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While most people associate the mathematical constant π (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.
"We didn't just find pi," 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. "We found the classic seventeeth century Wallis formula for pi, making us the first to derive it from physics, in general, and quantum mechanics, in particular."
The Wallis formula—developed by British mathematician John Wallis in his book Arithmetica Infinitorum—defines π as the product of an infinite string of ratios made up of integers. For Friedmann, discovering the Wallis formula for π in a quantum mechanics formula for the hydrogen atom's energy states underscores π's omnipresence in math and science.
"The value of pi has taken on a mythical status, in part, because it's impossible to write it down with 100 percent accuracy," said Friedmann, "It cannot even be accurately expressed as a ratio of integers, and is, instead, best represented as a formula."
(Score: 3, Interesting) by stormwyrm on Friday November 13 2015, @11:59PM
Gravitational and electromagnetic fields all have the property that the divergence of the field is zero except where the mass or charge is present, and as such in three-dimensional space the inverse-square law arises. The inverse-square law has spherical symmetry by definition and as such it shouldn't be so surprising to find π popping everywhere when you are dealing with fields with those properties, although that a specific representation of π should arise so naturally is indeed interesting. What is more remarkable is Euler's solution to the Basel problem [wikipedia.org], where he proved that the harmonic series where each term is raised to an even power sums to some rational multiple of a power of π. For example ζ(2) = 1 + (1/2)2 + (1/3)2 + (1/4)2 + ... = π2/6. Similarly, ζ(4) = 1 + (1/2)4 + (1/3)4 + (1/4)4 + ... = π4/90. Now here we have π popping up in a context where there are no obvious circles or spheres or even any clear connection to geometry, and it intrigued the mathematicians of Euler's day greatly.
Numquam ponenda est pluralitas sine necessitate.
(Score: 0) by Anonymous Coward on Saturday November 14 2015, @02:50AM
Interesting yes, shocking- not so much (perhaps, to the limited extent of my ability to add value to this conversation). A good parallel situation perhaps is a recent article I read on some insect, maybe it was a grasshopper like thing, that had evolved literal gears. Here seeing the human 'invention' of gears evolve naturally in the biosphere is interesting, but again, not so shocking. Though focusing on the specific representation, I would guess that the representation is part of the specific quantum mechanics situation going on here (similar to your mention of spherical em fields). I just imagine e.g. trying to use genetic algorithms to emerge small algorithms to represent pi, and presumably you'd see a large fraction hit the same sorts of simplest solutions humans have worked out thus far. In a similar fashion, i'm not surprised to see natural systems like the one this quantum model is representing, stumble/settle upon some simple solution.
(Score: 2) by opinionated_science on Saturday November 14 2015, @08:40PM
Yes, it is quite an intriguing idea that some of the greatest mathematicians had no computers, but what would they have used them for?
(Score: 0) by Anonymous Coward on Sunday November 15 2015, @08:08PM
pr0n?
(Score: 2) by Non Sequor on Sunday November 15 2015, @02:04AM
The homotopy group of the circle is the additive group of integers. Additionally, modular arithmetic can also be defined based on breaking a circle into an integer number of arcs. The integers already have an "in" with the circle, it just needs to be coaxed out.
Write your congressman. Tell him he sucks.