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5.74 Introductory Quantum Mechanics II
Spring 2009
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View Full Document Andrei Tokmakoff, MIT Dept. of Chemistry, 3/5/2009
71
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 frequencydomain representation of the
coupling between a transition and a continuous distribution of harmonic modes is equivalent to a
timedomain picture in which the transition energy gap fluctuates about an average frequency
with a statistical timescale and amplitude given by the distribution coupled modes.
7.1. FLUCTUATIONS AND RANDOMNESS: SOME DEFINITIONS
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“Fluctuations” is my word for the timeevolution 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 timedevelopment.
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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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.
 Spring '04
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