( t) ( t) ( t) ρ ψ ψ. (9.1)
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1 Adre Toaoff, MT Departet of Cestry, 3/19/29 p THE DENSTY MATRX Te desty atrx or desty operator s a alterate represetato of te state of a quatu syste for wc we ave prevously used te wavefucto. Altoug descrbg a quatu syste wt te desty atrx s equvalet to usg te wavefucto, oe gas sgfcat practcal advatages usg te desty atrx for certa te-depedet probles partcularly relaxato ad olear spectroscopy te codesed pase. Te desty atrx s forally defed as te outer product of te wavefucto ad ts cojugate. ( t ( t ( t ψ ψ. (9.1 Ts ples tat f you specfy a state χ, te tegral χ χ gves te probablty of fdg a partcle te state χ. ts ae derves fro te observato tat t plays te quatu role of a probablty desty. f you t of te statstcal descrpto of a classcal observable obtaed fro oets of a probablty dstrbuto P, te plays te role of P te quatu case: A ( AP A da (9.2 [ ] A ψ Aψ Tr A. (9.3 were Tr[ ] refers to tracg over te dagoal eleets of te atrx. Te last expresso s obtaed as follows. For a syste descrbed by a wavefucto te expectato value of a operator s ( ( ψ t c t, (9.4 ( ( (, At ˆ c tc t A ˆ (9.5 Also, fro eq. (9.1 we obta te eleets of te desty atrx as ( ( ( t c t c t,, ( t (9.6
2 Adre Toaoff, MT Departet of Cestry, 3/19/29 p. 9-2 We see tat, te desty atrx eleets, are ade up of te te-evolvg expaso coeffcets. Substtutg to eq. (9.5 we see tat ( ( At ˆ A t, Tr Aˆ practce ts aes evaluatg expectato values as sple as tracg over a product of atrces. So wy would we eed te desty atrx? t s a practcal tool we dealg wt xed states. Pure states are tose tat are caracterzed by a sgle wavefucto. Mxed states refer to statstcal xtures wc we ave perfect forato about te syste, for wc e ust perfor statstcal averages order to descrbe quatu observables. A xed state refers to ay case wc we subdvde a croscopc or acroscopc syste to a eseble, for wc tere s tally o pase relatosp betwee te eleets of te xture. Exaples clude a eseble at teral equlbru, ad depedetly prepared states. Gve tat you ave a statstcal xture, ad ca descrbe te probablty p of occupyg quatu state ψ, wt p 1, evaluato of expectato values s splfed wt a desty atrx: ( t ( ψ ( ˆψ ( (9.7 At ˆ p t A t (9.8 ( t p ( t ( t ψ ψ (9.9 ( ˆ ( At ˆ Tr A t. (9.1 Evaluatg expectato value s te sae for pure or xed states tese oly dffer te way eleets of are obtaed. Propertes of te desty atrx 1 s Hereta: ( Noralzato: Tr ( 1 (9.12
3 Adre Toaoff, MT Departet of Cestry, 3/19/29 p Tr ( 2 1 for pure state < 1 for xed state (9.13 Te last expresso reflects te fact tat dagoal atrx eleets ca be or 1 for pure states but le betwee ad 1 for xed states. addto, we worg wt te desty atrx t s coveet to ae ote of tese trace propertes: 1 Cyclc varace: Tr ( ABC Tr ( CAB Tr ( BCA ( varace to utary trasforato: Tr ( S AS Tr ( A (9.15 Desty atrx eleets Let s dscuss te desty atrx eleets for a xture. You ca t about ts as a eseble wc te dvdual olecules ( 1 to N are descrbed ters of te sae teral bass states, but te probablty of occupyg tose states ay vary fro olecule to olecule. We te expect tat we ca express te state of a certa olecule as were ψ c, (9.16 c s te coplex ad te-depedet apltude coeffcet for te occupato of bass state o olecule. Te te desty atrx eleets are cc, ψ c ψ ( c (9.17 Ts expresso states tat te desty atrx eleets represet values of te egestate coeffcets averaged over te xture: Dagoal eleets ( gve te probablty of occupyg a quatu state : cc p (9.18 For ts reaso, dagoal eleets are referred to as populatos.
4 Adre Toaoff, MT Departet of Cestry, 3/19/29 p. 9-4 Off-Dagoal Eleets ( tat descrbes te evoluto of coeret superpostos. ad are referred to as coereces. are coplex ad ave a te-depedet pase factor ( ( c t c t c c, (9.19 ωt e Desty atrx at teral equlbru Our wor wt statstcal xtures wll deal eavly wt systes at teral equlbru. Te desty atrx at teral equlbru eq (or s caracterzed by terally dstrbuted populatos te quatu states: β E e p (9.2 were s te partto fucto. Ts follows aturally fro te geeral defto of te equlbru desty atrx were te partto fucto β Ĥ e eq (9.21 ( We obta eq. (9.2 usg te specfc case H ˆ E, ( eq Tr e β Ĥ ( e β E e δ p δ β Hˆ. (9.23 Fro ts laguage oe ca also express a terally averaged expectato value as: 1 β E 1 A e A Tr( Aeq. (9.24
5 Adre Toaoff, MT Departet of Cestry, 3/19/29 p. 9-5 TME-EVOLUTON OF THE DENSTY MATRX Te equato of oto for te desty atrx follows aturally fro te defto of ad te te-depedet Scrödger equato. Usg ψ H ψ ψ ψ H ψ ψ ψ ψ + ψ ψ H ψ ψ + ψ ψ H (9.25 (9.26 [ H, ] (9.27 Equato (9.27 s te Louvlle-Vo Neua equato. t s soorpc to te Heseberg equato of oto for teral varables, sce s also a operator. Te soluto s ( t U( U Ts ca be deostrated by frst tegratg eq. (9.27 to obta. (9.28 t ( t ( dτ H( τ, ( τ (9.29 f we expad eq. (9.29 by teratvely substtutg to tself, te expresso s te sae as we we substtute t U exp + dτ H( τ (9.3 to eq. (9.28 ad collect ters by orders of H(τ. Note tat eq. (9.28 ad te cyclc varace of te trace ply tat te te-depedet expectato value of a operator ca be calculated eter by propagatg te operator (Heseberg or te desty atrx (Scrödger or teracto pcture:
6 Adre Toaoff, MT Departet of Cestry, 3/19/29 p. 9-6 ( ˆ ( At ˆ Tr A t Tr AU ˆ U Tr Aˆ ( t (9.31 For a te-depedet Haltoa t s stragtforward to sow tat te desty atrx eleets evolve as ( ( ( ( ψ ψ ψ ψ (9.32 t t t t U U ( t ( t t t (9.33 e ω Fro ts we see tat populatos, ( t ( t at te eergy splttg ω. (, are te-varat, ad coereces oscllate Te desty atrx te teracto pcture For te case wc we ws to descrbe a ateral Haltoa H uder te fluece of a exteral potetal V(t, ( ( H t H + V t (9.34 we ca also forulate te desty operator te teracto pcture. Fro our orgal defto of te teracto pcture wavefuctos We obta as ψ U ψ (9.35 S U U. (9.36 S Slar to te dscusso of te desty operator te Scrödger equato, above, te equato of oto te teracto pcture s V ( t, ( t (9.37 were, as before, V UVU. Ts expresso ca be wrtte sortad ters of te Lovlla superoperator L $
7 Adre Toaoff, MT Departet of Cestry, 3/19/29 p. 9-7 Here $ L s defed te Scrödger pcture as ˆ $ ˆ L. (9.38 $L Aˆ H, Aˆ (9.39 Equato (9.37 ca be tegrated to obta t ( t ( t dt V ( t, ( t t. (9.4 Repeated substtuto of ( t to tself ts expresso gves a perturbato seres expaso t ( t dt V ( t, ( t Here ( t 1 1 t 2 t t2 dt2 dt1 V ( t2, V ( t1, t t + + L t t t2 + dt dt 1 dt 1 V ( t, V ( t 1,, V ( t1, t t K t K K + L ( ( 1 ( 2 ( ( L+ + L (9.42 ad equato (9.41 ca also be expressed as ( s te t -order expaso of te desty atrx. Slar to eq. (9.28, ( t U ( U. (9.43 Ts s te soluto to te Louvlle equato te teracto pcture. t ca also be wrtte ters of a superoperator G $, te te-propagator: G $ s defed te teracto pcture as ( t G( t ( $ (9.44 GA $ ˆ U Aˆ U (9.45 For te case were te egestates of H are ow (o relaxato, te propagato for a partcular eleet of desty atrx
8 Adre Toaoff, MT Departet of Cestry, 3/19/29 p. 9-8 G$ t a b Ht/ + Ht/ ( ab e e abt e ω a b (9.46 Usg te Louvlle space te-propagator, te evoluto of te desty atrx to arbtrary order eq. (9.41 ca be wrtte as ( t t t2 dt dt dt Gˆ t t V t Gˆ t t V t Gˆ K t t V t ( ( ( ( L ( ( t t t. (9.47 Correlato Fuctos ad Respose Fuctos We ave prevously defed te correlato fucto as a equlbru average of te expectato value a product of operators: Sce p eq, AA ( ( ( C t A t A AA p At ( A( ( eq ( ( ( ( C Tr A t A ( eq Tr A t A Correlato fuctos ca be expressed ters of a te-propagator as AA ( eq ( eq ( equ ( ( ( C t Tr A t A Tr U AU A Tr AU A ( ˆ ( eq Tr AG t A. (9.48 (9.49. (9.5 Sce te lear respose fucto s related to te agary part of correlato fucto R( τ ( CAA ( τ CAA ( τ Tr A A { Tr( A( τ A( eq Tr( A( A( τ eq } ( ( τ, ( eq (9.51
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