Consider a wave packet with a Gaussian amplitude distribution A (k) = B exp [€“ σ (k €“ k0)2] where 2/√σ is equal to the 1/e width* of the packet. Using this function for A (k), show that




Sketch the shape of this wave packet. Next, expand w (k) in a Taylor series, retain the first two terms, and integrate the wave packet equation to obtain the general result ψ(x, t) = B √π/σ exp [€“(w0t €“ x) 2/4σ] exp [i (w0t €“ k0x)] Finally, take one additional term in the Taylor series expression w (k) and show that σ is now replaced by a complex quantity. Find the expression for the 1/e width of the packet as a function of time for this case and show that the packet moves with the same group velocity as before but spreads in width as it moves. Illustrate this result with a sketch.

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+00 B exp[-a(k- ko) Jexp(-ikx) dk V (x,0) = B exp(-x/4o)exp(-ikox) TT