PHYS 408 HOMEWORK 6 SOLUTIONS

Some of the algebra needed above would thus be

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Unformatted text preview: of only sine functions, as cosine function are even whilst sine functions are odd. Some of the algebra needed above would thus be avoided by starting out with a fourier decomposition in terms of 1 plus a sum over all possible sine functions. 4 3 3.1 Fourier transform, square pulse Outright calculation and plot of transform ˜ E0 (kx ) = ￿ E 0 E0 (x) = 0 +∞ −∞ −ikx x dx · e · E0 (x) = −L/2 < x < +L/2 (18) ￿ (19) otherwise +L/2 −L/2 dx · e−ikx x · E0 = E0 Lsinc(kx L/2) Figure 2: p3.jpg 5 4 Shifting a Fourier transform 4.1 inverse Fourier transform of delta functions f (x) = = = 1 2π ￿ +∞ ˜ dkx F (kx )e+ikx x −∞ ￿ +∞ ￿ ￿ 1 dkx δ (kx − (−a/2)) + δ (kx − (+a/2)) e+ikx x 2π −∞ ￿ ￿ 1 +i(−a/2)x +i(+a/2)x e +e 2π (20) (21) (22) “Makes sense” for the reason we have two plane waves in real space, each with a distinct wavenumber, so in a wavenumber representation we would expect contributions at only those two wavenumbers. 4.2 k -space shifting f (x) → f (x)eik0 x ˜ F (kx ) = δ (−kx +...
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This document was uploaded on 03/11/2014 for the course PHYS 408 at University of British Columbia.

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