Density matrix. c α (t)φ α (q)

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1 Densty matrx Note: ths s supplementary materal. I strongly recommend that you read t for your own nterest. I beleve t wll help wth understandng the quantum ensembles, but t s not necessary to know t n order to be able to solve problems. The same materal s covered n Chapter 10 of your book, wth more detals and examples. I recommend you have a look at that as well, after readng ths. There may be some dfferences of notaton (though I wll do my best to follow thers, where possble) but f you understand the man deas, that should not bother you too much. 1 Pure vs. mxed states In all the quantum mechancs that you have studed so far, you only looked at stuatons where there s certanly a wave-functon descrbng the system of nterest. If we call q the generalzed coordnates (spatal locatons, for smple atoms) then ths wavefuncton Ψ(q,t) gves the ampltude of probablty that at tme t, the generalzed coordnates have values near q. Ths wavefuncton can be found (at least n prncple) from the Schrödnger equaton, and once we have t we can calculate the expectaton value of any operator ( Ψ,ÂΨ). The way ths s usually done, s to fnd a complete bass set, let us call t {φ α (q)}, where α stand for some quantum numbers (how many are necessary vares from problem to problem). Because we are consderng fnte-sze systems, we can always consder these quantum numbers to be dscrete, but n more general cases some or all of them could be contnuous. I wll use α to denote a sum over all these bass states, wth the understandng that f one or more of the quantum numbers ncluded n α are contnuous, we wll actually ntegrate over them. If the bass s complete, then any wavefuncton can be decomposed as a lnear combnaton: Ψ(q,t) = α c α (t)φ α (q) and the problem s reduced to fndng the decomposton coeffcents c α (t). Once these are known, the expectaton value of any operator n ths state s: ( Ψ, ÂΨ ) = α,α c α(t)c α (t) ( φ α (q),âφ α (q)) = α,α c α(t)c α (t)a α,α where A α,α s the matrx element of the operator between those two bass states. So, n prncple, all thngs can be calculated. Cases such as ths, where the system s descrbed by a wavefuncton, are called pure states. Whle you mght thnk that the system s surely always descrbed by a wavefuncton, ths s actually not the case, n general. As you know, n many cases we look at our system of nterest beng coupled to a reservor. If the total system = system+reservor s solated, we can defne a wavefuncton for the total system, Ψ T (q,q R,t). If there were no nteractons, we could separate ths nto a product of two parts Ψ(q,t)Ψ R (q R,t) (ths smply shows that what the system s dong s ndependent of what the reservor s dong, f there are no nteractons between them) and n ths case we can dentfy Ψ(q,t) as the wavefuncton of the system. But f the system and the reservor nteract, whch s the case we are nterested n, we can no longer separate a system wavefuncton from a reservor wavefuncton. We do not want to work wth Ψ T (q,q R,t), because ths contans all the extra nformaton about what the reservor s dong, and we don t care/need to know that. So the queston s, how do we deal wth such a case, where there s no well-defned sngle wavefuncton for the system of nterest? Such cases are called mxed states. 1

2 What we do, s to say that the system may be wth probablty w n a state descrbed by a wavefuncton Ψ (q,t). In other words, nstead of sayng that the system s certanly n a state descrbed by a sngle wavefuncton, we allow t to be, wth dfferent probabltes, n dfferent states descrbed by dfferent wavefunctons. Of course, we must have w = 1. In ths mxed state, the expectaton value of any operator s gven by: ( w Ψ,ÂΨ ) snce wth probablty w the system s n state Ψ, n whch case the expectaton value s ( Ψ,ÂΨ ). Hopefully, ths looks lke a reasonable formula to you. Each of the possble wavefunctons can stll be decomposed n a complete bass set as: Ψ (q,t) = α c () α (t)φ α (q) and therefore, f we know these coeffcents and the probabltes w, we can rewrte: w c α () α,α (t)c () α (t)a α,α = α,α ρ α,α(t)a α,α (1) where we defne ρ α,α(t) = w c α () (t)c () α (t) Eq. (1) s very nterestng, because t shows that to calculate expectaton values, we only need these quanttes ρ α,α(t) (as opposed to needng to know each ndvdual Ψ,w ). Snce expectaton values are all we need to fnd at the end of a calculaton, t follows that all useful nformaton about the system s encoded n these matrx elements ρ α,α(t). We can defne the operator ˆρ(t) such that: (φ α (q), ˆρ(t)φ α (q)) = ρ α,α (t) (2) Ths s called the densty matrx or densty operator, and as I just sad, t contans all needed knowledge about the system. So ths densty operator (or equvalently, all ts matrx elements) replace the sngle wavefuncton that we had n the pure case. Wth ths defnton, we see that we can use the short-hand notaton: α,α ρ α,α(t)a α,α = Tr (ˆρ(t) Â) because α ρ α,α(t)a α,α gves us the matrx element (ˆρ(t) Â) α,α (just smple matrx multplcaton); and summng over all the dagonal matrx elements gves us the trace. You wll see ths trace notaton appear a lot; untl you get used to t, you should also wrte everythng explctly n terms of matrx elements, and convnce yourself that the formulae always agree. One bg advantage of usng the trace formulae s that they are correct no matter what bass one choose for the Hlbert space. Before dscussng what the densty matrx may be for our systems of nterest, let us quckly dscuss some general propertes of ths operator. (1) ρ αα s the probablty to fnd the system n state α of the bass. Ths comes from the defnton just followng Eq. (1): ρ α,α (t) = w c () α (t) 2 2

3 Snce c () α (t) 2 s the probablty for a system descrbed by Ψ to be found n the bass state α, summng over all possble states weghted by ther probabltes should ndeed gve us the total probablty to fnd the system n state α. (2) Tr(ˆρ) = α ρ α,α = 1. Ths can be shown ether usng (1) (the system s surely n some bass state); or usng the normalzaton of the Ψ states, whch means (Ψ, Ψ ) = α c α () (t) 2 = 1, whch combned wth the fact that w = 1, gves us agan ths; or smply, snce the average of the unt operator must be 1, we must have 1 = ˆ1 = Tr (ˆρ(t) ˆ1 ) = Tr(ˆρ). Ths condton bascally replaces the normalzaton condton that we have for the wavefuncton of a system n pure state. (3) ˆρ = ˆρ,.e. the densty matrx s a Hermtan operator. In terms of matrx elements ths means that ρ α,α = ρ α,α and s straghtforward to check from the defnton followng Eq. (1). (4) ˆρ s postve defnte, by whch we mean that all ts egenvalues are postve (or zero) numbers. Ths comes from a combnaton of the propertes above. Snce ˆρ s hermtan, we know for sure that there s a complete bass n whch ts matrx s dagonal. Moreover, the egenvalues must be real numbers. From (1), we know that these egenvalues (the dagonal elements n ths specal bass) are related to probabltes, so they must be not only real, but also postve numbers. Let us call ths specal complete bass set {ψ n (q)}, where n ndexes ts quantum numbers. Because ˆρ s dagonal n ths bass, we have ρ n,n = δ n,n ρ n, where δ n,n s the Kronecker symbol, equal to 1 f n = n and zero otherwse. ρ n s the correspondng egenvalue, and as we know from (1), t s also the probablty to fnd the system n bass state n. (5) Tr(ˆρ 2 ) 1. Snce the value of the trace does not depend on the bass we use to evaluate t, let s use the specal bass. In ths case, Tr(ˆρ 2 ) = n ρ 2 n (check ths!). Snce 0 ρ n 1 (they are probabltes), we have n ρ 2 n ( n ρ n ) 2 = 1, because n ρ n = Tr(ˆρ) = 1. The only way for Tr(ˆρ 2 ) = 1 s for the system to be n a pure state,.e. to be wth certanty descrbed by a sngle wavefuncton. You should check that f, ndeed, only one of the w = 1, whle all the other ones are zero, we fnd Tr(ˆρ 2 ) = 1. So the value of Tr(ˆρ 2 ) tells us f the system s n a pure state (result s 1) or a mxed state (result s strctly less than 1). (6) Fnally, we need the analog of the Schrödnger equaton. For a pure state, that allows us to fnd the wavefuncton at all tmes, f we know t at the ntal tme. We would lke a smlar equaton that allows us to fnd ˆρ(t), f we know ˆρ(0). Ths equaton s called the Louvlle-von Neumann equaton, and s: h d ˆρ(t) = [H, ˆρ(t)] dt where H s the Hamltonan of the system. Ths can be derved from the fact that each possble wavefuncton Ψ (t) satsfes the Schrödnger equaton, try t. However, we must be very careful wth ths, because t can only be used f there s a Hamltonan for the system (.e., the system s solated). If the system nteracts wth a reservor, then one needs to also nclude somehow the effects of the nteracton Hamltonan on the evoluton of the densty matrx. That s a very complcated problem, and some aspects of t are actually stll matters of actve research. So remember that ths equaton 3

4 holds only for solated systems. More about ths s dscussed on page 269 of your textbook. 2 Densty matrces for varous quantum statstcal ensembles Let H be the Hamltonan of our system of nterest, f t was solated. If the system was truly solated, then t would n prncple be descrbed by a sngle wavefuncton, and t would be n a pure state. The natural bass to use n ths case are the egenstates of the Hamltonan Hφ α (q) = E α φ α (q) where α are the needed quantum numbers. If the system was truly solated, ts energy could only be one of the allowed values E α. Even f the level α for the allowed energy E = E α s degenerate, f at t = 0 the system starts n one partcular egenstate correspondng to ths energy, let s say, Ψ(q,t = 0) = φ α0 (q), where E α0 = E, then the system wll forever stay n ths egenstate, wth Ψ(q,t) = e ī h Eα 0 t φ α0 (q). So we would always precsely know the pure state of the system. However, even n a mcrocanoncal ensemble, when the system s supposed to be solated, n fact we know that t stll has small nteractons wth other systems (for nstance, the quantum gas wll nteract wth the walls of the contaner, whch are made of atoms that vbrate about ther equlbrum postons. As gas atoms ht the wall, small amounts of energy can be transmtted back and forth between the gas and the wall). Generally these exchanges are very small, and they cannot change the system from one egenvalue E α to another one. But such nteractons are enough to randomly change the wavefuncton of the system between dfferent allowed egenfunctons. So even f we start n a partcular state α 0, as collsons take place, the system wll swtch randomly through all the other allowed states α. Let s call Ω ther multplcty (the degeneracy of the level wth the allowed energy, whch tells us how many possble canddates we have for these possble wavefunctons of the system). Clearly, n ths case the system s n a mxed state, and each of these Ω states are contrbutng. How much? Well, we ll plead agan total gnorance, and say that they must all be contrbutng equally, snce we have no reason to dscrmnate. Then, the probablty to be n any of these allowed states s 1/Ω, and the average of any operator must be: allowedα 1 ( φα Ω,Âφ 1 α) = allowedα Ω A α,α We can now use ths to construct the densty matrx. It s actually partcularly convenent to use ths bass to express ts matrx elements. If you follow the general dscusson for ths partcular case, you fnd that: { 1, f α s an allowed energy egenstate ρ α,α = Ω 0, otherwse and ρ α,α = 0 s α α. So n ths case, we fnd trvally that Tr(ˆρ Â). The same apples to other ensembles. For nstance, for a canoncal ensemble, we know that the probablty to be n energy egenstate α s w α = exp( βe α )/Z. Unlke n the mcrocanoncal state, now all possble energy egenstates get mxed n, but wth dfferent probabltes whch depend on the temperature of the system (fxed by that of the reservor). In ths case, followng the defnton, you can show that agan ρ α,α = 0 f α α,.e. the densty matrx s dagonal. Moreover: ρ α,α = 1 Z e βeα 4

5 Snce ths s the probablty to fnd system n state α, and ths must be normalzed, we fnd Z = α e βeα. The prevous equaton gves us the matrx elements. The queston s, can we extract the operator? The answer s: ˆρ = 1 Z e βĥ You should check that the matrx elements (φ α, ˆρφ α ) are ndeed equal to ρ α,α. Then, the average value of any operator s, by defnton: Tr(ˆρ Â) = α,α ρ α,α A α,α = α ρ αα A α,α snce the off-dagonal matrx elements are all zero. Ths leads to the expresson that we are actually usng for calculatons: 1 e βeα A α,α Z α The value of Z also can be re-expressed as a condton for the normalzaton of the densty matrx: 1 = Tr(ˆρ) = 1 Z Tr(e βh ) Z = Tr(e βh ) = α e βeα Smlar arguments apply for the grand-canoncal case, where one fnds ˆρ = 1 ˆN) e β(ĥ µ Z where ˆN s the number operator, countng how many partcles are n the system. Ths formula contans all needed knowledge, and we can derve the averages of operators, normalzaton condtons, etc, all from ths. You must admt that ths s a rather elegant and compact formulaton. The bottom lne s that ths notaton n terms of ˆρ and traces, etc. s just a condensed form for what we have been usng n class. Whle such smplfed notaton s not partcularly necessary when dealng wth smple systems of non-nteractng partcles, t becomes more and more mportant f one looks at more complcated problems (for nstance, nteractng systems). In those case, equatons generally get very long and complcated and any shorthand and compactfcaton s very useful. Those of you who wll contnue wth graduate studes n physcs are very lkely to encounter these densty matrces n the future (dependng on what area of physcs you choose to specalze n). 5