Scientists look into hydrogen atom, find old recipe for pi

A duo of scientists have spotted something unexpected in their calculations for the energy levels of a hydrogen atom: a 360-year old formula for pi.

Published in 1655 by the English mathematician John Wallis, the Wallis product is an infinite series of fractions that, when multiplied, equal pi divided by 2. It has not appeared in physics at all until now, when University of Rochester scientists Carl Hagen and Tamar Friedmann collaborated on a problem set that Dr. Hagen had developed for his quantum mechanics class.  

Instead of using Niels Bohr’s near-century-old calculations for the energy states of hydrogen, Hagen had his students use a method called the variational principle, just to see what might happen. Ultimately, the calculations demanded mathematical expertise, which came in the form of Dr. Friedmann, who is both a mathematician and a physicist.

“One of the things that I’m able to do is talk to both mathematicians and physicists, and that basically requires translating between two languages,” says Friedmann, who studied mathematics as an undergraduate student at Princeton, where she also earned a PhD. in Theoretical and Mathematical Physics.

Friedmann says that asking new questions in math from physics and seeking to understand the physical systems from a mathematical standpoint enriched her understanding of problems in both disciplines. Friedmann tends to take on problems that might not even have an answer; she thinks that’s the type of approach that can lead to new discoveries. “And when they happen,” Friedmann says, “it’s really amazing.”

Hagen and Friedmann weren’t looking for the Wallis formula, rather, they happened upon it while experimenting. As Hagen said in a release, “It just fell into our laps.”

Friedmann says she sees this discovery as a good example of keeping an open mind and letting the research tell you what’s there. "The special thing is that it brings out a beautiful connection between physics and math,” Friedmann said in a release. “I find it fascinating that a purely mathematical formula from the 17th century characterizes a physical system that was discovered 300 years later."

Of course, there has always been a strong relationship between physics and math, in that mathematics provides the language to describe and conduct the work of physics. But it is worth stopping now and again to ponder just how amazing it is that mathematics, a product of human thought whose rules are derived independently of any experience, can describe so neatly and precisely the physical world, a phenomenon that Nobel Prize-winning physicist Eugene Wigner called, in a 1959 lecture, the "unreasonable effectiveness of mathematics in the natural sciences," which he characterized as "a wonderful gift which we neither understand nor deserve.” 

[Editor's note: Because of an editing error, an earlier version of this article misspelled Eugene Wigner's name.]