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gaussian_dispersion

gaussian_dispersion - Dispersion of a Gaussian Wave Packet...

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Unformatted text preview: Dispersion of a Gaussian Wave Packet Physics 218 Spring 2007 This is a non-rigorous discussion. For a somewhat more formal discussion, see the recommended text, Elmore and Heald, Ch 12.15. It’s on reserve in the library if you did not pick one up in the book store. Consider a Gaussian wave packet with a wave number k . At time t = 0 its position will be described as η ( x, 0) = A exp[- x 2 / (2 σ 2 )] exp( ik x ) (1) We would like to know how this packet spreads out with time. We know that for a wave of the form η 1 ( x, t ) = A cos( k 1 x- ω 1 t ), the wave will propagate at its phase velocity c = ω/k . We also know that a Fourier transform tells us that how to decompose our Gaussian packet into sines and cosines with contributions at various wavelengths. To understand how it spreads with time, we need to know what components of different phase are the constituents in Eq. 1 and then see at what spread of phase velocities they propagate....
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