Let s treat the problem of the response of a system to an applied external force. Again,
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1 Page 33 QUANTUM LNEAR RESPONSE FUNCTON Le s rea he problem of he response of a sysem o an appled exernal force. Agan, H() H f () A H + V () Exernal agen acng on nernal varable Hamlonan for equlbrum sysem (rea exacly) f V() represens a small change, we can rea hs wh perurbaon heory n he neracon pcure. Now A s an operaor. We wan o descrbe A () whch we wll ge by ensemble averagng he expecaon value of A(). Remember he expecaon value for a sngle wavefuncon s A () () A ψ () ψ U (, ) AU (, ) ψ ψ ψ U A U ψ Where he propagaon n he neracon pcure s U (, ) + V ( )U (τ, ) dτ τ (exac) For he lnear response we use he frs order soluon: V ( τ ) V U () U () () f ( U ) () AU () f () A ( τ U, ) + dτ f ( ) A (τ)
2 Page 34 So, we can now calculae he value of he operaor A a me A () U A U d f ( ) A ( ) A ( ) + d f ( ) A ( ) ()+ A ( ) A ( ) d f ( ){A ( ) A A ( )} reanng lnear erms ()+ d f ( ) A A ( ), A ( ) Now, usng A () U () AU () and seng we can wre d f ( ) A A() A ()+ ( ), A () A ()+ d τ f ( τ) A (τ), A () where τ Now, wha we wan s he expecaon value of A, ha s ψ A ψ, averaged over he ensemble, whch we wll wre A () for he momen. Takng no accoun ha he force s appled equally o each member of ensemble we have () A A + d τ f ( τ) A ( τ), A ( ) The frs erm s ndependen of f, and so comes from an equlbrum ensemble average A p n n A n A n
3 Page 35 Comparng hs wh he expresson for he lnear response funcon, we fnd R () ( ) τ ( τ ), A ( ) A τ τ< or as s somemes wren wh he un sep funcon: R () ( ) Θ τ τ ( ) A ( τ ), A ( ) Noe ha he me developmen of he sysem wh he appled exernal feld s governed by he dynamcs of he equlbrum sysem. All of he me-dependence n he response funcon s under H. The response funcon s proporonal o he dfference of wo complex correlaon funcons: R () ( ) A ( ) A ( ) τ { τ A ( ) A (τ) } (C AA ( ) * τ C AA ( τ )) 2 C ( τ ) where C ( τ) m C ( τ ) f we express he correlaon funcon n he egensae descrpon: hen C ( ) 2 ω τ p n A jn e jn n,j R () ( τ ) 2 pn n,j A jn 2 sn ω jn Noe ha R () () τ s real!
4 Page 36 Alernavely, n he densy marx represenaon, τ * R () ( ) (C AA ( τ) C AA ( τ)) {Tr ( A ( ) τ Tr ( A ( τ ), A () ρ eq ) A ()ρ eq ) Tr ( A ( ) A (τ)ρ eq )} The response funcon and energy absorpon Le s nvesgae he relaonshp beween he lnear response funcon and he absorpon of energy from an exernal feld H H f ( ) A H µ E () Ths expresson gves he energy of he sysem, so he rae of energy absorpon averaged over he non-equlbrum ensemble s descrbed by: * H f A () The me-averaged rae of energy absorpon: T f q () T d A. T T f d () A + d τ R () ( τ ) f ( τ) () where R () ( ) τ ( ) ( ) µ τ, µ * See Wang (985).
5 Page 37 f we have a monochromac lgh source: () E cosω 2 [E e ω * + E e ω ] (2) f Lookng a he second erm n (): τ τ 2 d R () ( ) E e ω( τ) + E ( ) * e τ) * 2 E e ω χ ω + E e ω χ( ω) ω( (3) Dfferenang (2) and pluggng no () we have: q A T f ( T * * T 2 2 ( ) ) f () T d ωe e ω + ωe e ω E e ω χ ω+ E e ω χ( ω) Le s cycle average hs expresson (se T 2π / ω ). Frs erm vanshes. Cross erms n second negral vansh. T d e ω e + ω T T d e ω e ω T 2 q ω E 4 χ( ω) χ(ω) q ω 2 E χ ( ω ) 2 The absorpon of energy by he sysem s relaed o he magnary par of he suscepbly! α( ω ) q 4πω χ ( ) E n c c 2 ω E n 8π E
6 Page 38 The absorpon lneshape s relaed o he magnary par of χ and χ s relaed o he Fourer ransform of he correlaon funcon ha descrbes he flucuaons and dynamcs of he equlbrum sysem [C AA ()]. χ ( ω ) (χ( ω ) 2 χ( ω)) 2 { d e ω C AA () C AA ( ) d e ω C AA ( ) C AA ( ) } (C AA ( ) 2 ω C AA ( ω)) ) d e ω C AA ( C AA ( ) d e ω C AA ( ) C AA ( ) From problem se: he correlaon funcon obeys he dealed balance condon: C AA ( ω) e βω C AA (ω) Ths relaonshp reflecs he fac ha upward and downward ranson raes beween saes separaed by ω are relaed by he populaon dfference. Remember, from F.G.R. ha he raes k are drecly proporonal o C AA. Ths allows us o wre: C AA ( ω ) ± C AA ( ω) ( ± e βω ) C AA (ω) So χ ( ω ) ( e βω ) C AA ( ω ) 2 2 ( + e βω ) e ω A () A( ) d Ths s he resul from before he absorpon of energy s dcaed by he equlbrum flucuaons of he sysem. nserng no α(ω) n he prevous page we have he resul from earler: α ω + ( ) 2πω ( e βω ) e ω A c () A() d
7 Page 39 Relaxaon of a prepared sae The mpulse response funcon R () () descrbes he behavor of a sysem nally a equlbrum ha s drven by an exernal feld. To descrbe he relaxaon of a prepared sae, he sysem mus nally be n a non-equlbrum sae, and hen we wll wach he reurn o equlbrum. Ths behavor s descrbed by sep response funcon S AA, whch descrbes he behavor when a sysem held away from equlbrum by an exernal feld s suddenly released. Jus as we expec ha he mpulse response o rse from zero and be expressed as an odd funcon n me, he sep response should decay from a fxed value and look even n me. The sep response comes from holdng he sysem wh a consan feld H H fa unl a me S AA () R AA when he sysem s released, and relaxes o he sae H H. You expec o descrbe hs behavor by negrang he mpulse response over all mes <. Response Funcons are real. Quanum Correlaon Funcons are complex: C( ) C * () Classcal Correlaon Funcons are real and even: C ( ) C( ) For relaxaon n erms of a real observable ha s even n me, we consruc a symmerzed funcon: S AA () 2 { A () A () + A ( ) A ( ) } 2 2 {C AA ()+ C AA ( )} ( ) ( ) A, A + C AA ( ) S s relaed o he real par of he correlaon funcon, and defned for. The mpulse response s relaed o he me-dervave of he sep response, and n he classcal lm
8 Page 4 () R () d () (hgh T lm) kt d S AA f we defne S AA ( ω ) 2π d S AA () S AA ( ω ) ω + e ω, hen C AA ( ω) 2 ( + e βω ) C AA (ω) 2 C AA ( ) χ ( ω) anh β ω ω ω ω (classcal lm) 2 S AA ( ) 2kT S AA ( ) Ths s he flucuaon-dsspaon heorem (Chemsry Nobel Prze, 968; proven n 95 by Callen and Welon). Lars Onsager (93): The relaxaon of macroscopc non-equlbrum dsurbance s governed by he same laws as he regresson of sponaneous mcroscopc flucuaons n an equlbrum sae.
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