MIT5_74s09_lec05

MIT5_74s09_lec05 - 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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5-1 5.1. TIME-CORRELATION FUNCTIONS Time-correlation functions are an effective and intuitive way of representing the dynamics of a system, and are one of the most common tools of time-dependent quantum mechanics. They provide a statistical description of the time-evolution of a variable for an ensemble at thermal equilibrium. They are generally applicable to any time-dependent process for an ensemble, but are commonly used to describe random (or stochastic) and irreversible processes in condensed phases. We will use them in a description of spectroscopy and relaxation phenomena. This work is motivated by finding a general tool that will help us deal with the inherent randomness of molecular systems at thermal equilibrium. The quantum equations of motion are deterministic, but this only applies when we can specify the positions and momenta of all the particles in our system. More generally, we observe a small subset of all degrees of freedom (the “system”), and the time-dependent properties that we observe have random fluctuations and irreversible relaxation as a result of interactions with the surroundings. It is useful to treat the environment (or “bath”) with the minimum number of variables and incorporate it into our problems in a statistical sense – for instance in terms of temperature. Time-correlation functions are generally applied to describe the time-dependent statistics of systems at thermal equilibrium, rather than pure states described by a single wavefunction. Statistics Commonly you would describe the statistics of a measurement on a variable A in terms of the moments of the distribution function, P ( A ), which characterizes the probability of observing A between A and A + dA A dA A P A Average: = ( ) (5.1) A 2 Mean Square Value: = d A 2 P ( ) . (5.2) Similarly, this can be written as a determination from a large number of measurements of the value of the variable A : N A = 1 A i (5.3) N i = 1 A 2 1 N 2 = A i . (5.4) N i = 1 Andrei Tokmakoff, MIT Department of Chemistry, 2/24/2009
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5-2 The ability to specify a value for A is captured in the variance of the distribution 2 A 2 σ 2 = A (5.5) The observation of an internal variable in a statistical sense is also intrinsic to quantum mechanics. A fundamental postulate is that the expectation value A ˆ of an operator = ψ A ˆ is the mean value of A obtained over many observations. The 2 probability distribution function is associated with dr . To take this a step further and characterize the statistical relationship between two variables, one can define a joint probability distribution, P ( A,B ), which characterizes the probability of observing A between A and A + dA and B between B and B + dB . The statistical relationship between the variables can also emerges from moments of P ( A,B ). The most important measure is a correlation function C AB = A B A B (5.6) You can see that this is the covariance – the variance for a bivariate distribution. This is a
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MIT5_74s09_lec05 - MIT OpenCourseWare http:/ocw.mit.edu...

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