Suppose experimentalist A prepares a stream of silver atoms such that each atom is in the spin state ψ : ( ) = +. 2

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1 The Desty Matr Mchael Fowler /9/07 Pure States ad Med States Our treatmet here more or less follows that of Sakura, begg wth two maged Ster- Gerlach epermets. I that epermet, a stream of (o-ozed) slver atoms from a ove s drected through a homogeeous vertcal magetc feld, ad the stream splts to two. The slver atoms have ozero magetc momets, ad a magetc momet a homogeeous magetc feld epereces a ozero force, causg the atom to veer from ts straght le path, the magtude of the deflecto beg proportoal to the compoet of the atom s magetc momet the vertcal (feld) drecto. The observato of the beam splttg to two, ad o more, meas that the vertcal compoet of the magetc momet, ad therefore the assocated agular mometum, ca oly have two dfferet values. From the basc aalyss of rotato operators ad the propertes of agular mometum that follow, ths observato forces us to the cocluso that the total agular mometum of a slver atom s. Ordary orbtal agular mometa caot have half-teger values; ths epermet was oe of the frst dcatos that the electro has a sp degree of freedom, a agular mometum that caot be terpreted as orbtal agular mometum of costtuet parts. The slver atom has 47 electros, 46 of them have total sp ad orbtal mometa that separately cacel, the 47 th has o orbtal agular mometum, ad ts sp s the etre agular mometum of the atom. Here we shall use the Ster-Gerlach stream as a eample of a large collecto of quatum systems (the atoms) to clarfy ust how to descrbe such a collecto, ofte called a esemble. To avod uecessary complcatos, we oly cosder the sp degrees of freedom. We beg by eamg two dfferet streams: Suppose epermetalst prepares a stream of slver atoms such that each atom s the sp state : ( ) +. Meawhle, epermetalst B prepares a stream of slver atoms whch s a mture: half the atoms are state ad half are the state : call ths m B. Questo: ca we dstgush the stream from the B stream? Evdetly, ot by measurg the sp the z-drecto! Both wll gve up 50% of the tme, dow 50%. But: we ca dstgush them by measurg the sp the -drecto: the quatum state s fact ust that of a sp the -drecto, so t wll gve up the -drecto every tme

2 from ow o we call t -drecto oly 50% of the tme, as wll., whereas the state ( up the z-drecto) wll yeld up the The state s called a pure state, t s the kd of quatum state we ve bee studyg ths whole course. The stream B, cotrast, s a med state: the kd that actually occurs to a greater or lesser etet a real lfe stream of atoms, dfferet pure quatum states occurrg wth dfferet probabltes, but wth o phase coherece betwee them. I other words, these relatve probabltes B of dfferet quatum states do ot derve from probablty ampltudes, as they do fdg the probablty of sp up stream : the probabltes of the dfferet quatum states the med state B are eactly lke classcal probabltes. That beg sad, though, to fd the probablty of measurg sp up some such med state, oe frst uses the classcal-type probablty for each compoet state, the for each quatum state the m, oe fds the probablty of sp up that state by the stadard quatum techque. Theerefore, for a med state whch the system s state the epectato value of a operator  s wth probablty w, w, ˆ ˆ w ad we should emphasze that these do ot eed to be orthogoal (but they are of course ormalzed): for eample oe could be, aother z. (We put the usually omtted z for emphass.) The reaso we put a hat o  here s to emphasze that ths s a operator, but the w are ust umbers. The Desty Matr The equato for the epectato value  ca be wrtte: ( ˆ ˆ) ˆ Trace ρ where ˆ ρ w. To see eactly how ths comes about, recall that for a operator ˆB a fte-dmesoal vector space wth a orthoormal bass set, TrBˆ Bˆ B, where the repeated suff mples summato of the dagoal matr elemets of the operator. Therefore,

3 3 sce I, the detty. ( ˆ ˆ) Tr ρ w ˆ w w ˆ ˆ Ths ˆρ s called the desty matr: ts matr form s made eplct by cosderg states a fte N-dmesoal vector space (such as sps or agular mometa) ( V ) where the are a orthoormal bass set, ad ( V ) s the th compoet of a ormalzed vector V. It s coveet to epress ˆρ terms of kets ad bras belogg to ths orthoormal bass, ad evdetly w w ( V ) ( V ) k k ˆ k k,, k, k ρ ρ ( ˆ ) ˆ Trace ρˆ ρ k ˆ ρ k ˆ ρ. k k k k,, k, k, k (Sce ρ s ust a umber, ρ k ρk ρkδ.) k Trace( ˆ ρ ˆ ) s bass-depedet, the trace of a matr beg uchaged by a utary trasformato, sce t follows from TrBC TrBC that TrU U Tr UU Tr for UU. Note that sce the vectors V are ormalzed, ( ) ( ) w, t follows that ˆ Tr ρ (also evdet by puttg the equato for ). V V, wth the ot summed over, ad

4 4 For a system a pure quatum state, ˆρ, ust the proecto operator to that state, ad ˆ ρ ˆ ρ, as for all proecto operators. It s worth spellg out how ths dffers from the med state by lookg at the form of the desty matr. For the pure state, f a bass s chose so that s a member of the bass (ths ca always be doe), ˆρ s a matr wth every elemet zero ecept the oe dagoal elemet correspodg to, whch wll be uty. Obvously, ˆ ρ ˆ ρ. Ths s less obvous a geeral bass, where ˆρ wll ot ecessarly be dagoal. But the statemet ˆ ρ ˆ ρ remas true uder a trasformato to a ew bass. For a med state, let s say for eample a mture of orthogoal states,, f we choose a bass cludg both states, the desty matr wll be dagoal wth ust two etres w, w. Both these umbers must be less tha uty, so ˆ ρ ˆ ρ. m of oorthogoal states s left as a eercse for the reader. Some Smple Eamples Frst, our case above (pure state): all sps state ( / )( ) +. I the stadard, bass, / / / ˆ ρ ( / / ) / / / ad s / / 0 Tr( ˆ ρs) Tr / / 0 Notce that ˆ ρ ˆ ρ. s z / / 0 Tr( ˆ ρsz) Tr 0. / / 0

5 5 Now, case B (50-50 med up ad dow): 50% the state, 50%. The desty matr s ˆ ρ ( 0) + ( 0 ). 0 0 Ths s proportoal to the ut matr, so Tr ˆ ρs Trσ 0, ad smlarly for s y ad s z, sce the Paul σ-matrces are all traceless. Note also that ˆ ρ ˆ ρ ˆ ρ, as s true for all med states. Fally, a med state relatve to the -as: That s, 50% of the sps the state ( / )( ) ( / )( ), dow the -drecto. +, up alog the -as, ad 50% It s easy to check that / / / / 0 ˆ ρ + +. / / / / 0 Ths s eactly the same desty matr we foud for 50% the state, 50%! The reaso s that both formulatos descrbe a state about whch we kow othg we are a state of total gorace, the sps are completely radom, all drectos are equally lkely. The desty matr descrbg such a state caot deped o the drecto we choose for our aes. other two-state quatum system that ca be aalyzed the same way s the polarzato state of a beam of lght, the bass states beg polarzato the -drecto ad polarzato the y- drecto, for a beam travelg parallel to the z-as. Ordary upolarzed lght correspods to the radom med state, wth the same desty matr as the last eample above.

6 6 Tme Evoluto of the Desty Matr I the med state, the quatum states evolve depedetly accordg to Schrödger s equato, so d ˆ ρ wh [, ˆ w H H ]. dt ρ Note that ths has the opposte sg from the evoluto of a Heseberg operator, ot surprsg sce the desty operator s made up of Schrödger bras ad kets. The equato s the quatum aalogue of Louvlle s theorem statstcal mechacs. Louvlle s theorem descrbes the evoluto tme of a esemble of detcal classcal systems, such as may boes each flled wth the same amout of the same gas at the same temperature, but the postos ad mometa of the dvdual atoms are radomly dfferet each. Each bo ca be classcally descrbed by a sgle pot a huge dmesoal space, a space havg s dmesos for each atom (posto ad mometum, we gore possble teral degrees of freedom). The whole esemble, the, s a gas of these pots ths huge space, ad the rate of chage of local desty of ths gas, from Hamlto s equatos, s ρ/ t { ρ, H }, the bracket ow beg a Posso bracket (see Classcal Mechacs). yway, ths s the classcal precursor of, ad the reaso for the ame of, the desty matr. Thermal Equlbrum system thermal equlbrum s represeted statstcal mechacs by a caocal esemble. If the egestate of the Hamltoa has eergy E, the relatve probablty of the system beg E / that state s kt β E e e the stadard otato. Therefore the desty matr s: where β H β E e ˆ ρ e, Z Z β E β H Z e Tr e. Notce that ths formulato, apart from the ormalzato costat Z, the desty operator s / aalogous to the propagator U() t e Ht for a magary tme t β. Icdetally, for teractg quatum felds, the propagator ca be costructed as a set of Feyma dagrams correspodg to all possble sequeces of partcle scattergs by teracto. To fd the thermodyamc propertes of a feld theory at fte temperature, essetally the same set of dagrams s used to fd the free eergy: the dagrams ow descrbe the system propagatg for a fte magary tme, the same mathematcal tools ca be used. β E t zero temperature ( β ) the probablty coeffcets w e / Z are all zero ecept for the groud state: the system s a pure state, ad the desty matr has every elemet zero ecept for a sgle elemet o the dagoal. t fte temperature, all the w are equal: the desty

7 7 matr s ust /N tmes the ut matr, where N s the total umber of states avalable to the system. I fact, the etropy of the system ca be epressed terms of the desty matr: S k Tr ( ˆ ρ l ˆ ρ ). Ths s ot as bad as t looks: both operators are dagoal the eergy subspace.