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Shooting method for solving 1D quantum potentials
Developed by Andy Rundquist
Students will learn how to use the numeric shooting method to find the eigenenergies and eigenfunctions for 1D wells, including infinite square wells, finite square wells, and the Coulomb potential of the hydrogen atom.
Subject Areas  Modern Physics and Quantum Mechanics 

Level  Beyond the First Year 
Available Implementation  Mathematica 
Learning Objectives 
* Students will be able to describe the shooting method for solving for eigenenergies and eigenfunctions.
* Students will be able to use appropriate units for distance, energy, and time so that analytical and numeric results can be compared.
* Students will be able to model any shape potential well.
* Students will be able to calculate other parameters from the eigenfunctions (like expectation values).

Time to Complete  60 min 
1. Use the shooting method to determine the eigenenergies for an electron in an infinite square well of width 2 nm. Compare with the expected analytical result (with appropriate units).
2. Use the shooting method to plot the lowest few eigenfunctions for an infinite well with a shape determined by you (anything other than V=0 is fine inside the well).
3. For one of the solutions you've found in (2), determine the expectation values of:
* x
* $x^2$
* p (momentum)
* $p^2$
4. Using the shooting method, determine the lowest 10 energies of the hydrogen atom. Don't forget to try different values of the angular momentum.
5. For one of the solutions in (4) determine the expectation values of:
* r
* $r^2$
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Credits and Licensing
Andy Rundquist, "Shooting method for solving 1D quantum potentials," Published in the PICUP Collection, October 2016.
The instructor materials are ©2016 Andy Rundquist.
The exercises are released under a Creative Commons AttributionNonCommercialShareAlike 4.0 license