Old From New.

In 1655 the English mathematician John Wallis published a book in which he derived a formula for pi as the product of an infinite series of ratios. Now researchers from the University of Rochester, in a surprise discovery, have found the same formula in quantum mechanical calculations of the energy levels of a hydrogen atom.

“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 realised 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 characterises a physical system that was discovered 300 years later.”

The theory of quantum mechanics dates back to the early 20th century and the Wallis formula has been around for hundreds of years, but the connection between the two had remained hidden until now.

“Nature had kept this secret for the last 80 years,” Friedmann said. “I’m glad we revealed it.”

The researchers report their findings in the *Journal of Mathematical Physics*, from AIP Publishing.

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