The Road to Reality

Exercise [21.11]
Page 1 of 1

Author:  deant [ 01 Sep 2012, 17:50 ]
Post subject:  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.

Page 1 of 1 Archived: 07 Aug 2014
Powered by phpBB © 2000, 2002, 2005, 2007 phpBB Group