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Quantum Physicists Find Pi in a Very Strange Place

An accidental discovery.

A pair of scientists from the University of Rochester has discovered something unexpected lurking far down in the mathematical depths of quantum mechanics: pi.

To be sure, pi, the endlessly repeating constant describing the ratio between a circle's circumference and its diameter, underpins much of what we understand about the physical and mathematical worlds, including quantum mechanics. But it wasn't expected to appear just out of the blue in calculations for the excited energy states of electrons in an atom, which is just what the Rochester physicists describe in a new paper published in the Journal of Mathematical Physics.

"The existence of such a derivation indicates that there are striking connections between well-established physics and pure mathematics that are remarkably beautiful yet still to be discovered," the authors write.

Electron shells in various atoms. Image: Wikimedia Commons

They didn't quite find pi in the form you or I normally consider it, which is (probably) just the number 3.14 with an endless string of junk after it. Instead, they found what's known as the Wallis formula. Rather than looking at pi as decimal digits, the Wallis formula instead imagines it as the product of an endless string of ratios between two integers.

It looks like the expression above. The Greek symbol here means "the product of a sequence," e.g. we evaluate the formula for every value of n and then multiply it by the result of the formula for n + 1 and just keep going up like that. So, if the formula was just n, the product a sequence for n (from 1 to 4) would be 1 * 2 * 3 * 4. Cool?

As with the pi we all know and love, the Wallis formula winds up almost but never quite converging on a definite value. Every multiplication save for the first handful wind up changing the result by an infinitesimal amount and so that result just winds up wavering up and down by infinitely small degrees. This is as we'd expect for pi.

So, how do we get from quantum mechanics to that thing above?

Rochester physicist Carl Hagen had teamed up with mathematician and physicist Tamar Friedmann, a visiting professor at the school, to attempt to characterize the excited energy states of electrons in an atom using an alternative principle usually used to approximate only the ground (non-excited) states of electrons. Could it be done?

Indeed it could, they discovered. The principle usually only works if there is no lower energy state than the one being approximated, but Friedmann and Hagen found that by breaking up the problem into many subproblems with each one corresponding to the angular momentum of an electron in a given energy state they could make it work by finding the lowest energy state of each individual level and then putting it all back together.

The scientists took their results and compared them with the values derived by Neils Bohr all the way back in 1913 when the idea of an atom with discrete energy levels was in its formative stages. Bohr's values are the exact correct values for the energy states of electrons in a hydrogen atom, while the values offered by the variational principle are rather more blurry approximations. Blurriness is pretty common in quantum physics.

Taking the differences between their variational principle-approximated numbers and the real ones, Friedmann and Hagen found that, taken as a sequence, these differences begin to align with the Wallis formula for pi. This was unexpected.

Wallis formula calculations. Image: Google

"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," Hagen offered in a statement.

So, in the lowest energy level of the hydrogen atom, the variational principle gives an answer about 15 percent off from the exact solution. In the next highest energy level, the error drops to 10 percent, and as the energy levels increase, the error keeps going down until, as the energy levels approach infinity the error becomes infinitesimal.

Sound familiar? The error never goes away, but it becomes infinitely small, just as with pi. The sequence winds up looking a whole lot like the one given by the Wallis formula. (By the by, the error decreases in the atom because electrons of higher and higher energies have better defined orbits. Imagine seeing a baseball more clearly as it hurtles toward home plate than it appears in the pitchers stationary hand. Weird, eh?)

"It was a complete surprise," Friedmann said. "I jumped up and down when we got the Wallis formula out of equations for the hydrogen atom. 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."