Physics Professor Dreams Up a Homework Problem and then Discovers This

Carl Hagen, a particle physicist at the University of Rochester, decided to have his students apply a computational technique for approximating the energy states of quantum systems (such as molecules) to the hydrogen atom.  As he worked out the solution to the problem set himself, he noticed an interesting trend, and it turns out that he accidentally discovered a link between quantum physics and pure mathematics for pi.

John Wallis, an English mathematician, derived a pure mathematical formula for pi in his book “Arithmetica Infinitorum” published in 1655 (see featured image for this article, above). Until now, this derivation of pi has lived only in the world of mathematics, but, as a very interesting press release from EurekAlert! reports, almost 400 years later, the same formula he derived was discovered by Hagan in a happy accident:

“We weren’t looking for the Wallis formula for pi. It just fell into our laps,” said Carl Hagen, a particle physicist at the University of Rochester. Having noticed an intriguing trend in the solutions to a problem set he had developed for students in a class on quantum mechanics, Hagen recruited mathematician Tamar Friedmann and they realized this trend was in fact a manifestation of the Wallis formula for pi.

“It was a complete surprise – I jumped up and down when we got the Wallis formula out of equations for the hydrogen atom,” said Friedmann. “The special thing is that it brings out a beautiful connection between physics and math. I find it fascinating that a purely mathematical formula from the 17th century characterizes a physical system that was discovered 300 years later.”

“At the lower energy orbits, the path of the electron is fuzzy and spread out,” Hagen explained. “At more excited states, the orbits become more sharply defined and the uncertainty in the radius decreases.”

From the formula for the limit of the variational solution as the energy increased, Hagen and Friedmann were able to pull out the Wallis formula for pi.

Interestingly, the Wallis formula had been around for hundreds of years and the theory of quantum mechanics dates from the early 20th century, but the pi connection had remained hidden for all these years.  For additional details, check out the informative press release on Eureka Alert, and for the really adventurous, you can access the technical paper that was published in the Journal of Mathematical Physics on the AIP Scitation website.


Source: – “New derivation of pi links quantum physics and pure math” 

Featured Image Credit: Digitized by Google

The featured image is 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.