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Computing the Fourier Transform

We find the expansion coefficients, , by computing the Fourier transform of the wave packet in x-space. The Fourier transform is defined in terms of continuous variables x and q. To implement this in Matlab we need to approximate

by use of the discrete equation

To do this, we create a vector with all of our x values, i.e.:

Similarly, we define a vector with all the values of :

We then define a vector for our reciprocal space:

The values for are determined by the properties of the discrete Fourier transform. Note, in general, and do not contain the same number of elements, i.e. .

Now, the outer product of and is

Taking the exponential of i times each element of the above matrix yields:

Note, the Matlab command `` exp'' takes the exponent of each element in a matrix. Then,

i.e. a vector indexed by q which contains the sum we defined to be an approximation to the Fourier transform, known as the Discrete Fourier Transform (DFT).



Last modified: Mon Sep 25 16:11:02 EDT 1995