PHYS 408 HOMEWORK 6 SOLUTIONS

# 6 23 24 25 5 51 fourier transform of a gaussian the

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Unformatted text preview: k0 − a/2) + δ (−kx + k0 + a/2) = δ ((kx − k0 ) + a/2) + δ ((kx − k0 ) − a/2) ˜ Thus the eﬀect of multiplying f (x) by eik0 x is to shift the fourier transform to F (kx − k0 ). 6 (23) (24) (25) 5 5.1 Fourier transform of a gaussian the transform ˜ E (k ) = ￿ +∞ −∞ = E0 ￿ dx e−ikx E (x) +∞ 2 dx e−(x/σ+ikσ/2) e−k −∞ = E0 e−k 2 (26) σ 2 /4 σ ￿ +∞+i∞ 2 σ 2 /4 d(x/σ + ik σ /2) e−(x/σ+ikσ/2) −∞−i∞ ￿ = E0 σ π exp − k σ /4 √ 5.2 direct and inverse space widths ￿ 22 (27) 2 (28) (29) (30) Inversely proportional, σx = σ ∝ 1/σk = (1/(2/σ )) = σ /2. 6 6.1 matching functions and their fourier transforms Matching 1c, 4a : based on problem 5, gaussian in one space centered about origin is gaussian in the other also centered about zero, with no carrier wave modulation, and the width in one space inversely proportional to width in the inverse space. 2d, 3b: inclusion of a eikx multiplicative factor in real (direct) space will introduce a carrier wave into the real-space function,...
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