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# Numerical Considerations

The true relationships between the wave function in x-space and the amplitude function is the Fourier transform. By using discrete numerical techniques, we have changed the infinite integral to a finite sum. Since we only summed over a finite number of complex harmonics, we have implied that the x-space representation is periodic! This periodicity can come back to haunt us if we are not careful.

Because we estimated the Fourier transform by making the x-space function periodic, it also means that the reciprocal space amplitude function will be discrete, and harmonically related. Using this information, we can determine that the values for should be be separated by

where N is the number of points in , and is chosen to be odd to simplify matters. Furthermore, the maximum (most positive) and minimum (most negative) values of should be

Note, to take into account larger values of q (``higher frequencies''), we should decrease (``increase the sampling frequency''). To decrease (without increasing we should make our maximum value of x larger, and keep N the same.

In 6.011 you will learn more about the numerical issues concerning the Discrete Fourier Transform. You will also learn how to accelerate its computation by a method known as the Fast Fourier Transform (FFT).