Back in 2013, a student who was in multivariable calculus wanted to apply the math to something in physics. She was a real go-getter, so I suggested she try to do REAL Quantum Mechanics and use her MV to figure out the hard-core derivation of a hydrogen atom's energy states and wave function. Katie took that challenge and ran with it!
Check out her paper!! She hand-wrote a ~30 page paper as a 'how to do QM' guide for other students. She derived everything, learned the solutions to different partial differential equations (using things like Legendre polynomials, spherical harmonics, Laguerre polynomials, etc.). It is beautiful, impressive work! Katie is presently working on her PhD in physics, and continues to be amazing.
This paper shows the power of higher level mathematics in physics, as well as the level of difficulty of QM and how impressive the scientists who discovered all this were.
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