Dirichlet Characters and Particles in Boxes (Number Theory in Quantum Mechanics) | #SoME3

This video shows an application of some ideas from analytic number theory to quantum mechanics. I show how Dirichlet characters can be used to count how many states of the "particle-in-a-box" system have a certain energy. This video is also my submission to the third Summer of Math Exposition. Let me know if you have any suggestions for future topics or notice any mistakes in the video. I hope you enjoy the video! ------------------------------------------------------------------------------------------------------ 0:00 - Intro 0:36 - Particle in a 1D Box 2:57 - Particle in a 2D Box 5:30 - Connection to Number Theory 8:39 - Degeneracy Solution with Dirichlet Characters 10:44 - Examples of Degeneracy 13:23 - Which Energies are Allowed 17:43 - Examples of Checking Energies 19:08 - Conclusion