# 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.

## 27 Citations

### The Hydrogen Atom, Pi and Lerch's Transcendent

- Philosophy
- 2019

In this note, we extend the connection between the hydrogen atom and $\pi$ to the number $e$ via the Lerch's transcendent.

### On the Derivation of the Wallis Formula for $$\pi $$ in the 17th and 21st Centuries

- Chemistry
- 2017

The formula for \(\pi \) as an infinite product known as the Wallis formula was developed by John Wallis in 1655 by methods that pre-date the establishment of calculus. Herein I introduce said…

### Asymptotic Expansions Related to the Wallis Ratio Based on the Bell Polynomials

- MathematicsAsian Research Journal of Mathematics
- 2022

In this paper, we establish a new asymptotic expansion related to the Wallis ratio. By using the exponential Bell polynomials, we show that the coefficients of the asymptotic expansion can be…

### Improved variational method that solves the energy eigenvalue problem of the hydrogen atom

- PhysicsJournal of Physics: Conference Series
- 2018

In most quantum mechanics textbooks for graduate studies, the hydrogen atom is studied in an approximate way by means of the variational method. The type of trial functions commonly used are the…

### Pairwise entanglement and the Mott transition for correlated electrons in nanochains

- Physics
- 2016

Pairwise entanglement, calculated separately for charge and spin degrees of freedom, is proposed as a ground-state signature of the Mott transition in correlated nanoscopic systems. Utilizing the…

### Taking Physical Infinity Seriously

- Physics, EducationMartin Davis on Computability, Computational Logic, and Mathematical Foundations
- 2016

The concept of infinity took centuries to achieve recognized status in the field of mathematics, despite the fact that it was implicitly present in nearly all mathematical endeavors. Here I explore…

### A quantum mechanical well and a derivation of a $\pi^2 $ formula

- Physics
- 2017

Quantum particle bound in an infinite, one-dimensional square potential well is one of the problems in Quantum Mechanics (QM) that most of the textbooks start from. There, calculating an allowed…

### Transformation and summation formulas for basic hypergeometric series associated with the circumference ratio

- Mathematics
- 2019

In terms of the difference operators, we establish several curious transformation and summation formulas for basic hypergeometric series. When the parameters are specified, they produce $q$-analogues…

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