11 - p 72 MIT Department of Chemistry 5.74 Spring 2004...

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MIT Department of Chemistry p. 72 5.74, Spring 2004: Introductory Quantum Mechanics II Instructor: Prof. Andrei Tokmakoff TIME-CORRELATION FUNCTIONS > T.C.F.s are statistical measures of the time-evolution of an observable under equilibrium conditions. They describe an ensemble. > T.C.F.s can be used to describe spectroscopy and other time-dependent phenomena. > They are generally applicable to any time-dependent process for an ensemble, but are commonly used to describe random or stochastic processes in condensed phases. Qualitatively: A correlation function describes how long a given property of a system persists until it is averaged out by microscopic motions of system. It describes how and when a statistical relationship has vanished. T.C.F.s describes the dynamics associated with a dynamical variable A(t), averaged over the ensemble. A : Microscopic variable A : Equilibrium ensemble average Classical: A ∫∫ ( d p d q A p , q ; t ) f ( p ) f: equil. Prob. distribution function Quantum: A = p n A n p n = e −β E n / Z n n A i () : Expectation value of A for a member of ensemble as a function of time. t A i At t
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p. 73 If we look at this behavior there seems to be little information in the random fluctuations of A , but there are characteristic time scales and amplitudes to these changes. We can characterize these by comparing the value of A at time t with the value of A at time t’ later. Correlation functions are defined as a time-dependent quantity, At ) , multiplied by that quantity ( at some later time, At () , and averaged over ensemble: ( ) ( C AA ( t, t ) ≡ A t A t ) auto-correlation function C AB ( t, t ) ≡ A t ) B t ) cross-correlation function ( ( These are products of a pair of dynamical variables.
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11 - p 72 MIT Department of Chemistry 5.74 Spring 2004...

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