Quantum states in the 3D harmonic oscillator and planar Fock-Darwin systems realize the Wallis formula for pi through the scale-independent observable Q = <r><r^{-1}> that approaches 1 at high angular momentum.
Landau,Diamagnetismus der metalle,Zeitschrift für Physik64(1930) 629–637
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Quantum Realization of the Wallis Formula
Quantum states in the 3D harmonic oscillator and planar Fock-Darwin systems realize the Wallis formula for pi through the scale-independent observable Q = <r><r^{-1}> that approaches 1 at high angular momentum.