Students calculate the Fourier transform of the Dirac delta function.
1. << Normalization of the Gaussian for Wavefunctions | Fourier Transforms and Wave Packets | Fourier Transform of a Shifted Function >>
Find the Fourier Transform of the delta function.
If students know about the Dirac delta function, this is a great first example of the Fourier transform that students can work out in-class for themselves.
Students will need a short lecture giving the definition of the Fourier Transform \begin{equation} {\cal{F}}(f) =\tilde{f} (k)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ikx}\, f(x)\, dx \end{equation} We strongly suggest the convention of putting the exponential to the left of the function in the integrand to highlight the relationship between the Fourier Transform and the quantum mechanics notion of finding the projection of a quantum wavefunciton \(f(x)\) onto a plane wave (complete with the complex conjugate of the plane wave that turns the ket into a bra).
We always use the convention of putting a factor of \(\frac{1}{\sqrt{2\pi}}\) into the definitions of each of the Fourier Transform and its inverse to make the operations symmetric in this way. Warn students that this convention is NOT universal. They should use caution when using other resources.
This example highlights the inherent complex number nature of Fourier transforms. Because of the factor \(e^{-ikx}\), the Fourier transform of a real function is typically NOT real.
The answer here is trying to be the constant number \(\frac{1}{\sqrt{2\pi}}\), but this number is augmented by a complex phase \(e^{-ikx_0}\) that depends on \(x_0\)m where the peak of the delta function is.
You might want to ask students to act out the Fourier transform of the delta function using the arms representation.
This example is (almost) the inverse of Fourier Transform of a Plane Wave. If you really want the inverse problem, change one of the prompts to “Find the inverse Fourier transform of ...”
assignment Homework
format_list_numbered Sequence
accessibility_new Kinesthetic
30 min.
group Small Group Activity
10 min.
group Small Group Activity
30 min.
central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum energy hermitian operators probability superposition representations notations degeneracy
Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.assignment Homework
Set up the integrals for the Fourier series for this state.
Which terms will have the largest coefficients? Explain briefly.
Are there any coefficients that you know will be zero? Explain briefly.
Using the technology of your choice or by hand, calculate the four largest coefficients. With screen shots or otherwise, show your work.