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Quantum Systems: Introduction • to Further Apply Linear Response Theory to Spectroscopic Problems We Need to Extend the Formalism to Quantum Systems

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Quantum Systems: Introduction • to Further Apply Linear Response Theory to Spectroscopic Problems We Need to Extend the Formalism to Quantum Systems University of Washington Department of Chemistry Chemistry 553 Spring Quarter 2010 Lecture 21: Survey of Quantum Statistics 05/17/10 J.L. McHale “Molecular Spectroscopy” Prentiss-Hall, 1999. McQuarrie: Ch. 21-8 R. Kubo “The Fluctuation-Dissipation Theorem” Rep. Prog. Phys. 29, 255 1966. R. Kubo, Statistical Mechanical Theory of Irreversible Processes I. J. Phys. Soc. Jpn. 12(6), 570 1957. A. Summary of Linear Response Theory • In the last few lectures we established several key relationships. This includes the displacement of a property B of a system from equilibrium by the application of a weak field t ∆=B tBtB − = dsFstsφ − (21.1) () () 0 ∫ ()BA ( ) −∞ where the linear response or after-effect function is 1 φ tdXfBtA===−,0 BtA ,0 BtA 0 (21.2) BA ()∫ 0 {} () ( ) {}() ( ) () ( ) kTB • If the field is time dependent i.e. Ft( ) = Fω cosω tthen t ⎡ ∞ ⎤ ∆=B tdsFstsFeeφ −=Re itωωτ− i φττ d ()∫∫ ()BA ( )ω ⎢ BA ( ) ⎥ −∞ ⎣ 0 ⎦ (21.3) ⎡⎤itω ′′′ ==+FeωωRe⎣⎦χωBA () F() χω () cos ωχω t () sin ω t • In equation (21.3) χ ()ωχωχω=−′( ) i ′′ ( ) is the one-sided Fourier transform of the response function and is called the complex susceptibility. The real part is related to the cycle-averaged reversible work done by the field on the system Uω and the imaginary part is related to the dissipation of energy from the field by the system Dω . FF22ω U=⋅ωωχ′′′()ωχω and D = () (21.4) ωω22BA BA • A fundamental relationship between the imaginary part of the susceptibility and the correlation function is the fluctuation-dissipation theorem: ω ∞ χ′′ ωω= dt A0cos B t t (21.5) BA () ∫ () () kT 0 • Finally, as a result of the causality of the linear response function (φBA ()t = 0 if t<0) the real and imaginary parts of the complex susceptibility are related by the Kramers-Kronig relations: 2 +∞ yyχ′′ ( ) χω′()=℘ dy πω∫ y22− 0 (21.6) 2ω +∞ χ′()y χω′′ =− ℘ dy () ∫ 22 πω0 y − • The physical meaning of the real and imaginary parts of the complex susceptibility can be given further meaning by way of example. Suppose a static field E0 is applied to a polarizeable material. The polarization P comprises the induced dipole moments and aligned permanent dipoles of the system. The polarization opposes the static field so the average field in the material is: P EE= 0 − (21.7) ε 0 where ε 0 is the free space dielectric constant or the free space permittivity. • The polarization is proportional to the average field PEE=−=ε 00(εεχre1) (21.8) ε where the relative permittivity (also called the dielectric constant) is ε r = , ε 0 ε is the permittivity of the material, and χer= ε −1 is th electric susceptibility. • If the field is time dependent tt P() t=−=−εφ E ()() t'''Re'' t t dt ε E⎡ eitω ′ φ () t t dt ⎤ 000∫∫−∞⎣⎢ −∞ ⎦⎥ ∞ ⎡⎤itωωτ− i it ω ==εφττεχω00EeeRe() d 00 Ee Re ⎡⎤e ( ) (21.9) ⎣⎦⎢⎥∫0 ⎣⎦ ⎡⎤itω = εχω00EeRe ⎣⎦e () where now χωer()=− εω( ) 1, ε rr(ωεωεω) =+′ ( ) i r′′ ( ) and where the real and imaginary parts of ε r (ω) satisfy KK relations. • Note: The convention for the permittivity has it origins in the definition of the electric displacement D: PE==−εχωεεωRe⎡⎤ eitωω E Re⎡ e it 1 ⎤ 00⎣⎦er() 00 ⎣ ( ( ) )⎦ =−εεω0 Et()()r ( ) 1 (21.10) ∴Dt()=+ Pt ()εεωεεω00 Et () =r ( ) Et () = ( ) Et () • Now the refractive index is related to the dielectric permittivity by 2 n ()ω = εωr () so if the permittivity is complex so is the refractive index: nnin()ωω=+′′′ () ( ω) 22 (21.11) ′ ′ ′′ ′′ ′ ′′ ∴ε rr()ωω=−⎣⎦⎣⎦⎡⎤⎡⎤nnandnn () () ω εωωω () =2 () () • Consider an electromagnetic plane wave propagating through a medium with electric field E tEikxt=−Re⎡ exp ω ⎤ (21.12) () ⎣ 0 ⎣⎡ ( )⎦⎤⎦ 2πωn( ) 2π and the wave number is knin==()′′′()ω + ()ω . This puts (21.11) λλ into the form ⎡⎤⎡⎤2π x ⎛⎞′′′ Et()=+−Re⎢⎥ E0 exp ⎢⎥ i⎜⎟() n()ωωω in () t ⎣⎦⎣⎦⎝⎠λ (21.13) ⎡⎤ ⎡⎤⎛⎞⎛⎞22π xn′′′()ωπω xn () =−−Re⎢⎥Ei0 exp⎢⎥⎜⎟⎜⎟ω t exp ⎣⎦⎢⎥⎣⎦⎢⎥⎝⎠⎝⎠λλ • According to (21.12) n′(ω), which is commonly referred to as the refractive index, changes the wavelength of the wave as it propagates through the medium while the dispersive term n′′(ω) attenuates the amplitude of the wave. • Now consider now the wave intensity: 2 ⎡⎤−−4πωn′′( ) ⎡⎤ ωεωr ′′ ( ) −γ x I ∝=E I000exp⎢⎥ xI = exp ⎢⎥ xIe = (21.14) ⎣⎦λω⎣⎦nc′() where γ is the coefficient of absorption given in the Lambert-Beer Law as γ = ε M C (21.15) C is the concentration in molar units and ε M is the molar absorptivity. B. Quantum Systems: Introduction • To further apply linear response theory to spectroscopic problems we need to extend the formalism to quantum systems. • We will first define the quantum Liouville operator, the density operator, and the and the essential features of quantum mechanical linear response theory…which leads to Kubo’s theorem and the quantum mechanical fluctuation dissipation theorem. • We will find the Classical/Quantum Correspondences useful. o The classical distribution function f(X,t) corresponds to the quantum density matrix ρ(t). The equilibrium density operator is defined as −β H0 e −β H0 ρρeq ==0 …where Q = Tr() e (21.16) Q Which corresponds to the classical distribution function: e−β Hpq( , ) 1 f== f f() X,0 = where……… Q = dp dp dq dq e−β H ()pq, eq 011Qh3N ∫∫NN (21.17) • The equation of motion of the density operator is ∂ρ i =−[]HL, ρ =− ρ ∂t (21.18) where[]HHH , ρ =−ρρ • Classically: ∂f =−{}Hf, =− Lf ∂t (21.19) ∂Hf∂∂∂ Hf where{}Hf , =− ∂p ∂∂∂qqp • There is also correspondence between the classical ensemble average and the quantum mechanical average At=⇔= dXAXtf,, Xt At TrAtρ t (21.20) ()classical∫ ( ) ( ) ( ) quantum ( ( )()) • Finally the quantum correlation function is ⎪⎪⎧⎫⎛⎞Bt() A(00) + A( ) Bt( ) ⎧ ρ ⎫ gtTr==ρ TrBtA0 ⎡ 0 ⎤ (21.21) BA () ⎨⎬⎨⎬0 ⎜⎟⎣ () ( )⎦+ ⎩⎭⎪⎪⎝⎠22⎩⎭ • The quantum/classical correspondence is ⎧⎫ρ0 gBA() t=⇔= Tr⎨⎬⎣⎦⎡⎤ Bt() A (00 ) C BA () t dXfBtA0 () ( ) (21.22) ⎩⎭2 + ∫ • With (20.1)-(20.7) we can obtain the quantum fluctuation dissipation theorem…which is our first step toward a theory of spectroscopic line shapes. ∂ρ ii ii i =−[]H,,,,,ρρρρρρ =−⎡⎤ HAFt −() =−[] H +[] AFtL() =− + [] AFt() ∂t ⎣⎦00 0 i t ∴ρρteAFsds=+ −−Lts0 () , ρ () 0 ∫ []() −∞ (21.23) • The displacement of the operator B is therefore: i ⎛⎞t ∆=B t Tr Bρρ − Tr B ≈ Tr B e−−Lts0 () A, ρ F s ds () ( ) (0 ) ⎜⎟∫ []() ⎝⎠−∞ iitt ==−∫∫dsF() s Tr() Be−−Lts0 ()[] A,,ρ dsF() s Tr() B ( t s )[] A ρ −∞ −∞ (21.24) iitt dsF s Tr B t−= s A, ρφ dsF s t − s ∫∫()() ( )[] ()BA ( ) −∞ −∞ φρρBA ()tTrBtA=≈() ()[],, TrBtA()()[]0 o .
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