The universe as we know it seems to be governed by the logic and language of mathematics. Prominent examples of this fact includes Newtonian physics, which obeys the rules of calculus, and Einstein’s general theory of relativity, which is described by Reimann geometry. These laws describe our everyday world, and the more distant worlds of stars, planets, and black holes. But what about the invisible world of quantum mechanics where objects can behave both as waves and particles and the laws of physics at the atomic level are nothing like the laws of gravity and relativity? The answer is that even there, mathematics reign supreme. Perhaps the most famous demonstration of this fact is the Schrodinger equation that describes the evolution over time of a wave function, the key concept used by theoretical physicists to characterize the behavior of individual quantum mechanical systems such as light quanta. Apart from its sheer beauty, the Schrodinger equation has yielded predictions on the bound states of atoms that have been repeatedly confirmed by experimental evidence. Pause for a moment to ponder what this means. First, it testifies to the power of the human mind to conjure up highly abstract mathematical relations that can somehow describe our physical world to a high degree of precision. Second, Schrodinger’s equation also affirms that most likely, all aspects of the physical world obey mathematical logic, not just the world of stars, and planets, and black holes.
In case you think this is an isolated example, consider the one I am about to describe. This example concerns pi, the famous irrational number denoted by the Greek symbol π whose value is 3.14159… ad infinitum. π is famously linked to circles and trigonometry, but also pops up in diverse areas of science such as combinatorics, astrophysics, engineering (e.g., the Fourier transform), probability (Brownian motion, stochastic processes, Monte Carlo simulations) and so on. It is thus natural to ask whether π also lurk in the atomic world of quantum mechanics.
Until recently, the answer to this question would have been a tentative “maybe”. But maybe has been turned into a resounding yes, thanks to recent discovery by two mathematical physicists at the University of Rochester, Carl Hagen and Tamar Friedmann. The backstory to their discovery goes back to 17th-century England, when British mathematician, John Wallis, developed a formula that defines π as the product of an infinite string of ratios made up of integers. Wallis’s formula is as follows:
No one suspected that this formula would be hiding somewhere in quantum mechanics until Hagen and Friedmann derived a ratio to describe the energy states of the hydrogen atom. Hagen and Friedmann’s unexpected and beautiful result was published in the Journal of Mathematical Physics in 2015 (see link below).
Neither Hagen nor Friedmann set out to look for π nor for the Wallis formula. Their discovery began almost by accident, in a quantum mechanics course taught by Hagen, who by the way was one of the six physicists who predicted the existence of the Higgs boson. Early in the 20th century, the Danish physicist, Niels Bohr had developed calculations that give accurate values for the energy states of hydrogen, Hagen wanted his students to use an alternate method—called the variational principle—to approximate the value for the ground state of the hydrogen atom. Like the Wallis formula, the variational principle dates back to the 17th century, one of its first appearances being the Principle of Least Time of mathematician Pierre de Fermat, a contemporary of Wallis. Hagen also started thinking about whether it would be possible to apply the variational method to states other than the ground state. He then roped in Friedmann to see if they could come up with a formula to approximate various energy states of the hydrogen atom and to compare its accuracy with the exact formula derived by Bohr by way of a ratio of the two formulas. To their surprise, their derivations led them to a ratio which was in effect the Wallis formula for π! Here is the expression for the Friedmann and Hagen ratio:
which can be reduced to the classic Wallis formula shown above.
“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. “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.” Their beautiful result has been described by mathematician Moshe Machover of King’s College London as a “cunning piece of magic”. More than that, it is that another demonstration that the quantum world, like the world of stars, planets, and black holes, is fundamentally a mathematical one.
Here is the Hagen-Friedmann paper, published in the Journal of Mathematical Physics, 2015.