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# Solving the Energy Eigenvalue Equation for the Finite Well

**Keywords:**Quantum Mechanics, Finite Well, Middle Division, Eigenstates

### Highlights of the activity

- This small group activity is designed to help upper-division undergraduate students understand solutions of the energy eigenvalue equation for a finite potential well.
- Students solved the energy eigenvalue equation for each region of a finite well.
- The whole class discussion focuses on the form and continuity of the solution in each region and solving the time-dependent Schrödinger equation using the boundary conditions.

### Reasons to spend class time on the activity

One of the key problems in quantum mechanics is solving the energy eigenvalue equation for different potential profiles. The finite well is a nice example because it combines the solutions of the infinite well with exponentially decaying solutions. This example foreshadows a discussion of quantum tunneling effects.

Students are often confused about how to choose solutions to the energy eigenvalue equation (sines or exponentials) and need practice considering the boundary conditions. We divide the class up into three “regions” and ask students to solve the equation in their own region. We then ask students to make make the different solutions match at the boundaries.