G25.2651: Statstca Mechancs Notes for Lecture 11 I. PRINCIPLES OF QUANTUM STATISTICAL MECHANICS The probem of quantum statstca mechancs s the quantum mechanca treatment of an N-partce system. Suppose the correspondng N-partce cassca system has Cartesan coordnates and momenta and Hamtonan q 1,..., q 3N p 1,..., p 3N H = 3N =1 p 2 2m + U(q 1,..., q 3N ) Then, as we have seen, the quantum mechanca probem conssts of determnng the state vector Ψ(t) from the Schrödnger equaton H Ψ(t) = t Ψ(t) Denotng the correspondng operators, Q 1,..., Q 3N and P 1,..., P 3N, we note that these operators satsfy the commutaton reatons: [Q, Q j ] = [P, P j ] = 0 [Q, P j ] = Iδ j and the many-partce coordnate egenstate q 1...q 3N s a tensor product of the ndvdua egenstate q 1,..., q 3N : q 1...q 3N = q 1 q 3N The Schrödnger equaton can be cast as a parta dfferenta equaton by mutpyng both sdes by q 1...q 3N : [ 3N =1 h 2 2m 2 q 2 + U(q 1,..., q 3N ) q 1...q 3N H Ψ(t) = t q 1...q 3N Ψ(t) ] Ψ(q 1,..., q 3N, t) = t Ψ(q 1,..., q 3N, t) where the many-partce wave functon s Ψ(q 1,..., q 3N, t) = q 1...q 3N Ψ(t). Smary, the expectaton vaue of an operator A = A(Q 1,..., Q 3N, P 1,..., P 3N ) s gven by ( A = dq 1 dq 3N Ψ (q 1,..., q 3N )A q 1,..., q 3N, h,..., h ) Ψ(q 1,..., q 3N ) q 1 q 3N 1
A. The densty matrx and densty operator In genera, the many-body wave functon Ψ(q 1,..., q 3N, t) s far too arge to cacuate for a macroscopc system. If we wsh to represent t on a grd wth just 10 ponts aong each coordnate drecton, then for N = 10 23, we woud need 10 1023 tota ponts, whch s ceary enormous. We wsh, therefore, to use the concept of ensembes n order to express expectaton vaues of observabes A wthout requrng drect computaton of the wavefuncton. Let us, therefore, ntroduce an ensembe of systems, wth a tota of members, and each havng a state vector Ψ (α), α = 1,...,. Furthermore, ntroduce an orthonorma set of vectors φ ( φ φ j = δ j ) and expand the state vector for each member of the ensembe n ths orthonorma set: Ψ (α) = φ The expectaton vaue of an observabe, averaged over the ensembe of systems s gven by the average of the expectaton vaue of the observabe computed wth respect to each member of the ensembe: A = 1 Substtutng n the expanson for Ψ (α), we obtan Let us defne a matrx and a smar matrx Ψ (α) A Ψ (α) A = 1, = ( 1, ρ = ρ = 1 φ A φ ) φ A φ Thus, ρ s a sum over the ensembe members of a product of expanson coeffcents, whe ρ s an average over the ensembe of ths product. Aso, et A = φ A φ. Then, the expectaton vaue can be wrtten as foows: A = 1 ρ A = 1 (ρa) = 1 Tr(ρA) = Tr( ρa), where ρ and A represent the matrces wth eements ρ and A n the bass of vectors { φ }. The matrx ρ s nown as the densty matrx. There s an abstract operator correspondng to ths matrx that s bass-ndependent. It can be seen that the operator and smary ρ = Ψ (α) Ψ (α) ρ = 1 Ψ (α) Ψ (α) 2
have matrx eements ρ when evauated n the bass set of vectors { φ }. φ ρ φ = Note that ρ s a hermtan operator φ Ψ (α) Ψ (α) φ = ρ = ρ = ρ so that ts egenvectors form a compete orthonorma set of vectors that span the Hbert space. If w and w represent the egenvaues and egenvectors of the operator ρ, respectvey, then severa mportant propertes they must satsfy can be deduced. Frsty, et A be the dentty operator I. Then, snce I = 1, t foows that 1 = 1 Tr(ρ) = Tr( ρ) = w Thus, the egenvaues of ρ must sum to 1. Next, et A be a projector onto an egenstate of ρ, A = w w P. Then But, snce ρ can be expressed as P = Tr( ρ w w ) ρ = w w w and the trace, beng bass set ndependent, can be therefore be evauated n the bass of egenvectors of ρ, the expectaton vaue becomes P = j w j w w w w w w j =,j w δ j δ δ j = w However, P = 1 = 1 Ψ (α) w w Ψ (α) Ψ (α) w 2 0 Thus, w 0. Combnng these two resuts, we see that, snce w = 1 and w 0, 0 w 1, so that w satsfy the propertes of probabtes. Wth ths n mnd, we can deveop a physca meanng for the densty matrx. Let us now consder the expectaton vaue of a projector a a P a onto one of the egenstates of the operator A. The expectaton vaue of ths operator s gven by P a = 1 Ψ (α) P a Ψ (α) = 1 Ψ (α) a a Ψ (α) = 1 a Ψ (α) 2 But a Ψ (α) 2 P a (α) s just probabty that a measurement of the operator A n the αth member of the ensembe w yed the resut a. Thus, P a = 1 3 P P (α) a
or the expectaton vaue of P a s just the ensembe averaged probabty of obtanng the vaue a n each member of the ensembe. However, note that the expectaton vaue of P a can aso be wrtten as P a = Tr( ρp a ) = Tr( w w w a a ) =, w w w w a a w =, = w δ w a a w w a w 2 Equatng the two expressons gves 1 P a (α) = w a w 2 The nterpretaton of ths equaton s that the ensembe averaged probabty of obtanng the vaue a f A s measured s equa to the probabty of obtanng the vaue a n a measurement of A f the state of the system under consderaton were the state w, weghted by the average probabty w that the system n the ensembe s n that state. Therefore, the densty operator ρ (or ρ) pays the same roe n quantum systems that the phase space dstrbuton functon f(γ) pays n cassca systems. B. Tme evouton of the densty operator The tme evouton of the operator ρ can be predcted drecty from the Schrödnger equaton. Snce ρ(t) s gven by the tme dervatve s gven by ρ t = 1 [H, ρ] ρ t = = 1 ρ(t) = Ψ (α) (t) Ψ (α) (t) [( ) ( )] t Ψ(α) (t) Ψ (α) (t) + Ψ (α) (t) t Ψ(α) (t) = 1 (Hρ ρh) = 1 [H, ρ] [( ) ( )] H Ψ (α) (t) Ψ (α) (t) Ψ (α) (t) Ψ (α) (t) H where the second ne foows from the fact that the Schrödnger equaton for the bra state vector Ψ (α) (t) s t Ψ(α) (t) = Ψ (α) (t) H Note that the equaton of moton for ρ(t) dffers from the usua Hesenberg equaton by a mnus sgn! Snce ρ(t) s constructed from state vectors, t s not an observabe e other hermtan operators, so there s no reason to expect that ts tme evouton w be the same. The genera souton to ts equaton of moton s ρ(t) = e Ht/ h ρ(0)e Ht/ h = U(t)ρ(0)U (t) The equaton of moton for ρ(t) can be cast nto a quantum Louve equaton by ntroducng an operator 4
In term of L, t can be seen that ρ(t) satsfes L = 1 [..., H] ρ t = Lρ ρ(t) = e Lt ρ(0) What nd of operator s L? It acts on an operator and returns another operator. Thus, t s not an operator n the ordnary sense, but s nown as a superoperator or tetradc operator (see S. Muame, Prncpes of Nonnear Optca Spectroscopy, Oxford Unversty Press, New Yor (1995)). Defnng the evouton equaton for ρ ths way, we have a perfect anaogy between the densty matrx and the state vector. The two equatons of moton are t Ψ(t) = ī h H Ψ(t) ρ(t) = Lρ(t) t We aso have an anaogy wth the evouton of the cassca phase space dstrbuton f(γ, t), whch satsfes f t = Lf wth L = {..., H} beng the cassca Louve operator. Agan, we see that the mt of a commutator s the cassca Posson bracet. C. The quantum equbrum ensembes At equbrum, the densty operator does not evove n tme; thus, ρ/t = 0. Thus, from the equaton of moton, f ths hods, then [H, ρ] = 0, and ρ(t) s a constant of the moton. Ths means that t can be smutaneousy dagonazed wth the Hamtonan and can be expressed as a pure functon of the Hamtonan ρ = f(h) Therefore, the egenstates of ρ, the vectors, we caed w are the egenvectors E of the Hamtonan, and we can wrte H and ρ as H = ρ = E E E f(e ) E E The choce of the functon f determnes the ensembe. 1. The mcrocanonca ensembe Athough we w have practcay no occason to use the quantum mcrocanonca ensembe (we reed on t more heavy n cassca statstca mechancs), for competeness, we defne t here. The functon f, for ths ensembe, s f(e )δe = θ(e (E + δe)) θ(e E) where θ(x) s the Heavsde step functon. Ths says that f(e )δe s 1 f E < E < (E + δe) and 0 otherwse. The partton functon for the ensembe s Tr(ρ), snce the trace of ρ s the number of members n the ensembe: Ω(N, V, E) = Tr(ρ) = [θ(e (E + δe)) θ(e E)] 5
The thermodynamcs that are derved from ths partton functon are exacty the same as they are n the cassca case: etc. S(N, V, E) = n Ω(N, V, E) ( ) 1 n Ω T = E N,V 2. The canonca ensembe In anaogy to the cassca canonca ensembe, the quantum canonca ensembe s defned by ρ = e βh f(e ) = e βe Thus, the quantum canonca partton functon s gven by Q(N, V, T) = Tr(e βh ) = e βe and the thermodynamcs derved from t are the same as n the cassca case: etc. Note that the expectaton vaue of an observabe A s A(N, V, T) = 1 n Q(N, V, T) β E(N, V, T) = nq(n, V, T) β P(N, V, T) = 1 nq(n, V, T) β V A = 1 Q Tr(Ae βh ) Evauatng the trace n the bass of egenvectors of H (and of ρ), we obtan A = 1 E Ae βh E = 1 e βe E A E Q Q The quantum canonca ensembe w be partcuary usefu to us n many thngs to come. 3. Isotherma-sobarc and grand canonca ensembes Aso usefu are the sotherma-sobarc and grand canonca ensembes, whch are defned just as they are for the cassca cases: (N, P, T) = (µ, V, T) = 0 dv e βpv Q(N, V, T) = e βµn Q(N, V, T) = N=0 N=0 0 dv Tr(e β(h+pv ) ) Tr(e β(h µn) ) 6
D. A smpe exampe the quantum harmonc oscator As a smpe exampe of the trace procedure, et us consder the quantum harmonc oscator. The Hamtonan s gven by and the egenvaues of H are Thus, the canonca partton functon s E n = Q(β) = H = P 2 2m + 1 2 mω2 X 2 ( n + 1 ) hω, n = 0, 1, 2,... 2 e β(n+1/2) hω = e β hω/2 ( e β hω ) n n=0 Ths s a geometrc seres, whch can be summed anaytcay, gvng Q(β) = e β hω/2 1 e β hω = 1 e β hω/2 e β hω/2 = 1 2 csch(β hω/2) The thermodynamcs derved from t as as foows: 1. Free energy: The free energy s n=0 A = 1 β nq(β) = hω 2 + 1 β n( 1 e β hω) 2. Average energy: The average energy E = H s E = β nq(β) = hω 2 + hωe β hω 1 e β hω = ( ) 1 2 + n hω 3. Entropy The entropy s gven by S = nq(β) + E T = n ( 1 e β hω) + hω T e β hω 1 e β hω Now consder the cassca expressons. Reca that the partton functon s gven by ( ) Q(β) = 1 dpdxe β p 2 2m + 1 2 mω2 x 2 = 1 ( ) 1/2 ( ) 1/2 2πm 2π h h β βmω 2 = 2π βωh = 1 β hω Thus, the cassca free energy s A c = 1 β n(β hω) In the cassca mt, we may tae h to be sma. Thus, the quantum expresson for A becomes, approxmatey, n ths mt: and we see that A Q hω 2 + 1 β n(β hω) A Q A c hω 2 The resdua hω/2 (whch truy vanshes when h 0) s nown as the quantum zero pont energy. It s a pure quantum effect and s present because the owest energy quantum mechancay s not E = 0 but the ground state energy E = hω/2. 7