Quantum Mechanics Of PI

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 didn’t just find pi,” said Tamar Friedmann, a visiting assistant professor of mathematics and a research associate of high energy physics, and co-author of a paper published this week in the Journal of Mathematical Physics. “We found the classic seventeenth century Wallis formula for pi, making us the first to derive it from physics, in general, and quantum mechanics, in particular.”

"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."
 
“The value of pi has taken on a mythical status, in part, because it’s impossible to write it down with 100 percent accuracy,” said Friedmann, “It cannot even be accurately expressed as a ratio of integers, and is, instead, best represented as a formula.”

In quantum mechanics, a technique called the variational approach can be used to approximate the energy states of quantum systems, like molecules, that can't be solved exactly. Hagen was teaching the technique to his students when he decided to apply it to a real-world object: the hydrogen atom. The hydrogen atom is actually one of the rare quantum mechanical systems whose energy levels can be solved exactly, but by applying the variational approach and then comparing the result to the exact solution, students could calculate the error in the approximation.
 
Although applying the variational principle to calculate the ground state of a hydrogen atom is a relatively straightforward problem, its applicability to an excited state is far from obvious. This is because the variational principle cannot ordinarily be applied if there are lower energy levels. However, Friedmann and Hagen were able to get around that by separating the problem into a series of l problems, each of which focused on the lowest energy level for  a given orbital angular momentum quantum number, l.

When Hagen started solving the problem himself, he immediately noticed a trend. The error of the variational approach was about 15 percent for the ground state of hydrogen, 10 percent for the first excited state, and kept getting smaller as the excited states grew larger. This was unusual, since the variational approach normally only gives good approximations for the lowest energy levels.
 
Specifically, the calculation of Friedmann and Hagen resulted in an expression involving special mathematical functions called gamma functions leading to the formula.

 "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."

which can be reduced to the classic Wallis formula.

“What surprised me is that the formula occurred in such a natural way in the calculations, with no circles involved in determining the energy states,” said Hagen, the co-author of the paper. “And I am glad I didn’t think about this before Tamar arrived in Rochester, because it would have gone nowhere and we would not have made this discovery.”

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.

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.

Read more at: http://phys.org/news/2015-11-derivation-pi-links-quantum-physics.html#jC

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.

Read more at: http://phys.org/news/2015-11-derivation-pi-links-quantum-physics.html#jCp

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.

Read more at: http://phys.org/news/2015-11-derivation-pi-links-quantum-physics.html#jCp

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.

Read more at: http://phys.org/news/2015-11-derivation-pi-links-quantum-physics.html#jCp