Three hundred and sixty years ago, British mathematician John Wallis ground out an unusual formula for π, the famed number that never ends. Now, oddly, a pair of physicists has found that the same formula emerges from a routine calculation in the physics of the hydrogen atom—the simplest atom there is. But before you go looking for a cosmic connection or buy any crystals, relax: There is probably no deep meaning to the slice of π from the quantum calculation.
Defined as the ratio of the circumference of a circle to its diameter, π is one of the weirder numbers going. Its decimal representation, 3.14159265358979 …, never ends and never repeats. And π can be captured in many disparate formulas. For example, in 1655, Wallis figured out that π can be written as the product of an infinite number of ratios multiplied together: π/2=(2/1*2/3)*(4/3*4/5)*(6/5*6/7)*(8/7*8/9)* …
Deriving that formula didn't come easy for Wallis, says Tamar Friedmann, a mathematician and physicist at the University of Rochester (U of R) in New York. Roughly speaking, he started out by considering the ratio of the areas of a circle and a square that circumscribes it—which turns out to be π/4—she explains. Wallis found a way to write this ratio in terms of infinite sums, such as 1+23+33+43+ … After pages of arithmetic, he was able to replace the sums with the product and achieve his famous formula. Mathematicians have since found simpler ways to derive it involving techniques from probability theory, combinatorics, and trigonometry.
Now, Friedmann and Carl Hagen, a theoretical physicist at U of R in New York, have found a surprisingly easy way to derive the formula using a three-page calculation involving the hydrogen atom. The hydrogen atom consists of a single negatively charged electron bound to a single positively charged proton. According to quantum mechanics, the electron does not circle the proton the way the moon circles Earth but instead occupies cloudlike orbitals that give the probability of finding the electron here or there. Each orbital has a distinct energy.
In their calculation, reported this week in the Journal of Mathematical Physics, the researchers use a technique called the variational principle to come up with an upper limit for the energy of each orbital. They compare that estimate with the exact energy for the orbital, which can be deduced from a more precise calculation. Hagen had been assigning this problem in quantum mechanics class and had found that the approximate value approached the exact one more closely for higher energy states. That was odd, Friedmann says, as approximations tend to work better for lower energy states.
Friedmann proved that for orbitals in which the electron whizzes around the nucleus with a lot of "angular momentum," the ratio of the approximate and exact energies can be rewritten as the ratio of things called gamma functions. As angular momentum increases, the ratio of gamma functions narrows in on 1, explaining the efficacy of the approximation. Moreover, one of those gamma functions gives a value of π, whereas the other ones can be rewritten as the product of ratios in the Wallis formula. So with a bit of rearranging, the Wallis formula tumbles out. "I was completely surprised," Friedmann says. "I wasn't looking for it at all."
The emergence of the formula probably doesn't signal anything profound about quantum theory, cautions Bruno Nachtergaele, a mathematical physicist at the University of California, Davis, and editor of the journal in which the paper was published. "You are entitled to be delighted by this," he says, "but one shouldn't look too deep for meaning." In fact, the emergence of the formula may have more to do with the properties of gamma functions than the physics of the hydrogen atom, Nachtergaele says. Special functions such as gamma functions can often be written out in many ways as sums, products, integrals, etc., Nachtergaele says, so it's possible that Friedmann and Hagen's analysis could lead to other notable formulas, too.