Lecture 34: The `Density Operator. Phy851 Fall 2009


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1 Lecture 3: The `Deity Opertor Phy85 Fll 9
2 The QM `deity opertor HAS NOTHING TO DO WITH MASS PER UNIT VOLUME The deity opertor forli i geerliztio of the Pure Stte QM we hve ued o fr. New cocept: Mixed tte Ued for: Decribig ope qutu yte Icorportig our igorce ito our qutu theory Mi ide: We eed to ditiguih betwee `ttiticl ixture d `coheret uperpoitio Sttiticl ixture: it i either or b, but we do t kow which oe No iterferece effect Coheret uperpoitio: it i both d b t the e tie Qutu iterferece effect pper
3 Pure Stte qutu Mechic The gol of qutu echic i to ke predictio regrdig the outcoe of eureet Uig the forli we hve developed o fr, the procedure i follow: Tke iitil tte vector Evolve it ccordig to Schrödiger' equtio util the tie the eureet tke plce Ue the proector oto eigette of the obervble to predict the probbilitie for differet reult To cofir the predictio, oe would prepre yte i kow iitil tte, ke the eureet, the reprepre the e iitil tte d ke the e eureet fter the e evolutio tie. With eough repetitio, the reult hould how ttiticl greeet with the reult of qutu theory
4 Expecttio Vlue The expecttio vlue of opertor i defied (with repect to tte ) : A A The iterprettio i the verge of the reult of y eureet of the obervble A o yte prepred i tte. Proof: A = A = A = = = p( ) Thi i clerly the weighted verge of ll poible outcoe
5 Sttiticl ixture of tte Wht if we cot kow the exct iitil qutu tte of our yte? For exple, uppoe we oly kow the teperture, T, of our yte? Suppoe I kow tht with probbility P, the yte i i tte, while with probbility P, the yte i i tte. Thi i clled ttiticl ixture of the tte d. I thi ce, wht would be the probbility of obtiig reult of eureet of obervble A? Clerly, the probbility would be with probbility p, d with probbility p. P( ) = P( ) P( ) P( ) P( ) Thu the frequecy with which would be obtied over y repetitio would be p( ) = P P
6 The Deity `Opertor For the previou exple, Let u defie `deity opertor for the yte : = The probbility to obti reult could the obtied i the followig er: P Proof: P P ( ) = Tr{ I( )} I ( ) = P( ) = Tr{ I( )} = ( P P ) = = = = { } i coplete bi ( P P ) ( P P ) P P Thi will decribe the tte of the yte, i plce of wvefuctio = P P
7 Geeric Deity Opertor For ttiticl ixture of the tte { } with repective probbilitie {P }, the deity opertor i thu: = P The u of the P i Uity: P = The re required to be orlized to oe, but re ot ecerily orthogol For exple, we could y tht with 5% probbility, electro i i tte, d the other 5% of the tie it i i tte ( )/ = ( ) ( ) = 3 Thi tte i oly `prtilly ixed, eig iterferece effect re reduced, but ot eliited
8 Deity trix of pure tte Every pure tte h deity trix decriptio: = Every deity trix doe ot hve pure tte decriptio Ay deity trix c be teted to ee if it correpod to pure tte or ot: Tet #: If it i pure tte, it will hve exctly oe ozero eigevlue equl to uity Proof: = δ Strt fro: = Pick y orthoorl bi tht p the Hilbert pce, for which i the firt bi vector I y uch bi, we will hve the trix eleet δ,, = M M M L L L O
9 Tetig for purity cot. Tet #: I y bi, the pure tte will tify for every,: A prtilly ixed tte will tify for t let oe pir of, vlue: Ad totlly ixed tte will tify for t let oe pir of, vlue: Exple i pi/ yte: = < < = = d == 3 ( )( ) = = == 3 3 3
10 Probbilitie d`coherece I give bi, the digol eleet re lwy the probbilitie to be i the correpodig tte: The off digol re eure of the coherece betwee y two of the bi tte. Coherece i xiized whe: 3 3 =
11 Rule : Norliztio Coider the trce of the deity opertor = P Tr{ } = = Tr{ } = P P Sice the P re probbilitie, they ut u to uity
12 Rule : Expecttio Vlue The expecttio vlue of y opertor A i defied : A = Tr{ A} For pure tte thi give the uul reult: A = Tr{ A} = A For ixed tte, it give: A = Tr = p p A A
13 Rule 3: Equtio of otio For cloed yte: = d dt d d = dt dt i i = H H h h & = ih H, [ ] Pure tte will rei pure uder Hiltoi evolutio For ope yte, will hve dditiol ter: Clled ter equtio Exple: level to iterctig with qutized electric field. = i h H, ( ) Γ g e e g [ ] Γ e e e e Mter equtio decribe tte of yte oly, ot the `eviroet, but iclude effect of couplig to eviroet Pure tte c evolve ito ixed tte
14 Exple: Iterferece frige Coider yte which i i either coheret, or icoheret (ixtur uperpoitio of two oetu tte k, d k: = Coheret uperpoitio: k k = k Icoheret ixture: k ( k k ) k P(x) = Tr{ x x } = x x k k P ( x) = co(kx) Frige! k = NA = k k k k P( x) = No frige!
15 Etgleet Give the Illuio of decoherece Coider ll yte i pure tte. It i iitilly decoupled fro the eviroet: = c (, ( ) The tur o couplig to the eviroet: φ (, (, = U () (, Let the iterctio be odiiptive Syte tte do ot decy to lower eergy tte U (, ( ) ( ) φ = φ Strog iterctio: ue tht differet tte drive φ ito orthogol tte φ φ = δ,
16 The `reduced yte deity opertor Suppoe we wt to ke predictio for yte obervble oly Defiitio of yte obervble : A = A I ( ) Tke expecttio vlue: A = Tr{ (,e ) A () I (e ) }, ( ) = (, A ( ) I ( ) ( ) (, ( ) ( ) = A Defie the `reduced yte deity opertor : () = (e ) (,e ) (e ) = Tr e (, { } Phyicl predictio regrdig yte obervble deped oly o (): A = ( ) ( ) ( ) ( ) A { } A = Tr () A ()
17 Etgleet iic `collpe Retur to our etgled tte of the yte eviroet: (, ( ) = c Copute deity trix: = (, φ, ( ) ( ) c c φ = φ Copute the reduced yte deity opertor: () = Tr { e (, } = c c Tr { e () ( φ () φ ( }, = c, = c c ( ) ( ) φ φ
18 Collpe of the tte ( ) ( ) = c Cocluio: Ay ubequet eureet o the yte, will give reult if the yte were i oly oe of the, choe t rdo, with probbility P = c Thi i lo how we would decribe the `collpe of the wvefuctio Yet, the true tte of the whole yte i ot `collped : (, ( ) = c We ee tht the etgleet betwee yte d ev. iic `collpe I collpe durig eureet rel or illuio? Poiter Stte: for eurig device to work properly, the uptio, φ φ = δ, will oly be true if the yte bi tte, { }, re the eigette of the obervble beig eured φ
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