In order to appreciate the parallels between finite quantum systems (like Spin-1/2 systems) and infinite quantum systems (like the infinite potential well), students need to broaden their understanding of vectors spaces to include sets of functions. In particular, they need to extend their understanding of eigenstates to include eigenfunctions. Important new ideas related to the concept of eigenfunctions are differential operators and needing to consider boundary conditions.

We begin the exploration of eigenfunctions with an activity that has students consider the eigenfunctions of several relevant operators . Students evaluate whether or not a candidate function is an eigenfunction of the momentum and Hamiltonian operators.

The students are then asked to consider solving for the energy eigenstates of a particle in a finite well . In this activity, the class discusses the conditions needed to solve for the eigenstates (the boundary conditions and normalization) and the process of finding the eigenstates is outlined for the students. This activity includes a discussion of how to choose a representation for the eigenfunctions: with sinusoidal functions or complex exponential functions. The students are then shown a computer simulation that numerically solves for the wavefunction for a chosen energy value, and the students see graphically that the boundary and normalization conditions are only satisfied for certain discreet values of the energy.