Fourier Transforms and Wave Packets

This is a unit that introduces the Fourier transform and its properties and then applies the Fourier transform to free particle wave packets in non-relativistic quantum mechanics. The activities and homework are listed here. Appropriate text materials for mini-lectures can be found in the chapter Fourier Transforms and Wave Packets in the free online textbook The Geometry of Mathematical Methods.
1. Activity: Gaussian Parameters
Students use an applet to explore the role of the parameters \(N\), \(x_o\), and \(\sigma\) in the shape of a Gaussian \begin{equation} f(x)=Ne^{-\frac{(x-x_0)^2}{2\sigma^2}} \end{equation}
2. Activity: Normalization of the Gaussian for Wavefunctions
Students find a wavefunction that corresponds to a Gaussian probability density.
3. Activity: Fourier Transform of the Delta Function
Students calculate the Fourier transform of the Dirac delta function.
4. Activity: Fourier Transform of a Shifted Function
5. Activity: Fourier Transform of a Plane Wave
6. Activity: Fourier Transform of a Derivative
7. Activity: Fourier Transform of a Gaussian