MIT Department of Chemistry 5.74, Spring 5: Introductory Quantum Mechanics II Instructor: Professor Andrei Tokmakoff p. 8 LINEAR RESPONSE THEORY We have statistically described the time-dependent behavior of quantum variables in an equilibrium system through correlation functions. These can be used to describe deterministic (oscillatory) or random (stochastic) behavior. We ve also shown that spectroscopic lineshapes are related to correlation functions for the dipole moment. But it s not the whole story, and you ve probably sensed this from the perspective that correlation functions are complex, and how can observables be complex? We will now talk about linear response theory, which is a way of describing a real experimental observable: how does an equilibrium system change in response to an applied force? The quantity that will describe this is a response function, a real observable quantity. We will go on to show that it is related to sums of correlation functions. In this also is perhaps the more important type of observation. We will now deal with a nonequilibrium system, but we will show that when the changes are small away from equilibrium, the equilibrium fluctuations dictate the nonequilibrium response! Thus a knowledge of the equilibrium dynamics are useful in predicting non-equilibrium processes. So, how does the system respond if you drive it from equilibrium? > The system is moved away from equilibrium by external agent. > The system absorbs energy from external agent. What are the time-dependent properties of the system? H = H f (t) A Internal variable Time-dependence of external agent Hamiltonian of equilibrium state
p. 8 We average over an ensemble, each member of which is subject to same perturbation. average over equilibrium ensemble average over nonequilibrium ensemble external force applied at t moving system from A A A ()due t to interaction Let s develop At ()as an expansion in powers of f( t). A t ()=(terms f ( ) ) + (terms f () ) + + At A dt R t,t ) f (t ) + ()= + ( R (t,t ) : Linear Response Function The force is applied at t, and we observe the system at t. The linear response function is the quantity that contains the microscopic information that describes how the system responds to the ) applied force. We will look to find a quantum description of R (. Rationalization for an expansion of At ( ) in powers of f( t): Let s break time up into infinitesimal intervals: A ( t i )= A i = A i (,f i,f i,f i ) t i = i f (t i ) = f i
p. 83 Now, Taylor series expand about all f i = ( )= A ( ) + A i f j + At i i,, j i f j f j = A Value with no f applied Sum over change due to force at all times of application Linear (first order) term: j A i f j f j = j A i f j = f j j j f j lim = t i dt j R ( t i,t j ) f ( t j ) Properties of the Response Function Causality: The system cannot respond before the force has been applied. R( t,t ) = for t < t The time-dependent change in A is t δa()= t At () A = dt R t,t ) f ( t ( ) Stationarity: The time-dependence of the system only depends on the time interval between application of force and observation. R (t,t ) = R ( t t ) So, t δ A()= t dt R t t ( ) f ( t ) The response of the system is a convolution of the material response with the time-development of the applied force.
p. 84 Usually, we define the time interval τ = t t δa t ()= d τ R ( τ ) f (t τ) Impulse response. For a delta function perturbation: f ( t )= λδ (t t ) δ At ()= λr ( t t ) Thus, R describes how the system behaves when an abrupt perturbation is applied and is often referred to as the impulse response function. Frequency-Domain Representation δa t ()= d τ R ( τ ) f (t τ) Fourier Transform both sides: A( ) δ ω dt + d τ R ( ) f (t τ) e τ iωt insert (e iωτ e +iωτ ) A( ) + ( ω τ) R ( ) f (t τ) δ ω dt d τ e i t τ setting t = t τ dt = dt F.T. + iωt iωt = dt e f t τ ( ) d τ R ( ) e f ( ω ) χω Fourier-Laplace transform Susceptibility δa( ω ) =χ ( ω ) f ( ω ) spectral response
p. 85 A convolution of the force and response in time leads to the product of the force and response in frequency. This is a manifestation of the convolution theorem: A ( t ) B ( t ) dτ A ( t τ τ ) B ( )= ( where A( ω )=F A ( t ) and F [ ] is a Fourier transform. τ d A( )B t ) = F A(ω) B (ω) The susceptibility is the frequency domain representation of the linear response function. Spectrally the induced changes in the variable A is a product of the susceptibility with the spectral representation of the driving force f. Note that R ( ) τ is a real function, since the response of a system is an observable; however, the susceptibility χ( ω) is complex. We will relate C AA (τ) to R (τ ) and σ abs ( ω ) to χ( ω). χ( ω ) =χ ( ω ) + i χ ( ω) ( ) χ ω d τ R (τ) e iωτ ( )cos ωτ + i d R ( )sin ωτ = dτ R τ τ τ χ : even in frequency χ : odd in frequency χ ( ω ) = Re F (R(τ)) χ ( ω ) = Im F (R(τ)) χ ( ω)= χ ( ω) χ ( ω)= χ ( ω ) χ( ω) = χ * (ω ) Notice also χ ( ω)= [χ(ω)+ χ( ω )] ( )= [χ(ω) χ( ω )] χ ω i * χ (ω )
p. 86 KRAMERS-KRÖNIG RELATIONS Since they are cosine and sine transforms of the same function, χ (ω ) is not independent of χ ( ω). The two are related by the Kramers-Krönig relationship: ( ) + χ ω χ ( ω ) = dω π P ω ω + χ ω π P ω ω χ ( ω ) = ) dω ( These are obtained from and ( ) χ ω ) R (t cos ωt dt + R t χ ( ω )sin ω t dω ()= π Substituting: + ( ) dt cos ωt χ χ ω = ( )sin ω t dω π + lim dω χ cos ωt sin ω t dt = π L ( ) Using cos ax sin bx = sin ( a + b) x + sin ( b a) x L + χ ω = ( ) ( ) lim P dω χ ω ω + ω π L ω ω cos (ω + ω) L + cos (ω ω) L + If we choose L, the cosine terms are hard to deal with, but we expect they will vanish since they oscillate rapidly. This is equivalent to averaging over a monochromatic field. Alternatively, we can instead average over a single cycle: L = π/ ( ω ω), and obtain ( ) ( ) + χ ω χ ω = P dω π ω ω The other relation can be derived in a similar way. Note that these relationships are a consequence of causality, which dictate the lower limit of t initial = on the first integral evaluated above.
p. 87 Example: Classical Response Model absorption of radiation by dipoles with a forced damped harmonic oscillator: +γ +ω x x t x = F ( ) qe For an E.M. wave: F()= t F cos ω t = cos ωt m x t ()= qe m (ω ω ) +γ ω cos γω sin δ= (ω ω ) +γ ω (ω t +δ) An impulsive driving force gives the response function: x()= t d τ R (τ) f (t τ) if Ft ()= F δ(t t ), then x()= t F R ( t ) : R ( τ ) = γ exp τ sin Ωτ Ω= ω γ mω χ( ω ) = m (ω ω iγω) 4 χ ( ω ) = γω m (ω ω ) +γ ω <ω χ ω m ω ω ω + i γ / for γ< ( ) m χ= ω ω + i γω ( )( ω ) ω ( ω ω ) ω( ω ω ) ω ω = ω+ ω ω for ω ω χ = m ω ( ω ω ) + i γω m ω iγ ( ω ω ) +
p. 88 Nonlinear Response Functions If the system does not respond in a manner linearly proportional to the applied force, we can include nonlinear terms: the higher expansion orders in At ( ). Let s look at second order: () () dt dt R( ) (t;t δa t =,t ) f (t ) f (t ) Again we are integrating over the entire history of the application of two forces f and f, including any quadratic dependence on f. In this case, we will enforce causality through a time ordering that requires () that all forces must be applied before a response is observed and () that the application of f must follow f : t t t or R () (t;t,t ) R () Θ (t t ) Θ(t t ) () () t t δa t = dt dt R ( ) (t;t,t ) f (t ) f (t ) Now we will call the system stationary so that we are only concerned with the time intervals between interactions. t t () () δa t = dt dt R ( ) (t t,t t ) f (t ) f (t ) If we define the intervals between adjacent interactions τ = t t τ = t t = dτ dτ R ( ) (τ τ ) f (t τ, τ ) f (t τ )