# 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}
}
• Published 27 October 2015
• Physics
• Journal of Mathematical Physics
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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## References

SHOWING 1-10 OF 12 REFERENCES

### HYDROGEN ATOM AND RELATIVISTIC PI MESIC ATOM IN N SPACE DIMENSIONS

We derive in simple analytic closed form the eigenfunctions and eigenenergies for the hydrogen atom in N dimensions. A section is devoted to the specialization to one dimension. Comments are made on

### Group-theoretical derivation of angular momentum eigenvalues in spaces of arbitrary dimensions

• Physics, Mathematics
• 2012
The spectrum of the square of the angular momentum in arbitrary dimensions is derived using only group theoretical techniques. This is accomplished by application of the Lie algebra of the noncompact

### Elementary Combinatorial-Probabilistic Proof of the Wallis and Stirling Formulas

This short note provides a remarkably simple derivation of the Wallis and Stirling formulas based on elementary estimates applied to binomial coefficients. As a byproduct an elementary proof of the

### A Probabilistic Proof of Wallis's Formula for π

Using mostly elementary results and functions from probability, we prove Wallis's formula for pi: pi/2 = prod_n (2n * 2n) / ((2n-1) * (2n+1)). The proof involves normalization constants and the Gamma

### Calculus, Early Transcendentals

• Mathematics
• 2002
Chapter 0 Before Calculus 0.1 Functions 0.2 New Functions from Old 0.4 Families of Functions 0.5 Inverse Functions Inverse Trigonometric Functions 0.6 Exponential and Logarithmic Functions Chapter 1