Wallis formula from the harmonic oscillator
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We show that the asymptotic formula for $\pi$, the Wallis formula, that was related with quantum mechanics and the hydrogen atom in \cite{HF}, can also be related to the harmonic oscillator using a quantum duality between 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.
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Cited by 3 Pith papers
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Variational minimization on the 4D singular harmonic oscillator with quartic trial functions yields the Wallis product for odd effective angular momenta and its reciprocal for even ones in the large-nu limit via Gamma...
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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.
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