# 9 Atomic Wells

Keywords: Eigenstates, Wave vector, Time evolution, Periodic Systems

### Highlights of the activity

1. Nine students are selected from the class to represent the wave function of an atom, where the wave function is represented by $$\sum_{n=1}^{N} A\, e^{ikna}\; \phi (x-na) \; \; ,$$ $$k=\frac{-N\pi}{L}, … ,\frac{-2\pi}{L},0, \frac{2\pi}{L}, \frac{4\pi}{L}, … , \frac{N\pi}{L} \; \; .$$
2. Each student will then put their arm in a particular direction to represent the value of their $e^{ika}$ term.
3. The class is then asked to determine the wave vector of the envelope function.
4. A second group can be selected to perform the same demonstration with a different wave vector.
5. The energies of the eigenstates can then be compared using $$E=\alpha + 2\beta \, \cos{ka} \; \; .$$

### Reasons to spend class time on the activity

Students will oftentimes have a difficult time realizing that, given the spacing of the potential wells and the number of potential wells in the system, they can formulate the eigenstates of an electron in a potential landscape with little challenging computation. This exercise is designed to help students practice looking at the graphical representation of an eigenstate and determining the wave vector of the eigenstate.

### Instructor's Guide

Authors: Ethan Minot, Janet Tate, Teal Pershing

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