MIT5_74s09_lec09

MIT5_74s09_lec09 - 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 Dept. of Chemistry, 3/5/2009 7-1 7. OBSERVING FLUCTUATIONS IN SPECTROSCOPY 1 Here we will address how fluctuations are observed in spectroscopy and how dephasing influences the absorption lineshape. Our approach will be to calculate a dipole correlation function for a dipole interacting with a fluctuating environment, and show how the time scale and amplitude of fluctuations are encoded in the lineshape. Although the description here is for the case of a spectroscopic observable, the approach can be applied to any such problems in which the deterministic motions of an object under an external potential are modulated by a random force. We also aim to establish a connection between this picture and the Displaced Harmonic Oscillator model. Specifically, we will show that a frequency-domain representation of the coupling between a transition and a continuous distribution of harmonic modes is equivalent to a time-domain picture in which the transition energy gap fluctuates about an average frequency with a statistical time-scale and amplitude given by the distribution coupled modes. 7.1. FLUCTUATIONS AND RANDOMNESS: SOME DEFINITIONS 2 “Fluctuations” is my word for the time-evolution of a randomly modulated system at or near equilibrium. You are observing an internal variable to a system under the influence of thermal agitation of the surroundings. Such processes are also commonly referred to as stochastic. A stochastic equation of motion is one in which there is both a deterministic and a random component to the time-development. Randomness is a characteristic of all physical systems to a certain degree, even if the equations of motion that govern them are totally deterministic. This is because we generally have imperfect knowledge about all of the degrees of freedom for the system. This is the case when we look at a subset of particles which are under the influence of others that we have imperfect knowledge of. The result is that we may observe random fluctuations in our observables. This is always the case in condensed phase problems. It’s unreasonable to think that you will come up with an equation of motion for the internal determinate variable, but we should be able to understand the behavior statistically and come up with equations of motion for probability distributions 1 For readings on this topic see: Nitzan, A. Chemical Dynamics in Condensed Phases (Oxford University Press, New York, 2006), Chapter 7; C.H. Wang, Spectroscopy of Condensed Media: Dynamics of Molecular Interactions , Academic Press, Orlando, 1985.
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MIT5_74s09_lec09 - MIT OpenCourseWare http/ocw.mit.edu 5.74...

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