Recently there was a proof of the Wallis Product using quantum mechanics on the arXiv. However, there are many proofs of the result, Wikipedia has 4.

**Fine Print** the first proof has on Wikipedia, the Euler product as an input, $$\boxed{\sin \pi x = \pi x \prod_{n \in \mathbb{N}} \left(1 - \frac{x^2}{n^2}\right)}$$
and this follows from Weierstrass Factorization, which is a significant result. One proof by Oscar Ciaurri circumvents this. Another by Lars Holst.

**Quantum Mechanics** This derivation starts with the Hydrogen atom Schrödinger equation:

$$ H \psi = \left( - \frac{\hbar}{2m}\nabla^2 - \frac{e^2}{r} \right) \psi = E \psi $$

Since this equation is radially symmetric there is an $SO(3)$ action on the solutions to this equation preserving the energies $E$. So these eigenspaces can be indexed by the representations of $SO(3)$, $E_{\ell, m}$ and $\psi_{\ell , m}$. Can we guess the values of $E$ ?

A variational approach involves minimizing as best we can, using the point-to-line distance formula: $$ E_{\ell, 0} \leq \frac{\langle \psi | H | \psi \rangle }{\langle \psi| \psi \rangle} $$ This only works for the ground state energy.

They have a guesstimate of $\langle H_\ell \rangle_{min} = \frac{1}{\ell + \frac{1}{2}}\left[ \frac{\ell!}{(\ell + \frac{1}{2})!}\right]$ while the exact answer is $E_{0, \ell} = \frac{1}{(\ell + 1)^2}$

**Bohr correspondence principle** How do we know for $\ell \gg 1$ these two values are the same?
$$ \lim_{\ell \to \infty} \frac{\langle H_\ell \rangle_{min}}{E_{0, \ell}} = 1$$
This ultimately leads to the Wallis Product Formula using identities of the Gamma function

Is quantum mechanics necessary here? I think we are using Feynman-Hellman or Rayleigh-Ritz but I think there is something even more basic facts about Hermitian matrices as in these notes.

any bounded entire function is constant"gee that's nice" -- now I see it in your notes and I am like wow, we get $\zeta(2) = \frac{\pi^2}{6}$ for free! $\endgroup$ – john mangual Mar 31 '16 at 16:26useful, not static. $\endgroup$ – paul garrett Mar 31 '16 at 16:27