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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.

Activities

Students consider the dimensions of spin-state kets and position-basis kets.

Small Group Activity

60 min.

Quantum Calculations on the Hydrogen Atom

Students are asked to find eigenvalues, probabilities, and expectation values for \(H\), \(L^2\), and \(L_z\) for a superposition of \(\vert n \ell m \rangle\) states. This can be done on small whiteboards or with the students working in groups on large whiteboards.

Students then work together in small groups to find the matrices that correspond to \(H\), \(L^2\), and \(L_z\) and to redo \(\langle E\rangle\) in matrix notation.

In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.

Small Group Activity

30 min.

Representations for Finding Components
In this small group activity, students draw components of a vector in Cartesian and polar bases. Students then write the components of the vector in these bases as both dot products with unit vectors and as bra/kets with basis bras.