Wallis formula from the harmonic oscillator

We show that the asymptotic formula for π, the Wallis formula, that was related with quantum mechanics and the hydrogen atom in Friedmann and Hagen (2015) can also be related to the harmonic oscillator using a quantum duality that relate these two systems. As a corollary we show that this very interesting asymptotic formula is not related with the hydrogen atom or quantum mechanics itself but with a clever choice of a trial function and a potential in the Schroedinger equation when we use the variational approach to calculate the ground state energy associated with the given potential function.