Quantum Statistical Mechanics

Chapter 8 Quantum Statstcal Mechancs 8.1 Mcrostates For a collecton of N partcles the classcal mcrostate of the system s unquely specfed by a vector (q, p) (q 1...q N, p 1...p 3N )(q 1...q 3N,p 1...p 3N ) (8.1) n 6N dmensonal phase space. The quantum mcrostate s unquely specfed by a vector ψ n nfnte dmensonal Hlbert space. The vector can be wrtten n terms of ts components n the poston bases, ( ) ψ q q dq ψ q ψ q dq (8.2) where s the dentty operator Î q q dq Ψ(q) q ψ. (8.3) s a complex valued functon (wave functon) wth normalzaton dq Ψ(q) 2 dqψ(q) Ψ(q) 1. (8.4) In classcal statstcal mechancs observables O(q, p) are functons of q p. Inquantumstatstcalmechancsthecorrespondngobservables are the Hermtan operators usually obtaned by substtuton of operatorsˆq ˆp actng on ket-vectors (or q actng on wave functons) on place q 59

CHAPTER 8. QUANTUM STATISTICAL MECHANICS 60 of coordnates q p (.e. O(q, p) O(ˆq, ˆp)). However, because of the operator orderng ambguty the choce of a quantum observable mght not be unque, (e.g. pq ˆqˆp or ˆpˆq or 1 (ˆqˆp +ˆpˆq)...) the choce s usually 2 dctated by Hermtcty of the operator. Moreover, the value ofanoperator for a gven mcrostate s not unquely determned, but ts expectaton value can be obtaned from Born rule, ψ Ô ψ dqψ(q) ÔΨ(q). (8.5) 8.2 Densty matrx If the exact knowledge of the quantum mcrostate s not avalable, the system s sad to be n a not pure, butmxed state. Suchstatesarenotspecfedby aunquevectornhlbertspace,butbyacollectonofvector ϕ α } wth relatve probabltes p α },suchthat p α 1. (8.6) And the entropy of a mxed state s defned as α S α p α log p α. (8.7) Then, the ensemble average of a gven operator Ô s gven by Ô α p α ϕ α Ô ϕ α. (8.8) In a gven set of orthonormal bass, ψ n, the above expresson takes the followng form Ô p α ϕ α ψ n ψ n Ô ψ m ψ m ϕ α α,n,m n,m ( ) p α ψ m ϕ α ϕ α ψ n ψ n Ô ψ m α n,m ψ m ˆρ ψ n ψ n Ô ψ m n,m ψ m ˆρÔ ψ m Tr(ˆρÔ) (8.9)

CHAPTER 8. QUANTUM STATISTICAL MECHANICS 61 where the so-called densty matrx s defned as ˆρ α p α ϕ α ϕ α. (8.10) Ths s the Hermtan operator whch replaces the probablty dstrbuton functon n classcal phase space. Let ψ n be the energy egenstates, then ψ n t ˆρ(t) ψ m p α ψ n ϕ α (t) ϕ α (t) ψ m t α p α ( ψ n t ϕ α ϕ α ψ m + ψ n ϕ α ) t ϕ α ψ m α ( p α ( ψ n Ĥ ϕ α ϕ α ψ m + ψ n ϕ α ) ) t ψ m ϕ α α α p α (E n ψ n ϕ α ϕ α ψ m E m ψ n ϕ α ( ψ m ϕ α ) ) ψ n ˆρ (E n E m ) ψ m ψ n Ĥ ˆρ ˆρĤ ψ m (8.11) Thus, ndependently of bass we get the on Neumann equaton: ˆρ(t) [Ĥ, ˆρ], (8.12) t whch s a quantum verson of Louvlle s equaton (3.23) obtaned(once agan) by a formal substtuton of Posson brackets wth commutator,.e., } [, ]. 8.3 Statstcal ensembles In the equlbrum ˆρ(t) [Ĥ, ˆρ] 0 (8.13) t whch can be satsfed whenever ˆρ(Ĥ) s a functon of Hamltonan operator Ĥ. Thssuggestthefollowngstatstcalensembles 1. Mcrocanoncal ensemble: E) ρ(ĥ) δ(ĥ Ω(E) (8.14)

CHAPTER 8. QUANTUM STATISTICAL MECHANICS 62 In energy egenstates bass ψ n, ψ n ρ ψ m α p α ψ n ϕ α ϕ α ψ m 1 Ω(E) f E n E m n 0 otherwse (8.15) 2. Canoncal ensemble: where s the quantum partton functon. e βĥ ρ(ĥ) Z N (β) ( ) Z N (β) Tr e βĥ (8.16) (8.17) 3. Gr canoncal ensemble: where ˆN ρ(ĥ) e βĥ+βµ Z N (β) ( ) Z(z,, T) Tr e βĥ+βµ ˆN (8.18) (8.19) ˆN s an operator representng the number of partcles. For a sngle partcle n a box descrbed by Hamltonan The tme-ndependent Schrodnger equaton s the wave functons are gven by Ĥ 2. (8.20) 2m Ĥ ψ p E(p) ψ p (8.21) wth soluton 2 2m Ψ p(q) E(p)Ψ p (q) (8.22) Ψ p (q) ψ q ψ p 1 e p q. (8.23) The correspondng energes are E(p) 2 p 2 2m (8.24)

CHAPTER 8. QUANTUM STATISTICAL MECHANICS 63 the partton functon s Z 1 Tr(ρ) e β 2 p 2 2m (8.25) p dp β 2 p 2 (2π) 3 e 2m (8.26) ( ) 3/2 2m (2π) 3 dze z2 (8.27) 3 β ( ) 3/2 2πm (8.28) h 3 β λ 3 (8.29) where λ h 2πmkT. (8.30) 8.4 Indstngushable partcles The multpartcle Hlbert space s formed by a tensor product,.e. ψ q1,ψ q2,... ψ q1 ψ q2... (8.31) In quantum mechancs only certan states ψ correspondng to wave-functons Ψ(q 1, q 2,...) ψ q1,ψ q2,... ψ (8.32) are nterpreted as partcles n postons q 1, q 2,... For example, for a two partcles system the probablty of fndng the partcles n postons q 1 q 2 s gven by Born rule, Ψ(q 1, q 2 ) 2 ψ q1,ψ q2 ψ 2. (8.33) Clearly, f we exchange the two dentcal partcles the system would not be expermentally dstngushable. Ths leads to the dea of ndstngushable partcles. Indstngushable partcles: The square of wave functon s nvarant under nterchange of any par of partcles,.e. Ψ(..., q m,..., q n,...) 2 Ψ(..., q n,..., q m,...) 2. (8.34)

CHAPTER 8. QUANTUM STATISTICAL MECHANICS 64 There are two possble solutons (correspondng to bosons) (correspondng to fermons) Ψ(..., q m,..., q n,...) +Ψ(..., q n,..., q m,...) (8.35) Ψ(..., q m,..., q n,...) Ψ(..., q n,..., q m,...)., (8.36) but more generally we can have (correspondng to anyons) Ψ(..., q m,..., q n,...) e θ Ψ(..., q n,..., q m,...) wth θ 0nor θ π. For N partcles the total number of permutatons s N! whch forms a group S N of permutatons P S N actng on the Hlbert space. Clearly, (8.35) (8.36) putsomerestrctonsontheallowedstatesnthehlbert space allows only two types of ndstngushable partcles: Bosons, correspondng to symmetrc wave-functons P Ψ(q 1, q 2,..., q N ) ψ q1,ψ q2,..., ψ qn +Ψ(q 1, q 2,..., q N ) ψ q1,ψ q2,..., ψ qn (8.37) Fermons, correspondng to ant-symmetrc wave-functons P Ψ(q 1, q 2,..., q N ) ψ q1,ψ q2,..., ψ qn ( 1) p Ψ(q 1, q 2,..., q N ) ψ q1,ψ q2,..., ψ qn (8.38) where the party of permutatons s defned as ( 1) p +1 f P nvolves an even number of bnary exchanges 1 f P nvolves an odd number of bnary exchanges. Thus, to solve the system n addton to Hamltonan one must also specfy the allowed egenstates or the statstcs of partcles (.e. bosonsoffermons). Consder N partcles n a box of volume descrbed by Hamltonan Ĥ N N 2 Ĥ n 2m n. (8.39) n1 For each partcle separately t was already shown that the energy egenstates ψ p can be expressed n the poston bass as ψ p 1 e p q ψ q dq. (8.40) n1

CHAPTER 8. QUANTUM STATISTICAL MECHANICS 65 For non-nteractng partcles we can form a tensor product energy egenstate state,.e. ψ p1,ψ p2,..., ψ pn N 2 e N n1 pn qn ψ q1,ψ q2,..., ψ qn dq 1...dq N (8.41) such that Ĥ ψ p1,ψ p2,..., ψ pn 2 N n1 p2 n 2m ψ q1,ψ q2,..., ψ qn. (8.42) However, ths product vector does not satsfy bosonc (8.37), nor fermonc (8.38) statstcsmposedbythendstngushabltyassumpton. For ndstngushable partcles the state vector must satsfy ether p ψ p1,ψ p2,..., ψ pn + n p! N! P n ψ p1,ψ p2,..., ψ pn (8.43) N! for bosons or n1 n1 ψ p1,ψ p2,..., ψ pn 1 N! ( 1) Pn P n ψ p1,ψ p2,..., ψ pn (8.44) N! for fermons, where n p s the number of partcles wth wave vector p, n p N (8.45) p E p n p E. (8.46) p For two ndstngushable partcles the bosonc state vector s the fermonc state vector s ψ p1,ψ p2 + 1 2 ( ψ p1,ψ p2 + ψ p2,ψ p1 ) (8.47) ψ p1,ψ p2 + 1 2 ( ψ p1,ψ p2 ψ p2,ψ p1 ). (8.48) It should now be clear that two fermons cannot occupy the same state form condensate n contrast to bosons, 0, 1, 2,... for bosons n p (8.49) 0, 1 for fermons.

CHAPTER 8. QUANTUM STATISTICAL MECHANICS 66 8.5 Mcrocanoncal ensemble Let us dvde the energy spectrum nto cells contanng g 1, g 2,...levelswth average energes E 1, E 2,... occupaton numbers n 1, n 2,.... It s convenent to defne : w -thenumberofwaysn partcles can be assgned to -th cell wth g levels W n } as the number of states of the system correspondng to the set of occupaton numbers n }, Γ(E) as the number of states of the system wth energy egenvalue between E E +. Then, W n } w (8.50) Γ(E) n } W n } (8.51) where the sum s taken over all sets of occupaton numbers satsfyng n N (8.52) E n E. (8.53) For Bose gas each level can be occuped by an arbtrary number of partcles,.e. w (n + g 1)! (8.54) n!(g 1)! W n } (n + g 1)!. (8.55) n!(g 1)! For Ferm gas each level can be ether occuped by a sngle partcle or not,.e. g! w (8.56) n!(g n )! W n } g! n!(g n )!. (8.57)

CHAPTER 8. QUANTUM STATISTICAL MECHANICS 67 As before we approxmate the entropy as S k log Γ k log W n } (8.58) where n } s the set of occupaton numbers whch maxmzes W n }. We can fnd n by maxmzng (8.55) (8.57), subject to constrans (8.52) (8.53). The result for bosons s for fermons s n n g z 1 e βe 1 (8.59) g z 1 e βe +1, (8.60) where β (nverse temperature) z (fugacty) are determned from E n E (8.61) n N. (8.62) Usng Strlng s approxmaton g 1 we can calculate the entropy for a Bose gas S k log W n } (log( n + g 1)! log n! log(g 1)!) (( n + g ) log( n + g ) n log n g log g ) ( ( ) ( g n log +1 + g log 1+ n )) n g ( βe log z g log ( ) ) 1 ze βe (8.63) z 1 e βe 1

CHAPTER 8. QUANTUM STATISTICAL MECHANICS 68 for a Ferm gas S k log W n } (log(g )! log n! log(g n )!) (g log g n log n (g n ) log (g n )) ( ( ) ( g n log 1 g log 1 n )) n g ( βe log z g + log ( 1+ze ) ) βe. (8.64) z 1 e βe +1 Queston To Go: Can you defne a phenomenologcal gas (.e. W n p })whosestatstcssnetherbosoncnorfermonc? 8.6 Boltzmann gas The ndstngushablty of partcles assumpton was crucal for dervng equatons (8.63) (8.64). If we relax the assumpton then the countng goes as follows. There are N! n! ways to place N dstngushable partcles nto cells wth th cell havng n partcles. Each of the partcles can occupy g of the states n each cell. Thus, W n } N! g n n! f we use the correct Boltzmann countng (.e. 1 ), then N! W n } g n n! (8.65) the entropy of a Boltzmann gas would be gven by S k ( ) g n log z g e βe (βe log z) (8.66) n where n g ze βe. (8.67)

CHAPTER 8. QUANTUM STATISTICAL MECHANICS 69 In terms of energy levels n p ze βep where the constant z β are agan determned from (8.61) (8.62). The partton functon for the deal gas s Z N n p} gn p }e βenp} (8.68) where the number of states wth a gven set of occupaton numbers n p } s gven by 1 for Bose gas gn p } 1 for Ferm gas ( 1 ) (8.69) 1 N! N! p np! p n p! for Boltzmann gas. the occupaton numbers are also subject to constrans E p n p E (8.70) p n p N. (8.71) p For Boltzmann gas the partton functon can be re-expressedusngmultno- mal theorem Z N ( e βn p1 E p1 ) e βnp 2 Ep 2... 1 ( e βe p1 + e βe p2 ) N +... (8.72) n p1! n p2! N! n p} In a large volume lmt e βep p2 β dpe h 3 2m p Z N ( ) 3 ( ) 3/2 2m mkt dp h 3 β e p2 2πh 2 (8.73) ( ( ) ) 3/2 N mkt. (8.74) N 2πh 2 The gr partton functon for Boltzmann gas s gven by, ( ( ) ) 3/2 N Z(z,, T) z N z mkt Z N (,T). N 2πh 2 N0 N0

CHAPTER 8. QUANTUM STATISTICAL MECHANICS 70 8.7 Quantum gases The gr partton functons for quantum gases s estmated as follows, Z(z,, T) z N Z N (,T) N0 z N e β np Ep N0 n p} ( ze βe p ) np N0 n p}... np1 n p2 ( ze βe p ) np (8.75) whch can be further smplfed ( ) n0 ze βe p n Z(z,, T) ( ) n0,1 ze βe p n 1 1 ze βep The correspondng equaton of state s P kt log Z log ( ) 1 ze βep log ( ) 1+ze βep the average occupaton number s ( 1+ze βe p ) for Bose gas for Ferm gas. for Bose gas for Ferm gas. (8.76) (8.77) n p 1 β E p log Z ze βep 1 ze βep ze βep 1+ze βep The fugacty can now be elmnated usng N z 1 z log Z z 1 e βep 1 1 z 1 e βep +1 whch agrees wth the constrant (8.71) for Bose for Bose gas for Ferm gas. for Bose gas for Ferm gas. (8.78) (8.79) N 1 z 1 e βep 1 n p.

CHAPTER 8. QUANTUM STATISTICAL MECHANICS 71 Ferm N 1 z 1 e βep +1 n p. gases. In the lmt of nfnte volume the sum over energy levels n (8.77) (8.79) canbereplacedbyntegratonovermomentum,.e. dp, h 3 wth an excepton of p 0term whch mght dverge for a Bose gas when z 1. Ifweevaluatethepotentallydvergenttermseparately,then ) P kt dp4πp 2 p2 β log (1 ze h 3 2m log(1 z) for Bose gas 0 ) dp4πp 2 p2 β log (1+ze h 3 2m for Ferm gas. 0 dp4πp 2 1 h N 3 0 dp4πp 2 1 h 3 0 z 1 e βp2 /2m +1 Usng the followng denttes we obtan f 5/2 (z) f 3/2 (z) g 5/2 (z) g 3/2 (z) n1 n1 n1 n1 z 1 e βp2 /2m 1 + z n n 5/2 4 z n 4 n 3/2 ( 1) n+1 z n n 5/2 4 ( 1) n+1 z n n 3/2 4 z 1 z π 0 π 0 π 0 π 0 for Bose gas for Ferm gas. (8.80) (8.81) ( dxx 2 log 1 ze x2) (8.82) dxx 2 1 (8.83) z 1 e x2 1 ( dxx 2 log 1+ze x2) (8.84) dxx 2 1 z 1 e x2 +1 (8.85) P kt N g λ 3 5/2 (z) log(1 z) f λ 3 5/2 (z) g λ 3 3/2 (z)+ f λ 3 3/2 (z) z z 1 for Bose gas for Ferm gas for Bose gas for Ferm gas, (8.86) (8.87)

CHAPTER 8. QUANTUM STATISTICAL MECHANICS 72 where 2π 2 λ (8.88) mkt s the so-called thermal wavelength. Theequatonofstatecanbeobtaned by elmnatng fugacty from equatons (8.91) (8.93). Note, that for Bosonc systems at very low temperatures, when the thermal wavelength s comparable wth nter-partcle separaton,.e. λ (/N) 1/3, a fnte fracton of all partcles can occupy the ground state p 0 leadng to the Bose-Ensten condensaton. We have seen before that the classcal statstcal mechancscanbederved startng from the molecular dynamcs. The key assumpton n the knetc theory of gases s the molecular chaos assumpton whch breaks the tme reversal nvarance. For more general dynamcal systems the rreversblty can also be derved usng entropy producton defned as the sum ofpostve Lyapunov exponents. The queston that s often asked s whether t s possble to derve the rreversblty n a more general context of quantum physcs. At frst sght, the Schrodnger equaton s tme-symmetrc whch does not seem to help much. On the other h, there s the measurement postulate where the observer s able to break the tme-reversal symmetry at the tme when the observaton s made. Can the rreversblty be justfed n the context of quantum statstcal mechancs? 8.8 Quantum correctons To obtan the equaton of state we must elmnate fugacty z from for Bose gases, or from P kt λ 3 g 5/2(z) log(1 z) (8.89) N λ 3 g 3/2(z)+ z 1 z P kt (8.90) λ 3 f 5/2(z) (8.91) N λ 3 f 3/2(z) (8.92) for Ferm gases. Ths can be done perturbatvely ether n lowdenstes(large nter-partcle dstance) hgh temperature (small thermal wavelength).

CHAPTER 8. QUANTUM STATISTICAL MECHANICS 73 Snce f 3/2 (z) g 3/2 (z) are monotoncally ncreasng functon for small z, n the lmt (/N) 1/3 λ, wehave g 3/2 (z) z n n1 z + z2 f 3/2 (z) n1 wth approxmate solutons z n 3/2 ( 1) n+1 z n n 3/2 2 3/2 ( +2 3/2 ( +... for Bose gases 2 3/2 z z2 +... for Ferm gases 2 3/2 (8.93) ) 2 for Bose gases ) 2 for Ferm gases. (8.94) Then correspondng equaton of state can be obtaned by the so-called vral expanson (.e. expanson n the powers of densty) ( ( ( ) ) ( 2 ( ) ) ) 2 2 Nλ P 3 2 3/2 +2 5/2 2 3/2 +... λ 3 kt ) λ 3 ( ( +2 3/2 ( ) ) ( 2 ( 2 5/2 +2 3/2 ) ) 2 2 +... (8.95) by only keepng the leadng (quantum) correcton to the equaton of state of Boltzmann gas we obtan ( ) P 5/2 Nλ3 kt N 1 2 for Bose gases ( ) 5/2 Nλ3 1+2 for Ferm gases. (8.96) for Bose gases for Ferm gases