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MIT5_74s09_lec10

# MIT5_74s09_lec10 - MIT OpenCourseWare http/ocw.mit.edu 5.74...

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Unformatted text preview: 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 . 7-11 Andrei Tokmakoff, MIT Department of Chemistry, 3/15/08 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 + E e + H g + E g (7.48) H 0 = = ω eg + H eg + 2 H 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 ˆ . (7.52) 0 2 p m ω = = ¡ q = m ω 0 q ˆ (7.53) ¡ 2 = § See Mukamel, Ch. 8 and Ch. 7. 7-12 d = m ω 0 d (7.54) ¡ 2 = 2 H e = = ω 0 ( p 2 + ( q − d ) ) ¡ ¡ ¡ (7.55) 2 2 H g = = ω 0 ( p + q ) ¡ ¡ From (7.50) we have H eg = − 2 = ω d q + = ω d 2 ¡ ¡ ¡ (7.56) = − 2 = ω d q + λ ¡ ¡ So, we see that the energy gap Hamiltonian describes a linear coupling of the electronic system to the coordinate q. The slope of H eg versus q is the coupling strength, and the average value of H eg in the ground state, H eg ( q=0 ), is offset by the reorganization energy λ . To obtain the absorption lineshape from the dipole correlation function we must evaluate the dephasing function. 2 C μμ ( ) t = μ e − i ω eg t F t ( ) (7.57) eg e iH t e − e g iH t † F t ( ) = = U U (7.58) g e 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 = H 0 + 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 − iH t e e = = e − iH t g =...
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