New Physics Formula Links Quantum Mechanics And Pure Math

November 23, 2015 | by Alfredo Carpineti

he notorious number Pi has succeeded in becoming even more prevalent in science. Researchers from the University of Rochester discovered that a formula which approximates the energy levels of a hydrogen atom is the same as the one developed 360 years ago by English mathematician John Wallis to find the value of Pi.

The pair of physicists realized that when the variational method was employed to calculate the energy levels of the hydrogen atoms, the formula could be simplified into the Wallis Product, the one used for Pi. 

The variational method is used to approximate the lower energy states of atoms, when the electrons are close to the nucleus. It is not usually used on the hydrogen atom, as hydrogen is simple enough so scientists can compute the energy levels precisely. Energy levels give an indication of how tightly bound the electron is to an atom; the larger the energy level, the easier it is for the electron to break free. 

Professor Carl Hagen was teaching his students how to use the variational method and used the hydrogen atom as an example in the explanation. The students had to compute the expected values using the variational method and compare it with the exact solution and then calculate the error in the approximation.

When Hagen started to solve the problem he saw an unusual trend emerging. The errors were getting smaller as the energy level grew larger, which was surprising as the variational method only gives a good approximation for the smallest energy levels.

Hagen recruited Tamar Friedmann to study what would happen to the approximation as the energy increased, and they discovered that the formula was exactly the Wallis Product. 

"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 in a statement. "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 paper discussing the discovery is only three pages long and has been published in theJournal of Mathematical Physics.

Top Image Credit: Pi by J.Gabás Esteban, via Flickr. CC BY 2.0