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 University 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 Semiconductors (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 associated 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 symmetrybreaking 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 symmetrizations: 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