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MIT 5 74 - QUANTUM MECHANICAL TREATMENT OF FLUCTUATIONS

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MIT OpenCourseWare http ocw mit edu 5 74 Introductory Quantum Mechanics II Spring 2009 For information about citing these materials or our Terms of Use visit http ocw mit edu terms Andrei Tokmakoff MIT Department of Chemistry 3 15 08 7 11 7 4 QUANTUM MECHANICAL TREATMENT OF FLUCTUATIONS Introduction In describing fluctuations in a quantum mechanical system we will now address how they manifest themselves in an electronic absorption spectrum by returning to the Displaced Harmonic Oscillator model As previously discussed we can also interpret the DHO model in terms of an electronic energy gap which is modulated as a result of interactions with nuclear motion While this motion is periodic for the case coupling to a single harmonic oscillator we will look this more carefully for a continuous distribution of oscillators and show the correspondence to classical stochastic equations of motion Energy Gap Hamiltonian Now let s work through the description of the Energy Gap Hamiltonian more carefully Remember that the Hamiltonian for coupling of an electronic transition to a harmonic degree of freedom is written as H 0 H e Ee H g Eg 7 48 H 0 eg H eg 2H g 7 49 where the Energy Gap Hamiltonian is H eg H e H g 7 50 Note how eq 7 49 can be thought of as an electronic system interacting with a harmonic bath where H eg plays the role of the system bath interaction H 0 H S H SB H B 7 51 We will express the energy gap Hamiltonian through reduced coordinates for the momentum coordinate and displacement of the oscillator p q See Mukamel Ch 8 and Ch 7 2 0 m p m 0 q 2 7 52 7 53 7 12 m 0 d 2 d 7 54 H e 0 p 2 q d H g 0 p q 2 2 2 7 55 From 7 50 we have H eg 2 0 d q 0 d 2 2 0 d q 7 56 So we see that the energy gap Hamiltonian describes a linear coupling of the electronic system to the coordinate q The slope of Heg versus q is the coupling strength and the average value of Heg in the ground state Heg q 0 is offset by the reorganization energy To obtain the absorption lineshape from the dipole correlation function we must evaluate the dephasing function C t eg e F t e iH g t i eg t F t 7 57 e iHet U g U e 7 58 2 We now want to rewrite the dephasing function in terms of the time dependence to the energy gap H eg that is if F t U eg then what is U eg This involves a transformation of the dynamics to a new frame of reference and a new Hamiltonian The transformation from the DHO Hamiltonian to the EG Hamiltonian is similar to our derivation of the interaction picture Note the mapping H e H g H eg H H0 V 7 59 Then we see that we can represent the time dependence of H eg by evolution under H g The time propagators are 7 13 e iHet e iH g t exp i d H t eg 0 7 60 U e U gU eg and H eg t e iH g t H eg e iH g t U H egU g 7 61 g Remembering the equivalence between H g and the bath mode s H B indicates that the time dependence of the EG Hamiltonian reflects how the electronic energy gap is modulated as a result of the interactions with the bath That is U g U B Equation 7 60 immediately implies that i t U eg exp d H eg 0 F t e iH g t i e iH et exp 7 62 d H t 0 eg 7 63 Note Transformation of time propagators to a new Hamiltonian If we have eiH At Ae iH Bt and we want to express this in terms of Ae i H B H A t Ae iH BAt we will now be evolving the system under a different Hamiltonian H BA We must perform a transformation into this new frame of reference which involves a unitary transformation under the reference Hamiltonian H new H ref H diff e iH newt e iH ref t i exp H diff U ref H diff U ref d H t 0 diff 7 14 This is what we did for the interaction picture Now proceeding a bit differently we can express the time evolution under the Hamiltonian of H B relative to H A as H B H A H BA i e iH Bt e iH At exp d H t 0 BA where H BA e iH At H BA e iH At This implies i e iH At e iH Bt exp d H t 0 BA Using the second order cumulant expansion allows the dephasing function to be written as i t F t exp d H eg 0 i 2 t d 0 2 2 0 d 1 H eg 2 H eg 1 H eg 2 H eg 1 7 64 Note that the cumulant expansion is here written as a time ordered expansion here The first exponential term depends on the mean value of H eg H eg 0 d 2 7 65 This is a result of how we defined H eg Alternatively the EG Hamiltonian could also be defined relative to the energy gap at Q 0 H eg H e H g In fact this is a more common definition In this case the leading term in 7 64 would be zero and the mean energy gap that describes the high frequency system oscillation in the dipole correlation function is eg The second exponential term in 7 64 is a correlation function that describes the time dependence of the energy gap H eg 2 H eg 1 H eg 2 H eg 1 H eg 2 H eg 1 where H eg H eg H eg Defining the time dependent energy gap frequency in terms of the EG Hamiltonian as 7 66 7 67 7 15 eg we obtain H eg 7 68 Ceg 2 1 eg 2 1 eg 0 i F t exp t t d 2 0 2 0 7 69 d 1 Ceg 2 1 7 70 So the dipole correlation function can be expressed as C t eg e 2 t 2 0 0 i Ee Eg t g t e g t d 2 d 1 eg 2 1 eg 0 7 71 7 72 This is the correlation function expression that determines the absorption lineshape for a timedependent energy gap It is a perfectly general expression at this point The only approximation made for the bath is the second cumulant expansion Now let s look specifically at the case where the bath we are coupled to is a single harmonic mode Evaluating the energy gap correlation function Ceg t pn n eg t eg 0 n n 1 2 p n ne iH …


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