If you want to do #QuantumMechanics in #HigherDimensions then you need to know about associated Gegenbauer polynomials. Since there is no good reference on the web for these, I put together a presentation of Legendre, Gegenbauer and Jacobi polynomials to show how to derive their series expansions, Rodrigues formulas and differential equations:
https://analyticphysics.com/Special%20Functions/Hypergeometric%20Orthogonal%20Polynomials.htm
Lots of tedious detail that ultimately simplifies nicely. Suspect I'm missing something important here...
#higherdimensions #quantummechanics
#hypersphere #hypercube #higherdimensions
\[\dfrac{A_{circle}}{A_{square}}=\dfrac{\pi}{4}\ \ for\ \ n=2\]
\[\dfrac{V_{sphere}}{V_{cube}}=\dfrac{\pi}{6}\ \ for\ \ n=3\]
\[\dfrac{V_{hypersphere}}{V_{hypercube}}=\dfrac{\pi^{n/2}}{n2^{n-1}\Gamma(n/2)}\rightarrow0\ \ as\ \ n\rightarrow\infty\]
#higherdimensions #hypercube #hypersphere