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
This short activity lets student explore the role of the various parameters in the shape of a Gaussian function.
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
A quick small whiteboard question comparing the Gaussian for the wavefunction of free quantum wave packet to the Gaussian for the probability density.
Students find a wavefunction that corresponds to a Gaussian probability density.
3. Activity: Fourier Transform of the Delta Function
Use this activity first, immediately after the introduction of the definition of the Fourier transform. If students know about the delta function already, this activity will be quick and confidence building. Do NOT try to use this as an opportunity to introduce the delta function.
Students calculate the Fourier transform of the Dirac delta function.
4. Activity: Fourier Transform of a Shifted Function
Another quick and easy example of a Fourier transform. Students will practice identifying which variables matter and which don't. They will also practice basic exponential rules to identify an overall constant phase.
5. Activity: Fourier Transform of a Plane Wave
This activity confirms that the delta function and a constant phase are Fourier inverses. For this activity, students will need to recognize the exponential representation of the delta function.
6. Activity: Fourier Transform of a Derivative
The result of this calculation is important. But, as an activity, the logic is a little bit backwards in the sense that you find a formula in terms of the inverse Fourier transform and then need to formally take the Fourier transform of both sides. Consider doing this example as a mini-lecture and/or be prepared to interrupt the groups and offer lots of explicit support when groups get stuck.
7. Activity: Fourier Transform of a Gaussian
This, of course, is the calculation that the students most need. Since the solution is easy to find online, this makes a better classroom activity than homework problem. This activity will take longer than the others. Students will need to know how to “complete the square” in the exponential.