MIT 8 512 - Lecture 3: Properties of the Response Function (3 pages)

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Lecture 3: Properties of the Response Function



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Lecture 3: Properties of the Response Function

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Lecture Notes


Pages:
3
School:
Massachusetts Institute of Technology
Course:
8 512 - Theory of Solids II

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MIT OpenCourseWare http ocw mit edu 8 512 Theory of Solids II Spring 2009 For information about citing these materials or our Terms of Use visit http ocw mit edu terms Lecture 3 Properties of t h e Response Function In this lecture we will discuss some general properties of the response functions X and some uselul relations that they satisfy 3 1 General Properties of q w Recall that wit11 n F i E IR and U F i E IR Under Fourier transform this implies n w U f w n q w US gl ww As a result where X is the imaginary part of the response function X gl ww Consider the extension of w to the complex plane We can then rewrite the expression for x q 74 as Without the iq term in the energy denominator there would be siryularities poles on the real axis whenever w is equa 1to the spacing between the ground state and some excited state The presence of iq pushes these poles just into the lower l Bplane emuring that x q7 w is analytic in the entire upper l 2 w plane including the real axis Analyticity of x w in the upper 1 2 plane is needed to build causality into the theory Consider the response function in time t To evaluate this integral we perform a contour integral in the complex w plane For i 0 closing the contour in the upper 1 2 plane ensures that leJiwtl 0 on the curved portion of the Kramers Kronig 2 contour Since we have ensured that is analytic in the upper half plane Cauchy s residue theorem guarantees that the integral over the entire contour is 0 As a result the piece we need i e the integral from to along the real axis must also be 0 Thus t 0 for t 0 which means that the system cannot respond to a perturbation until after the perturbation has occurred What about the t 0 case In this case the contour must be closed in the lower 1 2 plane to prevent the exponential from blowing up However the i in the energy denominator has pushed the singularities into this region of the complex plane Thus the value of the contour integral will be nonzero and the system will respond



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