p()(t)=eift/hpe-iht/h(1.18): The two representations are equivalent:(b()/P)(t) =((0) [ent/pe-it/ (0)(1.19)=((0) ()(t) (0))(1.20)NO52.11.2Ouantum statistical mechanicsStatisticalensembles: Micro-canonical ensemble: fixed energy and particle number. The system is assumed tobe ergodic.. Canonical ensemble: fixed particle number, exchange energy with a thermal reservoirpxe-pa(1.21)where β = 1/kBT. Note that e-βi could be interpreted as an imaginary-time evolutionoperatorwitht=-ihβ:e-B =e-(-B)/h(1.22)·Grandcanonical ensemble:exchangeboth theenergyand particles,pxe-B(ti-uN).(1.23)K=H-μN is called grand-canonical Hamiltonian.ThermodynamiclimitN,V-→oo,N/V→p.Allthree ensembles are equivalentin thethermodynamic limit.Exceptwhen someobservablehasdivergentfluctuations-phasetransitionsandsymmetrybreaking systems.2

• Wave function ψ(r) ≡ ⟨r | ψ⟩. (1.12) • Momentum operator in the position basis: ⟨r |pˆ| ψ⟩ = Z dp ⟨r | p⟩ ⟨p |pˆ| ψ⟩ =  1 2πℏ 3/2 Z dpe ip·r/ℏp ⟨p | ψ⟩ (1.13) = −iℏ ∂ ∂r Z dp ⟨r | p⟩ ⟨p | ψ⟩ = −iℏ ∂ ∂r ⟨r| ψ⟩ ≡ −iℏ ∂ψ(r) ∂r . (1.14) • Schrödinger equation: iℏ d dt |ψ⟩ =  pˆ 2 2m + V (rˆ)  |ψ⟩, (1.15) iℏ ∂ψ(rt) ∂t = ⟨r|  pˆ 2 2m + V (rˆ)  |ψ⟩ = " 1 2m  −iℏ ∂ ∂r 2 + V (r) # ψ(r). (1.16) Heisenberg and Schrödinger representations • In the Schrödinger representation, states evolve with time: |ψ(t)⟩ = e −iHt/ ˆ ℏ |ψ(0)⟩. (1.17) • In the Heisenberg representation, operators (observables) evolve with time: pˆ (H)(t) = e iHt/ ˆ ℏpˆe −iHt/ ˆ ℏ . (1.18) • The two representations are equivalent: ⟨ψ(t)| pˆ| ψ(t)⟩ = D ψ(0)    e iHt/ ˆ ℏpˆe −iHt/ ˆ ℏ    ψ(0)E (1.19) = D ψ(0)    pˆ (H)(t)    ψ(0)E . (1.20) 1.2 Quantum statistical mechanics NO§2.1 Statistical ensembles • Micro-canonical ensemble: fixed energy and particle number. The system is assumed to be ergodic. • Canonical ensemble: fixed particle number, exchange energy with a thermal reservoir ρˆ ∝ e −βHˆ , (1.21) where β ≡ 1/kBT. Note that e −βHˆ could be interpreted as an imaginary-time evolution operator with t = −iℏβ: e −βHˆ = e −iHˆ (−iℏβ)/ℏ . (1.22) • Grand canonical ensemble: exchange both the energy and particles. ρˆ ∝ e −β(Hˆ −µNˆ) . (1.23) Kˆ ≡ Hˆ − µNˆ is called grand-canonical Hamiltonian. Thermodynamic limit N, V → ∞, N/V → ρ. • All three ensembles are equivalent in the thermodynamic limit . • Except when some observable has divergent fluctuations–phase transitions and symmetry￾breaking systems. 2

PartitionfunctionZ = Tre-β(H-μN)Grand canonical potential12(1.24)InZExpectationvaluesR) = TrpR,(1.25)e-β(H-μN),(1.26)pZThermodynamicrelationscanbeinferredfromthestatisticalmechanics2ITNe-8(I-μ) = N,(1.27)u(H-μN)2-2(1.28)=S,"OTT82(1.29)=P.OvNote that in the thermodynamic limit, 2 must be proportional to V.Therefore = -PV.NO51.21.3IdenticalparticlesThe quantum mechanics can be generalized for many-particle systems.Product states can be constructed from orthonormal single particle states a):(1.30)[α1...Qn)=[Q1)@[α2)@...@[Qn)NotethatweuseDtodenotetheproduct states.Overlapbetweenproductstates:(1.31)(α1...αN |Q1..QN) = (Q1 [Q1) (α2/α2) ..- (αN[α'N):a1..aN (ri...TN)=(ri...TNa1...aN)(1.32)=a (ri)ba(r2)...an(rN).ClosurerelationD(1.33)[α1..QN)(Q1...α/ = 1a1...anExchange symmetryOnly totally symmetric (Bosons) and anti-symmetric states (Fermions) are observed in nature:(rp1,Tp2...,rpN) =(r1,T2..,TN)(BoSons),(1.34)(1.35)(rp1,rp2,..,rpN)=(-1)P(r1,r2,...,r)(Fermions).Statistics theorem:Bosons (Fermions)haveinteger (half-integer)spinsNormalize symmetrized states are constructed from the product states by applying symmetriza-tions:(1.36)P(, ...TN) - EP (-P .. P),..N!(1.37)Q1...QN)=P[a1...QN),II.na!1cPba,(rp1)ba(rp2)...a(rpN)(1.38)bSyman (ri.. n) VN!IIana!43

Partition function Z = Tre −β(Hˆ −µNˆ) . Grand canonical potential Ω = − 1 β lnZ. (1.24) Expectation values D Rˆ E = TrˆρR, ˆ (1.25) ρˆ = 1 Z e −β(Hˆ −µNˆ) , (1.26) Thermodynamic relations can be inferred from the statistical mechanics − ∂Ω ∂µ = 1 Z TrNe ˆ −β(Hˆ −µNˆ) ≡ N, (1.27) − ∂Ω ∂T = − Ω − D Hˆ − µNˆ E T ≡ S, (1.28) − ∂Ω ∂V = P. (1.29) Note that in the thermodynamic limit, Ω must be proportional to V . Therefore Ω = −P V . 1.3 Identical particles NO§1.2 The quantum mechanics can be generalized for many-particle systems. Product states can be constructed from orthonormal single particle states |α⟩: |α1 . . . αN ) ≡ |α1⟩ ⊗ |α2⟩ ⊗ · · · ⊗ |αN ⟩. (1.30) Note that we use |) to denote the product states. Overlap between product states: (α1 . . . αN |α ′ 1 . . . α′ N ) = ⟨α1 | α ′ 1 ⟩ ⟨α2 | α ′ 2 ⟩. . .⟨αN | α ′ N ⟩, (1.31) ψα1.αN (r1 . . . rN ) ≡ (r1 . . . rN |α1 . . . αN ) = ψα1 (r1)ψα2 (r2). . . ψαN (rN ). (1.32) Closure relation X α1.αN |α1 . . . αN ) (α1 . . . αN | = 1. (1.33) Exchange symmetry Only totally symmetric (Bosons) and anti-symmetric states (Fermions) are observed in nature: ψ (rP 1, rP 2, . . . , rP N ) = ψ (r1, r2, . . . , rN ) (Bosons), (1.34) ψ (rP 1, rP 2, . . . , rP N ) = (−1)P ψ (r1, r2, . . . , rN ) (Fermions). (1.35) Statistics theorem: Bosons (Fermions) have integer (half-integer) spins. Normalize symmetrized states are constructed from the product states by applying symmetriza￾tions: Pˆψ (r1, r2, . . . , rN ) = 1 N! X P ζ P ψ (rP 1, rP 2, . . . , rP N ), (1.36) |α1 . . . αN ⟩ = s N! Q α nα! P | ˆ α1 . . . αN ), (1.37) ψ Sym. α1.αN (r1 . . . rN ) = 1 p N! Q α nα! X P ζ P ψα1 (rP 1)ψα2 (rP 2). . . ψαN (rP N ) (1.38) 3