Student handout: Visualizing Combinations of Spherical Harmonics

Central Forces 2023 (3 years)
Students observe three different plots of linear combinations of spherical combinations with probability density represented by color on the sphere, distance from the origin (polar plot), and distance from the surface of the sphere.
What students learn
  • Students use Mathematica to visualize states that are made up of linear combinations of spherical harmonics.
  • Students compare and contrast three different representations of the probability densities.
  • Students are asked to explore the geometric role of the relative phase between two states with the same value of \(L^2\)
  • Students are asked to explore states with linear combinations with different values of \(L^2\).

Use this Mathematica worksheet to explore the spatial properties of linear combinations of spherical harmonics \(Y_{\ell}^{m}(\theta, \phi)\).


Keywords
Spherical Harmonics Visualization Mathematica Sphere Rigid Rotor
Learning Outcomes