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 Exercise [21.11] 
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Joined: 12 Jul 2010, 07:44
Posts: 154
Post Exercise [21.11]
Replacing C with C+iD (C,D real) yields the wave packet formula:

\psi = Ae^{-B^2\left(x-C-i D\right)^2}

Expanding out the exponent gives B^2\left(D^2-2 i C D\right) + i\left(2B^2 D x\right) - B^2\left(x-C\right)^2, and so:

\psi = Ae^{B^2\left(D^2-2 i C D\right)}.e^{i\left(2B^2 D x\right)}.e^{-B^2\left(x-C\right)^2}

The first exponential is just a (complex) multiplying constant; the second controls the frequency, and the third shapes the amplitude.

Examining the second and third exponentials, we can see immediately that the wave packet has frequency \pi^{-1}B^2 D, and is centered at (i.e. peaks at) x = C.

01 Sep 2012, 17:50
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