Physics, Math and Pi

Tamar Friedmann and Carl R. Hagen,  ar University of Rochester, recently published a manuscript titled 'looked at 'Quantum Mechanical Derivation of the Wallis Formula for π'.'

They have been looking at the variational computation of the spectrum of the hydrogen atom. They start with Schrödinger equation for the hydrogen atom, given by:

H . ψ  =  ( - h².∇² / 2.m  -  e² / r) . ψ  =  E . ψ

and arrive at the famous pre-Newtonian formula for pi, derived by John Wallis in 1655.

360 years after a mathematical derivation of pi, physicists come up with a physical derivation for quantum energy levels that is identical to it. The universe never fails to amaze.

Two pages from the book "Arithmetica Infinitorum," by John Wallis. In the table on the left page, the square that appears repeatedly denotes 4/pi, or the ratio of the area of a square to the area of the circumscribed circle. Wallis used the table to obtain the inequalities shown at the top of the page on the right that led to his formula. Credit: Digitized by Google.

Here are some related links:

• Phys.org article
• Forbes/Science article