5.74 Introductory Quantum Mechanics II

Transcription

1 MIT OpenCourseWare ttp://ocw.mit.edu 5.74 Introductory Quantum Mecanics II Spring 9 For information about citing tese materials or our Terms of Use, visit: ttp://ocw.mit.edu/terms.

2 Andrei Tokmakoff, MIT Department of Cemistry, 3/3/ LINEAR RESPONSE THEORY We ave statistically described te time-dependent beavior of quantum variables in an equilibrium system troug correlation functions. We ave also sown tat spectroscopic linesapes are related to correlation functions for te dipole moment. But it s not te wole story. You ave probably sensed tis from te perspective tat correlation functions are complex, and ow can observables be complex? We will use linear response teory as a way of describing a real experimental observable. Specifically tis will tell us ow an equilibrium system canges in response to an applied potential. Te quantity tat will describe tis is a response function, a real observable quantity. We will go on to sow ow it is related to correlation functions. In tis also is peraps te more important type of observation. We will now deal wit a nonequilibrium system, but we will sow tat wen te canges are small away from equilibrium, te equilibrium fluctuations dictate te nonequilibrium response! Tus a knowledge of te equilibrium dynamics are useful in predicting non-equilibrium processes. So, te question is How does te system respond if you drive it from equilibrium? We will examine te case were an equilibrium system, described by a Hamiltonian H interacts weakly wit an external agent, V(t). Te system is moved away from equilibrium by te external agent, and te system absorbs energy from external agent. How do we describe te time-dependent properties of te system? We first take te external agent to interact wit te system troug an internal variable A. So te Hamiltonian for tis problem is given by H = H f (t) A. (.) Here f(t) is te time-dependence of external agent. We describe te beavior of an ensemble initially at termal equilibrium by assuming tat eac member of te ensemble is subject to te same interaction wit te external agent, and ten ensemble averaging. Initially, te system is at equilibrium and te internal variable is caracterized by an equilibrium ensemble average A.

3 8- Te external agent is ten applied at time t, and te system is moved away from equilibrium, and is caracterized troug a non-equilibrium ensemble average, A. A At ()as a result of te interaction. For a weak interaction wit te external agent, we can describe A( t ) by performing an expansion in powers of f ( t ). A()= t (terms f ( ) At ()= A + dt ) + (terms f () ) +K (.) ( ) ( ) R t,t f t +K (.3) In tis expression te agent is applied att, and we observe te system at t. Te leading term in tis expansion is independent of f, and is terefore equal to A. Te next term in (.3) describes te deviation from te equilibrium beavior in terms of a linear dependence on te external agent. R (t,t ) is te linear response function, te quantity tat contains te microscopic information tat describes ow te system responds to te applied agent. Te integration in te last term of eq. (.3) indicates tat te non-equilibrium beavior depends on te full istory of te application of te agent f ( quantum mecanical description of R. t ) and te response of te system to it. We are seeking a Rationalization for an expansion of At () in powers of f (t) : Let s break time up into infinitesimal intervals: t i = iδ f (t i ) = f i

4 8-3 A( t i )= A i = A i (K,f i,f i,f i ) Now, Taylor series expand about all f i = A At ( )= A (K,, ) + i i i f + K 444 j 4 3 j i f j f j = A Value wit no f applied Sum over cange due to force at all times of application Linear (first-order) term: A i f j = jδ Ai f j j f j f j = Δ t i j jδ f j... j ( j, i ) ( ) lim ( ) = dt R t t f t j Properties of te Response Function. Causality: Te system cannot respond before te force as been applied. Terefore R( t,t ) = for t < t, and te time-dependent cange in A is δa()= t At t () A = dt Rt, f t ( t ) ( ) (.4) Te lower integration limit as been set to to reflect tat te system is initially at equilibrium, and te upper limit is te time of observation. We can also make te statement of ( ) ( ) causality explicit by writing te linear response function wit a step response: Θ t t R t,t, were Θ (t < t ) (t t ) (t t ). (.5)

5 8-4. Stationarity: Similar to our discussion of correlation functions, te time-dependence of te system only depends on te time interval between application of potential and observation. Terefore we write R (t,t ) = R( t t ) and δ A()= t dt R ( t t ) f t t ( ) (.6) Tis expression says tat te observed response of te system to te agent is a convolution of te material response wit te time-development of te applied force. can write Rater tan te absolute time points, we can define a time-interval τ= t t, so tat we δa t ()= dτ R ( ) ( τ f t τ ) (.7) 3. Impulse response. Note tat for a delta function perturbation: We obtain f (t) = λδ (t t ) (.8) δ At ()= λr( t t ). (.9) Tus, R describes ow te system beaves wen an abrupt perturbation is applied and is often referred to as te impulse response function.

6 8-5 FREQUENCY-DOMAIN REPRESENTATION: THE SUSCEPTIBILITY Te observed temporal beavior of te non-equilibrium system can also be cast in te frequency domain as a spectral response function, or susceptibility. We start wit eq. (.7) and Fourier transform bot sides: i t + δ A( ω ) d t τ ( ) f ( t d R τ τ ) e ω (.) Now we insert e iωτ e +iωτ = and collect terms to give + ( ) dt d R δa ω = τ f t ) ( ) iω t τ iωτ τ ( ) ( τ e e (.) + ω ωτ dτ R ( τ ) e i (.) = dt e i t f t ( ) or δ A( ω ) = f % ( ω) χ ( ω ). (.3) In eq. (.) we switced variables, settingt = t τ. Te first term f % ( ω ) is a complex frequency domain representation of te driving force, obtained from te Fourier transform of f ( t ). Te second term χ ( ω ) is te susceptibility wic is defined as te Fourier-Laplace transform (single-sided Fourier transform) of te impulse response function. It is a frequencydomain representation of te linear response function. Switcing between time and frequency domains sows tat a convolution of te force and response in time leads to te product of te force and response in frequency. Tis is a manifestation of te convolution teorem: ( ) B ( t ) dτ A t ( τ ) B ( τ ) = dτ A τ ( ) B ( t τ ) = F A % ( ω) ( ) A t B % ω (.4) were A % ( ω ) = F A( t), F[L] is a Fourier transform, and F [L] is an inverse Fourier transform. Note tat R( τ ) is a real function, since te response of a system is an observable; owever, te susceptibility χ ( ω ) is complex: Since χ(ω ) = χ (ω ) + iχ (ω). (.5) We ave ( ) dτ R( ) χ ω = τ e iωτ, (.6)

7 8-6 χ= d R τ τ ) (.7) and τ ( )cos ωτ = Re F (R ( ) χ = d τ R τ sin τ ). ( ) ωτ = Im F (R ( ) (.8) χ and χ are even and odd functions of frequency: χ ( ω ) = χ ( ω ) (.9) χ ( ω )= χ ( ω ) (.) * so tat χ ( ω ) = χ ( ω). (.) Notice also tat eq. (.) allows us to write χ ( ω )= χ ( ω )+ χ ( ω ) (.) χ ( ω ) = χ ( ω ) χ ω. i ( ) (.3) KRAMERS-KRÖNIG RELATIONS Since tey are cosine and sine transforms of te same function, χ (ω) is not independent of χ ( ω ). Te two are related by te Kramers-Krönig relationsips: + χ ω χ ( ω)= P π ω ω ( ) dω (.4) ( ) + χ ω χ ( ω )= π P dω (.5) ω ω Tese are obtained by substituting te inverse sine transform of eq. (.8) into eq. (.7): ( )= + dt cosωt π ( ) sin ω t dω χ ω χ ω + L ( ) = lim dωχ ω cosωt sin ω t dt π L Using cos ax sin bx = sin (a + b) x + sin (b a) x, tis can be written as (.6)

8 8-7 cos(ω ω) L + cos(ω ω P ) + + χ ( ω)= lim L + dωχ ( ω ) π L ω + ω ω ω (.7) If we coose to evaluate te limit L, te cosine terms are ard to deal wit, but we expect tey will vanis since tey oscillate rapidly. Tis is equivalent to averaging over a monocromatic field. Alternatively, we can instead average over a single cycle: L = π /( ω ω) to obtain eq. (.4). Te oter relation can be derived in a similar way. Note tat te Kramers-Krönig relationsips are a consequence of causality, wic dictate te lower limit of t initial = on te first integral evaluated above.

9 8-8 Example: Classical Response Function and Susceptibility We can model absorption of ligt troug a resonant interaction of te electromagnetic field wit an oscillating dipole, using Newton s equations for a forced damped armonic oscillator: && x +γ & +ω x = F (t) (.8) x Here te x is te coordinate being driven, γ is te damping constant, and ω = k/m is te natural frequency of te oscillator. One way to solve tis problem is to take te driving force to ave te form of a monocromatic oscillating source Ten, equation (.8) as te solution x t qe F ()= t F cos ω t = cos ωt. (.9) m ()= qe m ((ω ω ) + 4γ ω ) sin (ωt + δ ) (.3) wit tanδ = ω ω. (.3) γω Tis sows tat te driven oscillator as an oscillation period tat is dictated by te driving frequency ω, and wose amplitude and pase sift relative to te driving field is dictated by te detuning (ω ω ). If we cycle average to obtain te average absorbed power from te field, te absorption spectrum is P ω F() t x& avg ( )= () t γω F = m (ω ω ) + 4γ ω A response function approac would be to find te solution to. (.3) x()= t dτ R ( τ ) f (t τ ), (.33) wic we can obtain by solving eq. (.8) using an impulsive driving force. If F()= t F δ (t t ), ten x() t = F R () t, and we obtain γ τ = exp τ sin Ωτ (.34) mω R ( ) Te reduced frequency is defined as

10 8-9 From tis we obtain te susceptibility χ ω = Ω= ω γ ( ) m ( 4 As we will see, te absorption of ligt by te oscillator is related to. (.35) i ). (.36) ω ω γω γω χ ( ω ) = m (ω ω ) + γ ω. (.37) For te case of weak damping γ<<ω, eq. (.36) is commonly written as a Lorentzian linesape by using te near-resonance approximation ω ω = (ω + ω )(ω ω ) ω ( ω ω ) χ( ω). (.38) mω ω ω + i γ /

11 8- Nonlinear Response Functions If te system does not respond in a manner linearly proportional to te applied potential but still perturbative, we can include nonlinear terms, i.e. iger expansion orders of At () in eq. (.3). Let s look at second order: δ A () t ( ) = ( ) ( ) ( ) ( dt dt R t;t,t f t f t (.39) Again we are integrating over te entire istory of te application of two forces f and f, including any quadratic dependence on f. In tis case, we will enforce causality troug a time ordering tat requires () tat all forces must be applied before a response is observed and () tat te application of f must follow f. Tat is t t t or wic leads to () R (t;t,t ) R () Θ(t t ) Θ ( t t ) (.4) δa() t () = t t dt dt R ( ) t;t,t ) f ( t ) f ( t ) (.4) ( Now we will call te system stationary so tat we are only concerned wit te time intervals between consecutive interaction times. If we define te intervals between adjacent interactions ) τ = t t (.4) τ = t t Ten we ave () () ) ( ) ( ) δa t = dτ dτ R ( ) (τ, τ f t τ τ f t τ (.43)

12 8-8.. QUANTUM LINEAR RESPONSE FUNCTIONS To develop a quantum description of te linear response function, we start by recognizing tat te response of a system to an applied external agent is a problem we can solve in te interaction picture. Our time-dependent Hamiltonian is H()= t H ( ) ˆ f t A = H + V ( t ) (.44) H is te material Hamiltonian for te equilibrium system. Te external agent acts on te equilibrium system troug A, ˆ an operator in te system states, wit a time-dependence f(t). We take V(t) to be a small cange, and treat tis problem wit perturbation teory in te interaction picture. We want to describe te non-equilibrium response A() t, wic we will get by ensemble averaging te expectation value of Â. Remember te expectation value for a pure state in te interaction picture is ( ) = ψ I (t) AI ( t ) ψ I (t) At = ψ UI AI U I ψ. (.45) Te interaction picture Hamiltonian for eq. (.44) is ( ) = VI t U t V t U t = f () t A () t ( ) ( ) ( ) I (.46) To calculate an ensemble average of te state of te system after applying te external potential, we recognize tat te non-equilibrium state of te system caracterized by described by ψ I (t) is in fact related to te initial equilibrium state of te system ψ, as seen in eq. (.45). So te non-equilibrium expectation value A() t is in fact obtained by an equilibrium average over te expectation value ofui AI U I : ()= pn nui AU I I n. (.47) n At Again n are eigenstates of H. Working wit te first order solution to U t U t I (,t ) = + i t t I () dt f ( t ) AI ( t ) (.48)

13 8- we can now calculate te value of te operator A at time t, integrating over te istory of te applied interaction f ( t ): At U ()= I A I U I = i t dt f + t ( t ) A I ( t ) AI ( t ) i t dt f t A t t ( ) ( I ) (.49) Here note tat f is te time-dependence of te external agent. It doesn t involve operators in H and commutes wit A. Working toward te linear response function, we just retain te terms linear in f ( t ) A t () A I ()+ t i t t d t f ( t ){ A ( t) A ( t ) A ( t ) A ( t ) I I I I () ( ) I ( ) A ( t ) = A t + i (.5) t dt f t A t, I t I } Since our system is initially at equilibrium, we set t = and switc variables to te time interval τ= t t and using A t = U t AU t obtain At I () ( ) ( ) ()= A t i ) A ( τ ), A I ( ) I () + d τ f ( t τ I (.5) We can now calculate te expectation value of A by performing te ensemble-average described in eq. (.47). Noting tat te force is applied equally to eac member of ensemble, we ave i At ()= A + dτ f( t τ ) A I ( ) A I ( ) τ, (.5) Te first term is independent of f, and so it comes from an equilibrium ensemble average for te value of A. n n At () = p n n = A (.53) Te second term is just an equilibrium ensemble average over te commutator in A I (t): n n A I A τ, A = p n A τ, A n. (.54) I ( ) I ( ) I ( ) I ( ) Comparing eq. (.5) wit te expression for te linear response function, we find tat te quantum linear response function is

14 8-3 i R ( τ ) = A τ, A τ = τ< I ( ) I ( ) (.55) or as it is sometimes written wit te unit step function in order to enforce causality: Te important ting to note is tat te time-development of te system wit te applied external potential is governed by te dynamics of te equilibrium system. All of te time-dependence in te response function is under H. (.56) Te linear response function is terefore te sum of two correlation functions wit te order of te operators intercanged, wic is te imaginary part of te correlation function C ( τ ) i τ = Θ( τ ){ A I ( τ ) A I ( ) A I ( ) A I ( τ ) } i * = Θ( τ )(C AA ( τ ) C AA ( τ )). (.57) = Θ( τ) C ( τ) R( ) As we expect for an observable, te response function is real. If we express te correlation function in te eigenstate description: ten i R τ = Θ τ A I τ, AI R( ) C t ( ) ( ) ( ) ω ()= p n A e i mn t (.58) n,m mn t = Θ() t pn A mn sinω mn t (.59) n,m R( τ ) can always be expanded in sines an odd function of time. Tis reflects tat fact tat te impulse response must ave a value of (te deviation from equilibrium) at t = t, and move away from at te point were te external potential is applied.

15 THE RESPONSE FUNCTION AND ENERGY ABSORPTION Let s investigate te relationsip between te linear response function and te absorption of energy from an electromagnetic field. We will relate tis to te absorption coefficient α α = E& / I wic we ave described previously. For tis case, H = H f ( t) A = H μ E( t ) (.6) Tis expression gives te energy of te system, so te rate of energy absorption averaged over te non-equilibrium ensemble is described by: E & = H = f A() t (.6) t t We will want to cycle-average tis over te oscillating field, so te time-averaged rate of energy absorption is E & T = T dt f t A() t () = T dt f t A + dτ R τ f t τ ( ) ( ) T t Here te response function is R(τ ) = i μ τ electromagnetic field, we can write ( ), ( ) () wic leads to te following for te second term in (.6): (.6) μ /. For a monocromatic f t = E cos ωt = E e iω t + E * e i ω t, (.63) iω( t τ τ ( ) E e ) * iω ( t τ ) d R τ + = E e i t E e ω χ ( ω )+ E e By differentiating (.63), and using it wit (.64) in eq. (.6), we ave T E & = A f( T ) f ( ) * iω t χ ( ω) (.64) T d t iω E e iω t + iω E * e iωt E e ω χ ω + E e i t χ ( 4T i t ( ) * ω ω) (.65) We will now cycle average tis expression, setting T = π ω. Te first term vanises and te cross terms in second integral vanis, because T So, te rate of energy absorption from te field is T i ω t +i t iω t i t dt e e ω = and T dt e e ω =.

16 8-5 i E ( ) ( ) E & = ω χ ω χ ω 4 ω = E χ ( ω) (.66) So, te absorption of energy by te system is related to te imaginary part of te susceptibility. Now, from te intensity of te incident field, I E = c /8π, te absorption coefficient is ( ) E& 4πω α ω = = χ ( ω ). (.67) I c Now, let s sow tat tis is consistent wit te expression we found earlier ( ) 4 ω ( e β ω i t α ω = π c Starting wit te imaginary part of te susceptibility ( ) ( ( ) ( )) i χ ω = χ ω χ ω ) dt e ω C μμ () t. (.68) ω ω = { dt e i t C () t C AA ( t ) dt e i t AA C AA ( t ) C AA ( t ) } = { dt e ω i t i t C AA ( t ) C AA ( ) = ( C% ω % AA ( ) C AA ( ω ) ) t dt e ω C AA ( t ) C AA ( ) t } We ave also establised tat te correlation functions obey te detailed balance condition: C % ( ω ) =e β ω C % (ω) = C % * AA (ω) (.69) AA AA (.7) Tis relationsip reflects te fact tat upward and downward transition rates between states separated by ω are related by te population difference. Tis allows us to write: So C % AA ( ω )± C % AA ( ω) = ( ± e β ω )C % AA (ω) (.7) β ω ) ( ) χ ( ω )= ( e C AA ω + ω = ( e β ω ) e i t A t Inserting into eq. (.67), we ave te result from earlier: πω α( ω )= ( + β ω iωt e ) c e μ () () ( ) A dt t ( ) (.7) μ dt (.73)

17 8-6 So te absorption of energy from an external force, tat is te time-evolution of a nonequilibrium system, is related to te imaginary part of χ. In turn, witin te weak perturbations allowed by linear response, χ is related to te Fourier transform of te correlation function tat describes te fluctuations and dynamics of te equilibrium system C AA (t). Relationsips of tis form tat relate non-equilibrium dynamics of te system driven away or relaxing toward equilibrium to te fluctuations about te equilibrium state are known as fluctuation-dissipation relationsips.

18 RELAXATION OF A PREPARED STATE Te impulse response function R () t describes te beavior of a system initially at equilibrium tat is driven by an external field. Alternatively, we may need to describe te relaxation of a prepared state, in wic we follow te return to equilibrium of a system initially eld in a non-equilibrium state. Tis beavior is described by step response function S( t). Te step response comes from olding te system wit a constant field H = H fa until a time t wen te system is released, and it relaxes to te equilibrium state governed by H = H. We can anticipate tat te form of tese two functions are related. Just as we expect tat te impulse response to rise from zero and be expressed as an odd function in time, te step response sould decay from a fixed value and look even in time. In fact, we migt expect to describe te impulse response by differentiating te step response. Response Functions are real. ( ) * ( ) Quantum Correlation Functions are complex: C t =C t Classical Correlation Functions are real and even: C( t) = C ( t) For relaxation in terms of a real observable tat is even in time, we construct a symmetrized function:

19 8-8 S AA () t = { AI () t A I ( ) + A I () AI ( t) } = A I ( t), A I () + = { C () t + C ( t) } AA AA = C AA () t S is related to te real part of te correlation function, and defined for t. Te impulse response is related to te time-derivative of te step response, and in te classical limit R t ()= d S AA () t kt dt (ig T limit) If we define S AA ( ω ) = dt S AA ( ) ω t e i t, ten S ω = C ω + C ω = + e β ω AA ( ) AA ( ) AA ( ) ( ) C AA ( ω) β ω ( ) tan χ ω = S ( ω ) S AA ( ω ) AA kt ω (classical limit) Tis is te fluctuation-dissipation teorem (Cemistry Nobel Prize, 968; proven in 95 by Callen and Welton). Lars Onsager (93): Te relaxation of macroscopic non-equilibrium disturbance is governed by te same laws as te regression of spontaneous microscopic fluctuations in an equilibrium state.