MIT5_74s09_lec11

MIT5_74s09_lec11 - MIT OpenCourseWare http/ocw.mit.edu 5.74...

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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 .
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Andrei Tokmakoff, MIT Department of Chemistry, 3/12/2009 p. 11-1 11.1 VIBRATIONAL RELAXATION * Here we want to address how a quantum mechanical vibration undergoes irreversible energy dissipation as a result of interactions with other intra- and intermolecular degrees of freedom. Why is this process important? It is the fundamental process by which non-equilibrium states thermalize. As chemists, this plays a particularly important role in chemical reactions, where efficient vibrational relaxation of an activated species is important to stabilizing the product and not allowing it to re- cross to the reactant well. We will be looking specifically at vibrational couplings and relaxation, but the principles are the same for spin-lattice relaxation and electronic population relaxation through electron-phonon coupling. For an isolated molecule with few vibrational coordinates, an excited vibrational state must relax by interacting with the remaining internal vibrations or the rotational and translational degrees of freedom. If a lot of energy must be dissipated, radiative relaxation may be more likely. In the condensed phase, relaxation is usually mediated by the interactions with the environment, for instance, the solvent or lattice. The solvent or lattice forms a continuum of intermolecular motions that can absorb the energy of the vibrational relaxation. Quantum mechanically this means that vibrational relaxation (the annihilation of a vibrational quantum) leads to excitation of solvent or lattice motion (creation of an intermolecular vibration that increases the occupation of higher lying states). For polyatomic molecules it is common to think of energy relaxation from high lying vibrational states ( kT << h ω 0 ) in terms of cascaded redistribution of energy through coupled modes of the molecule and its surroundings leading finally to thermal equilibrium. We seek ways of describing these highly non- equilibrium relaxation processes in quantum systems.
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Andrei Tokmakoff, MIT Department of Chemistry, 3/12/2009 p. 11-2 Classically vibrational relaxation reflects the surroundings exerting a friction on the vibrational coordinate which damps its amplitude and heats the sample. We have seen that a Langevin equation for an oscillator experiencing a fluctuating force f ( t ) describes such a process: Q t ( ) + ω 0 Q γ Q f t ( ) && 2 2 & = / m (12.1) This equation ascribes a phenomenological damping rate γ to the vibrational relaxation; however, we also know in the long time limit, the system must thermalize and the dissipation of energy is related to the fluctuations of the environment through the classical fluctuation-dissipation relationship: ( ) ( 0 ) f t f 2 γ T δ ( t ) (12.2) = m k We would also like to understand the correspondence between these classical pictures and quantum relaxation.
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