Here is a link to the powerpoint presentation from this session.
In this workshop, we explored a number of different activities that help students understand the geometry of quantum mechanics and the special role played by eigenstates of the Hamiltonian. Examples were drawn from two-level systems and the unperturbed hydrogen atom.
We started with a compare and contrast activity in which students explored the geometric meaning of
- Eigenvalues and Eigenvectors. This is an important prerequisite to the study of eigenstates in quantum mechanics.
Then we did two Maple activities that allowed students to visualize probabily densities for two simple quantum systems.
First we looked at the probability densities for eigenstates confined to a ring:
- The Ring. In this system, the vertical direction on the graph was used to represent probability density. We discovered that the probability density was independent of time for single eigenstates of the Hamiltonian, but linear superpositions of eigenstates can exhibit time dependent probability densities.
Similarly, we looked at eigenstates confined to the surface of a sphere.
- Spherical Harmonics (visualization). In this case, the probability densities corresponding to single eigenstates all had axial symmetry. I also showed an example from another Maple worksheet that showed that Linear combinations of Spherical Harmonics (visualization). do not have to have axially symmetric probability densities.
We ended with a java program that allows students to simulate successive