New derivation of pi links quantum physics and pure math

November 10, 2015
Two pages from the book "Arithmetica Infinitorum," by John Wallis. In the table on the left page, the square that appears repeatedly denotes 4/pi, or the ratio of the area of a square to the area of the circumscribed circle. Wallis used the table to obtain the inequalities shown at the top of the page on the right that led to his formula. Credit: Digitized by Google

In 1655 the English mathematician John Wallis published a book in which he derived a formula for pi as the product of an infinite series of ratios. Now researchers from the University of Rochester, in a surprise discovery, have found the same formula in quantum mechanical calculations of the energy levels of a hydrogen atom.

"We weren't looking for the Wallis formula for pi. It just fell into our laps," said Carl Hagen, a particle physicist at the University of Rochester. Having noticed an intriguing trend in the solutions to a problem set he had developed for students in a class on , Hagen recruited mathematician Tamar Friedmann and they realized this trend was in fact a manifestation of the Wallis formula for pi.

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

The researchers report their findings in the Journal of Mathematical Physics.

In quantum mechanics, a technique called the variational approach can be used to approximate the energy states of quantum systems, like molecules, that can't be solved exactly. Hagen was teaching the technique to his students when he decided to apply it to a real-world object: the hydrogen atom. The hydrogen atom is actually one of the rare quantum mechanical systems whose energy levels can be solved exactly, but by applying the variational approach and then comparing the result to the exact solution, students could calculate the error in the approximation.

Discovery of classic pi formula a 'cunning piece of magic'
Figure 1: Calculations derived by Tamar Friedmann and Carl Hagen resulted in an expression of special mathematical functions called gamma functions. Credit: Tamar Friedmann and Carl Hagen/University of Rochester

When Hagen started solving the problem himself, he immediately noticed a trend. The error of the variational approach was about 15 percent for the ground state of hydrogen, 10 percent for the first excited state, and kept getting smaller as the grew larger. This was unusual, since the variational approach normally only gives good approximations for the lowest .

Hagen recruited Friedmann to take a look at what would happen with increasing energy. They found that the limit of the variational solution approaches the model of hydrogen developed by physicist Niels Bohr in the early 20th century, which depicts the orbits of the electron as perfectly circular. This would be expected from Bohr's correspondence principle, which states that for large radius orbits, the behavior of quantum systems can be described by classical physics.

"At the lower energy orbits, the path of the electron is fuzzy and spread out," Hagen explained. "At more excited states, the orbits become more sharply defined and the uncertainty in the radius decreases."

Discovery of classic pi formula a 'cunning piece of magic'
Figure 2: The classic 18th century Wallis formula defining pi. Credit: University of Rochester

From the formula for the limit of the variational solution as the increased, Hagen and Friedmann were able to pull out the Wallis formula for pi.

The theory of quantum mechanics dates back to the early 20th century and the Wallis has been around for hundreds of years, but the connection between the two had remained hidden until now.

"Nature had kept this secret for the last 80 years," Friedmann said. "I'm glad we revealed it."

Explore further: Good quantum states and bad quantum states

More information: "Quantum mechanical derivation of the Wallis formula for pi," by Tamar Friedmann and C.R. Hagen, Journal of Mathematical Physics , November 10, 2015. DOI: 10.1063/1.4930800 ,

Related Stories

Good quantum states and bad quantum states

July 6, 2015

It is impossible to obtain all information about a large quantum system consisting of hundreds or thousands of particles. A new technique allows to describe such systems in terms of 'continuous matrix product states.' With ...

Calcium-48's 'neutron skin' thinner than previously thought

November 2, 2015

An international team led by Gaute Hagen of the Department of Energy's Oak Ridge National Laboratory used America's most powerful supercomputer, Titan, to compute the neutron distribution and related observables of calcium-48, ...

Parity effect observed in graphene

August 4, 2015

Researchers have theoretically projected and successfully proven through experimentation the parity effect of the quantum Hall edge transport in graphene antidot devices with pn junctions (PNJs). Graphine, or single-layered ...

Team extends the lifetime of atoms using a mirror

October 13, 2015

Researchers at Chalmers University of Technology have succeeded in an experiment where they get an artificial atom to survive ten times longer than normal by positioning the atom in front of a mirror. The findings were recently ...

Recommended for you

Laser cooling a polyatomic molecule

April 26, 2017

(—A team of researchers at Harvard University has successfully cooled a three-atom molecule down to near absolute zero for the first time. In their paper published in Physical Review Letters, the team describes ...

New CERN results show novel phenomena in proton collisions

April 25, 2017

In a paper published today in Nature Physics , the ALICE collaboration reports that proton collisions sometimes present similar patterns to those observed in the collisions of heavy nuclei. This behaviour was spotted through ...


Adjust slider to filter visible comments by rank

Display comments: newest first

1.7 / 5 (18) Nov 10, 2015
This was never a secret. It has just been recently perceived.
For all the talk about godlessness all humans personify the mysterious.
4.4 / 5 (7) Nov 10, 2015
Thus prior to perception it was a secret.
4.7 / 5 (7) Nov 10, 2015
I suppose I should look at the J Math Phys article. However, their variational bound approaches the Bohr classical model in the limit of large n with circular orbits. Why is it surprising that such a limit would involve pi?
5 / 5 (6) Nov 10, 2015
Only a secret til' we find it.
Nov 10, 2015
This comment has been removed by a moderator.
5 / 5 (2) Nov 10, 2015
The equations describing quantum states frequently contain Pi. Why is it surprising that an equation calculating approximations to quantum states contains an equation that is an approximation to Pi?

I am reminded of an experiment I did years ago at University to measure c, the speed of light. All was going very well and we had a very good value for c. Then we realized that one of the constants used in the calculation was a derived constant that had folded in the value of c. So, if you looked at it mathematically and cancelled out all the other terms you ended up with:

c(E) = c(T)

Where c(E) is the value "measured" and c(T) is the accepted textbook value.
5 / 5 (3) Nov 10, 2015
In other words, the guys just found a way to prove Wallis' formula with quantum mechanics. Cool.
1.7 / 5 (6) Nov 10, 2015
Yes. Secret between Apple & Pie were hidden from humanity for several millennia.
Dinosaurs never heard of pie. The Great Missing Recipe!
If you don't believe, just Do Google IMAGE Search for Apple Pie.
e was The Secret between Pi & Pie!
3 / 5 (2) Nov 10, 2015
Physics and Math have always been intimately entwined from the Fibonacci numbers in the arrangement of the patterns on sunflowers to Heisenberg's quantum matrices of the Hydrogen atom.
5 / 5 (6) Nov 10, 2015
Isn't it because the hydrogen atom is spherical?

Atoms are not spherical in mathematical terms, they are more like clouds or swarms. Because electron trajectories and positions cannot be computed for any given point in time, they can't be said to have clear boundaries (except at absolute zero).

That said they are spherical in layman's terms (because superimposition of electron trajectories over time average out to a round shape), so this mathematical relationship is very intuitive. A nice change for quantum physics :-)
Steve 200mph Cruiz
4.6 / 5 (9) Nov 10, 2015
Odin, jayded

A discovery does not mean you create something, it means you notice something that was already there

No it is not spherical, this is what the electron shell looks like around a hydrogen atom:
The shape of the electron bubble changes dramatically depending on the state and how much energy the electron contains.
Just imagine how complicated it gets in atoms with more electrons which are repelling each other while doing all this as well, and then in molecules where electrons and even entire atoms start getting shared between other atoms and nuclei.
5 / 5 (1) Nov 11, 2015
It takes very good teachers to present new knowledge so that minds grab the concept and respond "I knew it all along" as evidenced well in these blogs.
This neurological response is called the Hindsight Bias: https://en.wikipe...ght_bias
Very good gentlemen, this advances our modeling of atoms by directly showing the transition from quantum to classical in one small atom. Statistically the Heisenberg uncertainty becomes smaller the more massive or energetic the numbers are. Lovely!
3 / 5 (2) Nov 11, 2015
Job001 I really want to know what is surprising. The researcher was apparently jumping for joy - suggesting that he had found something important and totally unexpected. What was that? The semi-classical behavior of electrons for a large principal quantum number, n, is well known. They are called Rydberg atoms and they have long been found to make the transition to classical Bohr atom behavior in the limit that n becomes large. So that is not the surprise. What is it? As has been pointed out, it is not surprising to find pi when you are examining an orbit that is circular (planetary) in the classical limit.
5 / 5 (2) Nov 11, 2015
As I understand the article, the surprise was not caused by finding pi in their calculation, rather that they got this particular formula (Wallis formula).
5 / 5 (1) Nov 11, 2015
"At the lower energy orbits, the path of the electron is fuzzy and spread out," Hagen explained. "At more excited states, the orbits become more sharply defined and the uncertainty in the radius decreases."

Now I thought it was a case closed that electrons do not orbit the atom's nucleus; they just pops here and there in a "bubble" of probability. Is he referring to an "orbit" because he mentioned Bohr or am I missing something? By orbit does he refer to the probability bubble being more "sharply defined" or he says 20th century physics are right?
5 / 5 (1) Nov 11, 2015
del2 OK but if they are doing an iterative approximation as n increases they have to find some series. Why is Wallis' series so neat from a physic's point of view Or is this just sort of an aesthetic surprise -- rather than some an interesting physical result?

Bulbuzor: For large n, the velocity and radius both increase. Hence, the uncertainty principal allows for more and more localized wave packets. With the right combination of wave functions, one can find the Bohr atom electron as a limiting case. Both quantum and classical theory are correct in the this limit.
5 / 5 (1) Nov 11, 2015
have found the same formula in quantum mechanical calculations of the energy levels of a hydrogen atom
Isn't it because the hydrogen atom is spherical? The formulas for sphere geometry contain pi number often...

YES! The measurements of Quantum Mechanics show this to be the case. It also supports my new spherical model of the hydrogen atom, which was described in my CSIE2015 technical paper: "Computer Modeling of Atomic Forces". Whereas the Bohr model has an unlimited size, my model conforms with the Rydberg wavelengths/frequencies (as does the Bohr model) and also conforms to the measurements of QM. In addition, I have provided a method of analyzing these dynamic forces utilizing computer analysis and have actually calculated these forces at two different separations between two hydrogen atoms.
not rated yet Nov 11, 2015
Or is it more like throwing darts at a line?
vlaaing peerd
not rated yet Nov 12, 2015
Only a secret til' we find it.

It's a secret if something is kept intentionally hidden, which it wasn't. So it was rather just unknown.
5 / 5 (1) Nov 12, 2015
I ran a bit of PHP code, and let it do 20 million iterations, then multiplied by TWO and got an apx value of pie as

Final Result =

Where as the accepted value is

Apparently if you use every second ratio, then you get the square root of 2!!!!

5 / 5 (1) Nov 13, 2015
Hi big hairy jimbo,

Indeed: http://math.stack...equal-pi

5 / 5 (1) Nov 15, 2015
"The special thing is that it brings out a beautiful connection between physics and math."

More specifically, I think Id say it was a connection between physics and geometry.
not rated yet Nov 16, 2015
"The special thing is that it brings out a beautiful connection between physics and math."

More specifically, I think Id say it was a connection between physics and geometry.

I always thought geometry was a branch of maths?

Quite clearly, our mathematical discoveries are constraint by the world we live in. No matter what it is that some mathematician discovers in his/her wildest imagineerings, it always comes down to something connected to the whole set of things we are allowed to discover.

It's as if there's a prescribed path of discovery to which we are constrained, no matter what we do or claim to do as creative and original thought..
1 / 5 (2) Nov 16, 2015
Is this only for the ground energy state, the 1s1 orbital? circular. If the electron can be anywhere in that orbital, why wouldn't its characteristic equations match an infinite set of circular ratios? I'm missing how this isn't obvious.

If so, it illustrates the effects of population growth. One mind doing the work of 50 is better because the 50 can't directly correlate their insights. Intelligence is becoming diluted. Which would explain many of the legends in their own mind on this site.
not rated yet Nov 18, 2015
@animah Thanks for the link.

That series of square roots of two's looks like it might generate pi more accurately with fewer iterations. I'm going to have to compare the two now :-)))))
not rated yet Dec 31, 2015
The derivation is not for the exact description of the hydrogen atom, but for an approximate model to it. This model is closely related to Bohr's atom model, which is circular, hence intimately related to Pi. The approximated model is expressed in terms of the Gamma function, which has a factorial form (hence products of successive values, just like the Wallis product). If you look it up, there are well-known relations between the Wallis integral (closely related to the Wallis product) and the Gamma function. In fact, if you look up the formula for W(2p+1) in the Wikipedia entry on the Wallis integrals, it closely resembles the square root of the expression reported in the paper.

A fun fact? Maybe. But the title "New derivation of pi links quantum physics and pure math" is totally overblown.

Please sign in to add a comment. Registration is free, and takes less than a minute. Read more

Click here to reset your password.
Sign in to get notified via email when new comments are made.