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Lecture 10 Linear - time last From 10-1 out c . want we : to linear study in rcspouie Hamiltonian with system quantum a response H=HotV Towards this picture end use to relevant The . [ ) id+A±lt these an preliminaries for expression system The . notation Ho , in ) we : ] 1 subscript the quantum a done be will drop we in response derive to ready are , calculation following the in the from I . the classical H= we ,=lH A are . } ,=lH linear the picture interaction As [ = is , eats response With an ✓ appropriate Re is evolution time iot9±= Linear picture interaction the , are case , Hot interested Perturbation 8<131+77 V in Ho = at = Tr { Bitch ) t time %) - } t = f to Tr ( Bltlglt ' ) ) described ) B observable in on system a fltlalt - response switched V study we ' dt = to . with the y 10 t 8431+17 = if - Trf [ BHI ' vlt ) , )} 't glt - z ' dt to NOW Trl use d) A[ B. get ABC ( Tr = To ( Tr = ) ACB - = Tr ( ABC - BAC ) the equilibrium C) [ Ai B) t Tr{ }dt = if - [ Bit ) 's vet , 91T ] ' 't to ifotdt = API to first in g- ° 'd [ ' BH + i 790 < [ del ) Alt , 't FHYSIEY ] ,Alt 't ] 7 Bltl ' oflt } ) to which ( write we to taking -7 b - ) t 8<131+17 ) = - where cfpalt Because of DA and commentator A = < - dtl ) A 7 by given Ali ) , we ] ) for A sB= and ° B < - then ¢ , alt ) = it [ oalo DBIH , ) ] ) . t subtract also can , and B from averages [ BIH it = the ' fit 1 is response ) ' t p linear the 4,3 alt - 1370 > o 10-3 form Alternative had We d. a iTr{BH[a. = ] } ÷zTr{ ↳ # 't % = - )[ Bit ] ) Hfettlo PHOA )etHo ipltia PHO 13*0 commututor the Rewrite EP , A ] follows as pH . e- = Ae - . : ( = - A) e- EPHAEPHO tltoaetko §dt = = = of §dt §de e- if = di . , it P PH e- e- ( HA - AH + o " e- [ A. ni ( e- ) ii e- 0 0 where in the picture ⇒ Alt %a 't ) step last ) eiltota = Tr{ = = = Bet § 4 i { Bit e- [ BH in the interaction ittot )§dt )A' that used we ni lit )o ( it ) ,A6 ) ] ) . )% de } Kubo formula § 10 Recap - y D 8<131+17 / = X ,3a( t ' t - ) ' flt dtl 1 Ict b - Where It ) XBA the ensures Note ¢ , ,n ) th < [ = fs Then : p Alo , =) ] > ) < BH ) A. Lite ) ) A limit of { 0 → the 0 into goes quantum second result classical the observable an )dt Alttitt ¥§ = tame "tde " e- ~ ¢Balt Since Note " { so Bltlitlol ) p = ) { equilibrium Blttslnts XBAH ) . correlation ) > < Blt ) = = - de ° . manifestly = BH formula Kubo A. 13<131+7 = It ) transform Kubo ) classical formula The the by given is . 1072g ! the In : +70 Causal is lopalt : quantum a + it 1 response classical it ° response linear The admittance the is ) function ,={ at OH ) step Heaviside The .lt 4,3 = Pok Alo ) ) )a¥ ( Bh ⇒ 4 functions Bit ) ) Flo ) ) stationary are ) > Bk nilo . . - < ) Alo ) > . 10-5 i Electric ) B J = = SE - where V . Ehl I d3r - = volume = it tr § ) qiri : F. E✓ polarization frill electric uniform , , I field electric applied to Elw olw ) density current = response V jlw7= conductivity D= note qq.fi I d- e T Examples = = , = - =) A VP = get then we Oijltl Xjpct = The d- ) C jiltlisjlo jikljnjlo < ) > alt ) ) ) that means oijlw < pv = PVOIH = This ) = prop jiltlj < Conductivity . is ; ( o eiwtdt > ) obtained in the w→o a cry . = Oijlw → o ) = PV ) . < Iiltljjlo ) > dt . limit 10.6 ii Dynamic magnetic susceptibility ) B= ✓ = at vk.is - , a , VK = p ) Xlw Xmmlw ) = pv ) = ) Mit < kilo ) ) eiwtdt 0 In classical static the case : p Xlw - 01 = = PVS ( pvf < - wit 0 = since <Mcas ( pu = )Ml° ) ) uncorrelated This result may be - ( o Mlo ) mlo ) 472 ( pu ( in ) Mit obtain we ) ) ) ) dt )ml° ) min - dt ) > mt ) < become = hula as familiar ) t > < mlo → ) > since s . to you . Mlt ) and Mo ) 10-7 , perturbation has , the response , of propertiei analytical Namely . the after comes the for consequences susceptibility dynamical the that fact the Causality properties of analytical and Causality X if t S< ) Bit 7 ftp.alt = ' - t fplt ) 1 ' dt ' is - that such XBACW ) = Blw ) ) < 8 Then • Xlw ) ) iwt 4h S = falw ) dt e 0 We function this continue can a Xlz ) izt f = freqnuccic complex to , dt date 0 Inz . - For In ? 't 0 , function y analytic in upper the - and o → e as half plane that means §dw with behaves integrand therefore is This the t -70 ' ×#-iy ' w - w this see To , . integration ÷ @ into =o + the upper - need just we half of extend to the complex the plane { dtI¥+I=o , since the only pole is at 2- = w . in this 10-8 Now the use can we , Note : P the To see real § The " ××I÷y defines ,¥yfH d . part principal the ' # is it ° { limit the Y -70 ex o it P X =o xtiyldx - 1°×¥f←= .§×¥,,←iId×=÷fsT¥I = - Ziti Iz I = thrfore and in t¥'dxt§±I' '¥d×= ¥ ! =P ¥IIi¥i = part imaging ×÷f value ) write this part - tiitocx principal ×÷j= The (k ) =P ×±÷j with expression hypo ¥j = Tax ) and him y -70 ×t÷j = Ptx Iitck ) , III Introducing the XCW ) Note get we double " ( w ) . the dw here are "lw Xiw ) ) = Kramer Generally response through 's - and the " gives ) ' ' w - ' ( w 1 - IT X the , equation w ' - w p × Kramer ' w this ' W - part real of - X relations kronig the ' '( X ' Tt dw Pf = dw ) p - - ( part imaginary is X "ii - = and - dispersion or ' is reactive the dissipative krouig ' 9 derivatives not :o) itslwwiflxtwytix ' , real 10 as ) w Prime X( .§[pC± : o= Taking { it + of part imaginary ' X = and pine : and real relations part . . relations part These of are the related : ) ] l Time reversal - Most symmetry quantitie time reversal → 5 last → reversal Ea ± = T operator . ( T x [ - = that A transform ) " T Ea = Al - t ) . anti An F) , Tale transformation The a → ) - operators consider we Alt where Fxl -7 FXF = time under an transtrm should therefore and is 5 since understood be and generally : ( spin ) 5 , L Note - momentum one , under I - can one angular So sign I → I like change or : I The invariant either are desciped is T operator unity 147+131+7 ) x*Tl9 = anti is ) + p× antiuuikz an by linear . It > satisfies and < Warning It's : Ttlte The safe adjoints ) notation Dirac to that use +147 < = the involve is * designed for equations above T . but linear be operators way of . o -10 Q What Uk T= where basis K , usually taken ( conjugation complex is be IF ? 4=1 and to fermions spintess For gives write always Can that operator the is . ) K spintnl with o I - acts that U For . spin Pauli ?t ( - = a th , , )k , need freedom of =(%) IB - we operators ox where KC = degrees spin the K some ( 1<2--1 ) r = fermions spin the # K to respect with . > 5 K For Lo -11 ? T - tz only we ) represent can ,og,sz ) Cox E= matrices on a=⇐ ;) are need We ' uE*i T2= while tr spin T2= There therefore are systems . Those K . 5 2 = does isy U= the job . fomious spin less for that Note - that yourself Convince = 1 tz igkiogk two wwe = classes T2=+| # iogti of time and = . - oj2= reversal those - 1 . symmetric with T2= - 1 .