MIT5_74s09_lec07

MIT5_74s09_lec07 - MIT OpenCourseWare http:/ocw.mit.edu...

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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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8-1 Andrei Tokmakoff, MIT Department of Chemistry, 3/03/09 8.1. LINEAR RESPONSE THEORY We have statistically described the time-dependent behavior of quantum variables in an equilibrium system through correlation functions. We have also shown that spectroscopic lineshapes are related to correlation functions for the dipole moment. But it’s not the whole story. You have probably sensed this from the perspective that correlation functions are complex , and how can observables be complex? We will use linear response theory as a way of describing a real experimental observable. Specifically this will tell us how an equilibrium system changes in response to an applied potential. The quantity that will describe this is a response function, a real observable quantity. We will go on to show how it is related to correlation functions. In this also is perhaps the more important type of observation. We will now deal with a nonequilibrium system, but we will show that when the changes are small away from equilibrium, the equilibrium fluctuations dictate the nonequilibrium response! Thus a knowledge of the equilibrium dynamics are useful in predicting non-equilibrium processes. So, the question is “How does the system respond if you drive it from equilibrium?” We will examine the case where an equilibrium system, described by a Hamiltonian H 0 interacts weakly with an external agent, V ( t ). The system is moved away from equilibrium by the external agent, and the system absorbs energy from external agent. How do we describe the time-dependent properties of the system? We first take the external agent to interact with the system through an internal variable A . So the Hamiltonian for this problem is given by HH = 0 f ( t ) A . (0.1) Here f ( t ) is the time-dependence of external agent. We describe the behavior of an ensemble initially at thermal equilibrium by assuming that each member of the ensemble is subject to the same interaction with the external agent, and then ensemble averaging. Initially, the system is at equilibrium and the internal variable is characterized by an equilibrium ensemble average A .
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8-2 The external agent is then applied at time t 0 , and the system is moved away from equilibrium, and is characterized through a non-equilibrium ensemble average, A . A At () as a result of the interaction. For a weak interaction with the external agent, we can describe A t by performing an expansion in powers of f t . 1 A t () = ( terms f 0 ) + ( terms f ) + K (0.2) A + d t 0 R t , 0 f t 0 + K (0.3) ( ) ( ) In this expression the agent is applied at t 0 , and we observe the system at t . The leading term in this expansion is independent of f , and is therefore equal to A . The next term in (0.3) describes the deviation from the equilibrium behavior in terms of a linear dependence on the external agent. R ( t,t 0 ) is the linear response function, the quantity that contains the microscopic information that describes how the system responds to the applied agent.
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MIT5_74s09_lec07 - MIT OpenCourseWare http:/ocw.mit.edu...

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