pset_2

pset_2 - MIT OpenCourseWare http/ocw.mit.edu 5.72...

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5.72 Problem set #2 (Feb. 2008) Cao I. 1) For a harmonic oscillator, show Ct ( ) = xt ( ) x0 ( ) satisfies . 2 0 0 CC ω += && 2) Solve for C(t) and its Fourier transform ˜ C ω ( ) = ( ) e i ω t dt . 3) The forced oscillator obeys the equation of motion ( ) 2 0 it mx m x f e ωω derive the expression for χ ( ω ) from the above equation. 4) Write the formula for K(t). 5) Verify [i.e. () () Kt β =− & () βω C x ~ 2 = ] 6)* Verify the Kramers-Kronig relations. II. The relaxation of rotation motions can be described by the rotational diffusion equation p t = D R 2 p , where 2 is the angular part of the Laplacian operator 2 = 1 sin θ ∂θ sin θ ∂θ + 1 sin 2 θ 2 ∂φ 2 1) Show that the average orientation u(t) = cos θ (t) satisfies . ( ) 2 ( ) R ut Dut & 2) Show the orientational correlation function is given by C(t) = cos θ (t)cos θ (0) = 1 3 e 2D R t .
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pset_2 - MIT OpenCourseWare http/ocw.mit.edu 5.72...

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