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MIT 5 74 - LECTURE NOTES

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MIT OpenCourseWarehttp://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.Andrei Tokmakoff, MIT Dept. of Chemistry, 3/5/2009 7-1 7. OBSERVING FLUCTUATIONS IN SPECTROSCOPY1 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 DEFINITIONS2 “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. 2 Nitzan, Ch. 1.5 and Ch. 7.7-2 When we introduced correlation functions, we discussed the idea that a statistical description of a system is commonly formulated in terms of probability distribution functions P. Observables are commonly described by moments of a variable obtained by integrating over P, for instance x =∫dx x Ρ(x) (7.1)2x = dx x2 Ρ x∫() For time-dependent processes, we use a time-dependent probability distribution x (t) =∫dx x t()Ρ(x t, ) . (7.2) xt = dx x t Ρ x t, 2 () ∫ 2 () ( ) Correlation functions go a step further and depend on joint probability distributions Ρ(′′,;′BtAt , ) that give the probability of observing a value of A at time t” and a value of B at time t’: A t() ()B t′′ ′ = dA dB AB Ρ(t ′,;, B)∫∫ ′A t′ . (7.3) Random fluctuations are also described through a time-dependent probability distribution, for which we need an equation of motion. A common example of such a process is Brownian motion, the fluctuating position of a particle under the influence of a thermal environment. It’s not practical to describe the absolute position of the particle, but we can formulate an equation of motion for the probability of finding the particle in time and space given that you know its initial position. This probability density obeys the well known diffusion equation, here written in one dimension: 2∂Ρ(xt, )=D ∂Ρ x t, (7.4)( )∂t ∂x2 Here D is the diffusion constant which sets the time-scale and spatial extent of the random motion. Note the similarity of this equation to the time-dependent Schrödinger equation for a free particle if D is taken as imaginary. Given the initial condition Ρ(x,t0 )=δ(x − x0 ), the solution is a conditional probability density 0Ρ(xt, x ,t )= 2π1 Dt exp⎜⎜⎝⎛−(xx 4 − Dt )2 ⎟⎟⎠⎞ (7.5)0 07-3 The probability distribution describes the statistics for fluctuations in the position of a particle averaged over many trajectories. Analyzing the moments of this probability density in eq. (7.2) we find that x = 0 (7.6)2x =2Dt So, the distribution maintains a Gaussian shape centered at x0 , and broadening with time as 2Dt. Brownian motion is an example of a Gaussian-Markovian process. Here Gaussian refers to cases in which we describe the probability distribution for a variable ()as a GaussianΡxnormal distribution. Here in one dimension: ()= Ae (xx0 )2/2Δ2Ρ x −− (7.7)22 2Δ= x −x The Gaussian distribution is important, because the central limit theorem states that the distribution of a continuous random variable with finite variance will follow the Gaussian distribution. Gaussian distributions also are completely defined in terms of their first and second moments, meaning that a time-dependent probability density P(x,t) is uniquely characterized by a mean value in the observable variable x and a correlation function that describes the fluctuations in x. Gaussian distributions for systems at thermal equilibrium are also important for the relationship between Gaussian distributions and parabolic free energy surfaces: ()=−kT (x) (7.8)Gx ln ΡIf the probability density is Gaussian along x, then the system’s free energy projected along this coordinate (often referred to as a potential of mean force) has a harmonic shape. Thus Gaussian statistics are effective for describing fluctuations about an equilibrium mean value x0. Markovian means that, given the knowledge of the state of the system at time t1, you can exactly describe P for a later time t2 . That is, the system has no memory of the behavior at an earlier time t0.7-4 ( x,txtx,t)=Ρ x, ; ,t)Ρ( x,;Ρ ;,; ( tx tx,t)2 1 0 2 1 1 2 Ρ( ;; )=Ρ tt )( ;


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