16 - Page 110 QUANTUM MECHANICAL TREATMENT OF FLUCTUATIONS...

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Page 110 QUANTUM MECHANICAL TREATMENT OF FLUCTUATIONS * Introduction and Preview Now the origin of frequency fluctuations is expected to be interactions of our molecule (or more appropriately our electronic transitions) with its environment. This we can treat with our D.H.O. model, which is a general approach to coupling to nuclear vibrations. We found that iH t 0 µ e 0 µ e iH t t µ ( ) µ ( 0 ) = p n n n n i 2 − ω eg t H t iH t e g e e e = µ eg We can write this in terms of the Hamiltonian that describes the electronic energy gap’s dependence on Q (deviation relative to ω eg ): H eg = H e TOT H TOT = ω = H e H g (Energy Gap Hamiltonian) g eg 2 i ω t iH t e eg eg t C µµ ( ) = e µ eg Now if we believe there are interactions that lead to fluctuations in the energy group—variations in d or ω 0 , then our H eg is now time-dependent! t i t + C µµ ( ) = e ω eg t exp i d τ H eg ( ) τ = 0 Performing the cumulant expansion: i t exp = 0 τ − i t d τ H eg ( ) 1 eg τ 2 eg d τ H eg ( ) = exp = 0 τ +  − i 2 0 t d τ 2 0 τ 2 d τ H ( ) H ( τ 1 ) + + = * See Mukamel, Ch. 8 and Ch. 7
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Page 111 t t Defining δω eg ( ) H eg ( ) = F t g t ( ) ( ) = e ( ) = t d τ 2 d τ δω ( τ δω ( 0 ) g t 1 ) 1 eg eg 0 0 τ 2 So we have an expression for how the time-dependence of the energy gap Hamiltonian leads to the lineshape. Also note: H 0 = H e + E e + H g + E g 2 2 Q 2 H = p + 1 g 2 m 2 m ω D = = ω + H + 2 H eg eg g 2 2 1 2 2 H = H H = 1 m ω 0 ( Q d ) 2 m ω Q eg e g 2 0 2 2 2 = − ω 0 m d Q + 1 m ω 0 d ±²³² 2 ²³² ´ ± ´ linear in Q const Note that this looks very much like a Hamiltonian that describes the coupling of an electronic system to a bath [one degree of freedom here] of H.O. with a linear coupling between the two! H H S + H + H SB = B = e E + λ e + g E g g H S e 2 2 H B = p + 1 m ω 0 Q 2 2 m 2 2 H SB ( H ) = m ω 0 d 2 Q ²³² eg ± ´ coupling strength Fluctuations in coupling to bath could lead to line broadening! Equivalently, coupling to a bath of many harmonic oscillators should lead to line-broadening.
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Page 112 Time-Dependent Energy Gap Hamiltonian Let’s work through this more carefully. Start by defining reduced coordinates E λ e H g H TOT TOT = ω 0 m p = p ~ 2 m ω 0 E D q = q ~ 2 = m ω 0 d = d ~ 2 = E A H = ω 0 p 2 + q + d 2 0 d Q = e ~ ~ ~ H eg 2 H g = ω 0
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