A Slice of Quantum

Pi or "π" is commonly found in geometry and math. It's the ratio of a circle's circumference to its diameter and it can't be expressed exactly as a fraction or a decimal: No fractional expression is correct and the decimal neither ends nor repeats.

But π can be expressed using a mathematical formula, first written by English mathematician John Wallis in 1655. It's an infinite string of integers derived from a calculus equation. Wallis was on pretty shaky ground when he developed it, because the actual calculus it uses was still about 30 years in the future from him. But his formula bore out, and was later shown to be a corollary of a formula for the sine function developed by Leonhard Euler.

And now π has shown up in, of all places, quantum mechanics. A physicist at the University of Rochester had his students figure out the energy states for the energy states of hydrogen atoms. Adding energy to the atoms changes their energy states, but it doesn't happen in a smooth ascending curve. There are only certain levels of energy the atoms can have, and they jump from one to another the way a car shifts from one gear to another. The physicist, Carl Hagen, wanted his students to use a different method than the most common one in order to learn some different things.

Tamar Friedman, a visiting assistant professor, worked with Hagen on the project and they discovered that the different formula they used created reduced to the Wallis formula. In other words, the mathematical relationship between the different energy states of a hydrogen atom involved good ol' π.  And, as another professor commenting on the work noticed, almost nothing involved in learning this connection required modern machinery, computers or testing. It could have been found some time ago and proven just as easily.

It's certainly a neat connection between the quantum world and classical scientific disciplines, whether it ever takes researchers anywhere or not. And makes one wonder what other little hidden linkages lurk behind the next classroom experiment.