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## Introducing Eigenfunctions

In order to appreciate the parallels between finite quantum systems, such as spin-1/2 systems, and infinite quantum systems, such as the infinite potential well, students need to extend their understanding of vector spaces to include sets of functions. This requires transitioning from a system with a discrete set of eigenstates to a continuous system where the eigenstates are eigenfunctions. Important new ideas related to the concept of eigenfunctions are differential operators and needing to consider boundary conditions.

### Activities

**Operators and Functions***(Estimated time: )*: This small group activity 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.

**Solving the Energy Eigenvalue Equation for the Finite Well***(Estimated time: 40 minutes)*: In this small group activity students are asked to consider solving for the energy eigenstates of a particle in a finite well. 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 discrete values of the energy.