Exercise [21.11]

deant · 01 Sep 2012, 17:50

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

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