A quantum derivation of a classic math formula

When a quantum mechanical Hamiltonian cannot be solved exactly, one can estimate system energies with a technique called the variational method. The idea is to calculate the energy expectation value for a trial wavefunction with one or more tunable parameters and determine the minimum energy that results from varying the parameters. As an exercise to help his students get a feel for the approach, the University of Rochester’s Carl Hagen applied it to a solvable system—the hydrogen atom. With the help of Rochester colleague Tamar Friedmann, he found to his surprise that the exercise yielded a representation for $\pi$ published in 1655 by mathematician John Wallis:

$$\frac{\pi }{2}=\frac{2·2}{1·3}\frac{4·4}{3·5}\frac{6·6}{5·7}\cdots .$$

Hagen and Friedmann considered a trial wavefunction that had the same angular behavior as the hydrogen atom but different radial behavior. They calculated how the minimum variational energy of their trial form depended on the angular momentum quantum number $\mathrm{\ell }$ and divided that energy by the exact energy of a hydrogen atom with angular momentum $\mathrm{\ell }$ and principal quantum number $n=\mathrm{\ell }+1$. According to the variational principle, the ratio of energies

$$\frac{{(\mathrm{\ell }+1)}^2}{(\mathrm{\ell }+\frac{3}{2})}{\left[\frac{\mathrm{\Gamma }(\mathrm{\ell }+1)}{\mathrm{\Gamma }(\mathrm{\ell }+\frac{3}{2})}\right]}^2$$

must be $\le 1$. (The $\mathrm{\Gamma }$ function is the famous generalization of the factorial.) In fact, as $\mathrm{\ell }$ approaches infinity, the ratio approaches 1. The Wallis representation then follows from the recursion property of the $\mathrm{\Gamma }$ function, $\mathrm{\Gamma }(z+1)=z\mathrm{\Gamma }\left(z\right)$, and the specific values $\mathrm{\Gamma }\left(1\right)=1$ and $\mathrm{\Gamma }(1/2)={\pi }^{1/2}$. Given the different radial behavior of the trial and exact wavefunctions, it may seem surprising that the ratio of variational to exact energies tends to 1 for large $\mathrm{\ell }$. The authors note, however, that in the infinite-$\mathrm{\ell }$ limit, both wavefunctions describe electrons on sharply defined trajectories; the quantum fuzziness that distinguishes the electron orbits for finite $\mathrm{\ell }$ goes away. (T. Friedmann, C. R. Hagen, J. Math. Phys. 56, 112101, 2015, doi:10.1063/1.4930800 .)