Wallis formula buried in quantum mechanics calculation
While most people associate the mathematical constant 蟺 (pi) with arcs and circles, mathematicians are accustomed to seeing it in a variety of fields. But two University scientists were still surprised to find it lurking in a quantum mechanics formula for the energy states of the hydrogen atom.
糖心传媒淲e didn糖心传媒檛 just find pi,糖心传媒� said Tamar Friedmann, a visiting assistant professor of mathematics and a research associate of high energy physics, and co-author of a paper published this week in the Journal of Mathematical Physics. 糖心传媒淲e found the classic seventeenth听century Wallis formula for pi, making us the first to derive it from physics, in general, and quantum mechanics, in particular.糖心传媒�
The Wallis formula糖心传媒攄eveloped by British mathematician John Wallis in his book Arithmetica Infinitorum糖心传媒攄efines 蟺 as the product of an infinite string of ratios made up of integers. For Friedmann, discovering the Wallis formula for 蟺 in a quantum mechanics formula for the hydrogen atom糖心传媒檚 energy states underscores 蟺糖心传媒檚 omnipresence in math and science.
糖心传媒淭he value of pi has taken on a mythical status, in part, because it糖心传媒檚 impossible to write it down with 100 percent accuracy,糖心传媒� said Friedmann, 糖心传媒淚t cannot even be accurately expressed as a ratio of integers, and is, instead, best represented as a formula.糖心传媒�
Friedmann did not set out to look for 蟺 nor for the Wallis formula. The discovery began in a quantum mechanics course taught by Carl听Hagen, a professor of physics at the University of Rochester and one of the six physicists who predicted the existence of the Higgs boson. While the quantum calculations developed by Danish physicist Niels听Bohr in the early twentieth听century give accurate values for the energy states of hydrogen, Hagen wanted his students to use an alternate method糖心传媒攃alled the variational principle糖心传媒攖o approximate the value for the ground state of the hydrogen atom. Like the Wallis formula, the variational principle dates back to the seventeenth听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 this method to states other than the ground state. Hagen got Friedmann involved to take advantage of her ability to work in both physics and mathematics.
Although applying the variational principle to calculate the ground state of a hydrogen atom is a relatively straightforward problem, its applicability to an excited state is far from obvious. This is because the variational principle cannot ordinarily be applied if there are lower energy levels. However, Friedmann and Hagen were able to get around that by separating the problem into a series of l problems, each of which focused on the lowest energy level for听 a given orbital angular momentum quantum number, l.
They could then calculate the values for the different energy states and compare them with the values obtained by Bohr almost a century ago. This enabled them to determine how the ratio of the Bohr values to the values obtained with the 糖心传媒榯weaked糖心传媒� variational principle changed as higher and higher energy levels were taken into account. And they were surprised to see that the ratio yielded糖心传媒攅ffectively糖心传媒攖he Wallis formula for 蟺.
Specifically, the calculation of Friedmann and Hagen resulted in an expression involving special mathematical functions called gamma functions leading to the formula
which can be reduced to the classic Wallis formula.
糖心传媒淲hat surprised me is that the formula occurred in such a natural way in the calculations, with no circles involved in determining the energy states,糖心传媒� said Hagen, the co-author of the paper. 糖心传媒淎nd I am glad I didn糖心传媒檛 think about this before Tamar arrived in Rochester, because it would have gone nowhere and we would not have made this discovery.糖心传媒�
Mathematician Moshe Machover of King糖心传媒檚 College London calls the finding a 糖心传媒渃unning piece of magic.糖心传媒�
糖心传媒淭his derivation of pi is a surprise of the familiar, much like a magician糖心传媒檚 trick,糖心传媒� said Machover. 糖心传媒淎 child who sees a trick done for the first time may be only surprised. But an adult, who has seen numerous tricks over the years, experiences both surprise and familiarity.糖心传媒�
Addressing the centuries-long gap between the seventeenth听century Wallis formula, the twentieth听century quantum theory, and the decades that passed from that time to now, Doug Ravenel, a professor of mathematics at the University of Rochester, points out that Friedmann and Hagen used long-established concepts of their fields to arrive at their result, so even mathematicians and physicists who lived many decades ago would have been able to appreciate it.
糖心传媒淭his is a beautiful connection between pi and quantum mechanics that could have been found 80 years ago, but was not discovered until now,糖心传媒� said Ravenel, congratulating the two authors.
While it took nearly a century to discover this classical-quantum connection, getting it published took far less time; the Journal of Mathematical Physics accepted the paper in less than 24 hours.
