Quantum mechanical derivation of the Wallis formula for π

@article{Friedmann2015QuantumMD,
  title={Quantum mechanical derivation of the Wallis formula for $\pi$},
  author={Tamar Friedmann and Carl R. Hagen},
  journal={Journal of Mathematical Physics},
  year={2015},
  volume={56},
  pages={112101}
}
A famous pre-Newtonian formula for π is obtained directly from the variational approach to the spectrum of the hydrogen atom in spaces of arbitrary dimensions greater than one, including the physical three dimensions. 

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For example, the ratio is