Solutions+5.nb

Solutions+5.nb - Homework 5 Solutions vx v0 x D = v0 H 3.0...

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Unformatted text preview: Homework 5, Solutions vx @ v0_, x_ D = - v0 H 3.0 ê 4 + Cos @ 2 Pi x D + 1.0 ê 4 Cos @ 4 Pi x DL- v0 H 0.75 + Cos @ 2 p x D + 0.25 Cos @ 4 p x DL Plot @ vx @ 1, x D , 8 x, 0, 5 <D 1 2 3 4 5- 2.0- 1.5- 1.0- 0.5 (b) The RL vectors are 2 p n, n any integer H c L TrigToExp @ vx @ v0, x DD- 0.75 v0- 1 2 ‰- 2 Â p x v0- 1 2 ‰ 2 Â p x v0- 0.125 ‰- 4 Â p x v0- 0.125 ‰ 4 Â p x v0 Since vx is an expansion in a finite number of plane waves, its fourier transform has delta functions at those wavevectors, and that’s all. FourierTransform @ vx @ v0, x D , x, k D- 1.87997 v0 DiracDelta @ k D- 0.313329 v0 DiracDelta @ k- 4 p D- 1.25331 v0 DiracDelta @ k- 2 p D- 1.25331 v0 DiracDelta @ k + 2 p D- 0.313329 v0 DiracDelta @ k + 4 p D (d) K_ij = (2 delta_ij - delta_ {i, j + 1} - delta_ {i, j - 1}) / (a^2 * 2) where j + 1 and j - 1 are interpretted as modulo N = 640 V_ij = delta_ij V(x_i = i / 32) (e) kemat @ n_ D : = DiagonalMatrix @ Table @ 2, 8 n <DD- DiagonalMatrix @ Table @ 1, 8 n- 1 <D , 1 D- DiagonalMatrix @ Table @ 1, 8 n- 1 <D ,- 1 D- DiagonalMatrix @8 1 < , n- 1 D- DiagonalMatrix @8 1 < ,- n + 1 D MatrixForm @ kemat @ 6 DD 2- 1- 1- 1 2- 1- 1 2- 1- 1 2- 1- 1 2- 1- 1- 1 2 vmat @ n_ D : = DiagonalMatrix @ Table @ vx @ v0, i ê 32 D , 8 i, n <DD MatrixForm @ vmat @ 6 DD- 1.96176 v0 0. 0. 0. 0. 0....
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This note was uploaded on 12/12/2011 for the course PHYS 238 taught by Professor Staff during the Fall '11 term at UC Irvine.

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Solutions+5.nb - Homework 5 Solutions vx v0 x D = v0 H 3.0...

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