A. Te densty matrx and densty operator In general, te many-body wave functon (q 1 ; :::; q 3N ; t) s far too large to calculate for a macroscopc syste

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1 G : Statstcal Mecancs Notes for Lecture 13 I. PRINCIPLES OF QUANTUM STATISTICAL MECHANICS Te problem of quantum statstcal mecancs s te quantum mecancal treatment of an N-partcle system. Suppose te correspondng N-partcle classcal system as Cartesan coordnates and momenta q 1 ; :::; q 3N p 1 ; :::; p 3N and Hamltonan H = 3N p 2 2m + U(q 1 ; :::; q 3N ) Ten, as we ave seen, te quantum mecancal problem conssts of determnng te state vector (t) from te Scrodnger equaton H (t) = t (t) Denotng te correspondng operators, Q 1 ; :::; Q 3N and P 1 ; :::; P 3N, we note tat tese operators satsfy te commutaton relatons: [Q ; Q ] = [P ; P ] = 0 [Q ; P ] = I and te many-partcle coordnate egenstate q 1 :::q 3N s a tensor product of te ndvdual egenstate q 1 ; :::; q 3N : q 1 :::q 3N = q 1 q 3N Te Scrodnger equaton can be cast as a partal derental equaton by multplyng bot sdes by q 1 :::q 3N : "? 3N 2 2m 2 q 2 + U(q 1 ; :::; q 3N ) q 1 :::q 3N H (t) = t q 1:::q 3N (t) # (q 1 ; :::; q 3N ; t) = t (q 1 ; :::; q 3N ; t) were te many-partcle wave functon s (q 1 ; ::::; q 3N ; t) = q 1 :::q 3N (t). Smlarly, te expectaton value of an operator A = A(Q 1 ; :::; Q 3N ; P 1 ; :::; P 3N ) s gven by A = dq 1 dq 3N (q 1 ; :::; q 3N )A q 1 ; :::; q 3N ; ; :::; (q 1 ; :::; q 3N ) q 1 q 3N 1

2 A. Te densty matrx and densty operator In general, te many-body wave functon (q 1 ; :::; q 3N ; t) s far too large to calculate for a macroscopc system. If we ws to represent t on a grd wt ust 10 ponts along eac coordnate drecton, ten for N = 10 23, we would need total ponts, wc s clearly enormous. We ws, terefore, to use te concept of ensembles n order to express expectaton values of observables A wtout requrng drect computaton of te wavefuncton. Let us, terefore, ntroduce an ensemble of systems, wt a total of members, and eac avng a state vector, = 1; :::;. Furtermore, ntroduce an ortonormal set of vectors ( = ) and expand te state vector for eac member of te ensemble n ts ortonormal set: = Te expectaton value of an observable, averaged over te ensemble of systems s gven by te average of te expectaton value of te observable computed wt respect to eac member of te ensemble: A = 1 Substtutng n te expanson for, we obtan Let us dene a matrx and a smlar matrx A = 1 = ;l ;l 1 l = ~ l = 1 A l A l l! l l A l Tus, l s a sum over te ensemble members of a product of expanson coecents, wle ~ l s an average over te ensemble of ts product. Also, let A l = A l. Ten, te expectaton value can be wrtten as follows: A = 1 ;l l A l = 1 (A) = 1 Tr(A) = Tr(~A) were and A represent te matrces wt elements l and A l n te bass of vectors f g. Te matrx l s nown as te densty matrx. Tere s an abstract operator correspondng to ts matrx tat s bass-ndependent. It can be seen tat te operator and smlarly = ~ = 1 2

3 ave matrx elements l wen evaluated n te bass set of vectors f g. l = Note tat s a ermtan operator l = y = l = l so tat ts egenvectors form a complete ortonormal set of vectors tat span te Hlbert space. If w and w represent te egenvalues and egenvectors of te operator ~, respectvely, ten several mportant propertes tey must satsfy can be deduced. Frstly, let A be te dentty operator I. Ten, snce I = 1, t follows tat 1 = 1 Tr = Tr(~) = w Tus, te egenvalues of ~ must sum to 1. Next, let A be a proector onto an egenstate of ~, A = w w P. Ten But, snce ~ can be expressed as P = Tr(~w w ) ~ = w w w and te trace, beng bass set ndependent, can be terefore be evaluated n te bass of egenvectors of ~, te expectaton value becomes P = w w w w w w w = ; w = w However, P = 1 = 1 w w w 2 0 Tus, w 0. Combnng tese two results, we see tat, snce P w = 1 and w 0, 0 w 1, so tat w satsfy te propertes of probabltes. Wt ts n mnd, we can develop a pyscal meanng for te densty matrx. Let us now consder te expectaton value of a proector a a P a onto one of te egenstates of te operator A. Te expectaton value of ts operator s gven by P a = 1 P a = 1 a a = 1 But a 2 P a s ust probablty tat a measurement of te operator A n te t member of te ensemble wll yeld te result a. Tus, P a = 1 3 P P a a 2

4 or te expectaton value of P a s ust te ensemble averaged probablty of obtanng te value a n eac member of te ensemble. However, note tat te expectaton value of P a can also be wrtten as P a = Tr(~P a ) = Tr( w w w a a ) = ;l w l w w w a a w l = ;l = w l w a a w l w a w 2 Equatng te two expressons gves 1 P a = w a w 2 Te nterpretaton of ts equaton s tat te ensemble averaged probablty of obtanng te value a f A s measured s equal to te probablty of obtanng te value a n a measurement of A f te state of te system under consderaton were te state w, wegted by te average probablty w tat te system n te ensemble s n tat state. Terefore, te densty operator (or ~) plays te same role n quantum systems tat te pase space dstrbuton functon f(?) plays n classcal systems. B. Tme evoluton of te densty operator Te tme evoluton of te operator can be predcted drectly from te Scrodnger equaton. Snce (t) s gven by te tme dervatve s gven by t = 1 [H; ] t = = 1 (t) = t (t) H = 1 (H? H) = 1 [H; ] (t) (t) (t) (t) + (t)? (t) t (t) (t) (t)h were te second lne follows from te fact tat te Scrodnger equaton for te bra state vector (t) s? t (t) = (t)h Note tat te equaton of moton for (t) ders from te usual Hesenberg equaton by a mnus sgn! Snce (t) s constructed from state vectors, t s not an observable le oter ermtan operators, so tere s no reason to expect tat ts tme evoluton wll be te same. Te general soluton to ts equaton of moton s (t) = e?ht= (0)e Ht= = U(t)(0)U y (t) Te equaton of moton for (t) can be cast nto a quantum Louvlle equaton by ntroducng an operator 4

5 In term of L, t can be seen tat (t) satses L = 1 [:::; H] t =?L (t) = e?lt (0) Wat nd of operator s L? It acts on an operator and returns anoter operator. Tus, t s not an operator n te ordnary sense, but s nown as a superoperator or tetradc operator (see S. Muamel, Prncples of Nonlnear Optcal Spectroscopy, Oxford Unversty Press, New Yor (1995)). Denng te evoluton equaton for ts way, we ave a perfect analogy between te densty matrx and te state vector. Te two equatons of moton are t (t) =? H (t) (t) =?L(t) t We also ave an analogy wt te evoluton of te classcal pase space dstrbuton f(?; t), wc satses f t =?Lf wt L = f:::; Hg beng te classcal Louvlle operator. Agan, we see tat te lmt of a commutator s te classcal Posson bracet. C. Te quantum equlbrum ensembles At equlbrum, te densty operator does not evolve n tme; tus, =t = 0. Tus, from te equaton of moton, f ts olds, ten [H; ] = 0, and (t) s a constant of te moton. Ts means tat t can be smultaneously dagonalzed wt te Hamltonan and can be expressed as a pure functon of te Hamltonan = f(h) Terefore, te egenstates of, te vectors, we called w are te egenvectors E of te Hamltonan, and we can wrte H and as H = = E E E f(e )E E Te coce of te functon f determnes te ensemble. 1. Te mcrocanoncal ensemble Altoug we wll ave practcally no occason to use te quantum mcrocanoncal ensemble (we reled on t more eavly n classcal statstcal mecancs), for completeness, we dene t ere. Te functon f, for ts ensemble, s f(e )E = (E? (E + E))? (E? E) were (x) s te Heavsde step functon. Ts says tat f(e )E s 1 f E < E < (E + E) and 0 oterwse. Te partton functon for te ensemble s Tr, snce te trace of s te number of members n te ensemble: (N; V; E) = Tr = [(E? (E + E))? (E? E)] 5

6 Te termodynamcs tat are derved from ts partton functon are exactly te same as tey are n te classcal case: etc. S(N; V; E) =? ln (N; V; E) 1 ln T =? E N;V 2. Te canoncal ensemble In analogy to te classcal canoncal ensemble, te quantum canoncal ensemble s dened by = e?h f(e ) = e?e Tus, te quantum canoncal partton functon s gven by Q(N; V; T ) = Tr(e?H ) = e?e and te termodynamcs derved from t are te same as n te classcal case: A(N; V; T ) =? 1 ln Q(N; V; T ) E(N; V; T ) =? ln Q(N; V; T ) P (N; V; T ) = 1 ln Q(N; V; T ) V etc. Note tat te expectaton value of an observable A s A = 1 Q Tr(Ae?H ) Evaluatng te trace n te bass of egenvectors of H (and of ), we obtan A = 1 Q E Ae?H E = 1 Q e?e E AE Te quantum canoncal ensemble wll be partcularly useful to us n many tngs to come. 3. Isotermal-sobarc and grand canoncal ensembles Also useful are te sotermal-sobarc and grand canoncal ensembles, wc are dened ust as tey are for te classcal cases: (N; P; T ) = (; V; T ) = dv e?p V Q(N; V; T ) = e N Q(N; V; T ) = 1 N=0 N=0 1 0 dv Tr(e?(H+P V ) ) Tr(e?(H?N) ) 6

7 D. A smple example { te quantum armonc oscllator As a smple example of te trace procedure, let us consder te quantum armonc oscllator. Te Hamltonan s gven by and te egenvalues of H are Tus, te canoncal partton functon s E n = Q = 1 n=0 H = P 2 n m m!2 2!; n = 0; 1; 2; ::: e?(n+1=2)! = e?!=2 Ts s a geometrc seres, wc can be summed analytcally, gvng Q = e?!=2 1? e = 1?! e!=2? e?!=2 Te termodynamcs derved from t as as follows: 1. Free energy: Te free energy s 1? e?! n n=0 = 1 2 csc(!=2) A =? 1 ln Q =! ln? 1? e?! 2. Average energy: Te average energy E = H s E =? ln Q =! 2 +!e?! 1? e?! = n! 3. Entropy Te entropy s gven by S = ln Q + E T =? ln? 1? e?! +! T e?! 1? e?! Now consder te classcal expressons. Recall tat te partton functon s gven by Q = 1 dpdxe? p 2 2m m!2 x 2 = 1 2m Tus, te classcal free energy s A cl = 1 ln(!) 1=2 2 m! 2 1=2 = 2! = 1! In te classcal lmt, we may tae to be small. Tus, te quantum expresson for A becomes, approxmately, n ts lmt: and we see tat A Q?!! ln(!) A Q? A cl?!! 2 Te resdual!=2 (wc truly vanses wen! 0) s nown as te quantum zero pont energy. It s a pure quantum eect and s present because te lowest energy quantum mecancally s not E = 0 but te ground state energy E =!=2. 7

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by 6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng