Posted on 11/10/2015 10:54:08 AM PST by Red Badger
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
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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 quantum mechanics, 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.
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
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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 excited states grew larger. This was unusual, since the variational approach normally only gives good approximations for the lowest energy levels.
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."
Figure 2: The classic 18th century Wallis formula defining pi. Credit: University of Rochester
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From the formula for the limit of the variational solution as the energy 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 formula 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."
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 , http://scitation.aip.org/content/aip/journal/jmp/56/11/10.1063/1.4930800
Plenty of research is going on for that newly discovered subatomic particle, the moron. It’s found only in buffoonium.
That and the Par-Mesan..................found chiefly in cheese.....................
Pie are squared.
Yep, there is an order to the universe that I refuse to believe is accidental or was left to random entropy, but was created by design IMHO.
I love math and physics and what they mean and describe and reveal. I just wish I had the brain of a genius to truly understand it. These higher math principals give me fits. Can’t get my brain around them. Calculus was bad enough. Thank goodness there are true geniuses out there who can discover and process the math and physics of the universe. It is fascinating. Just wish I truly deeply understood it.
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One of the comments from the source....
“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?”
I guess what he is saying is that since the model for hydrogen has electrons in circular orbits it shouldn’t be surprising to find pi in the calculations.
Being an irrational number, pi cannot be expressed exactly as a fraction (equivalently, its decimal representation never ends and never settles into a permanent repeating pattern). Still, fractions such as 22/7 and other rational numbers are commonly used to approximate pi. The digits appear to be randomly distributed; however, to date, no proof of this has been discovered.
Also, pi is a transcendental number - a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of pi implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge.
Ancient civilizations needed the value of pi to be computed accurately for practical reasons. It was calculated to seven digits, using geometrical techniques, in Chinese mathematics and to about five in Indian mathematics in the 5th century CE.
The historically first exact formula for pi, based on infinite series, was not available until a millennium later, when in the 14th century the Madhava-Leibniz series was discovered in Indian mathematics.[1][2]
In the 20th and 21st centuries, mathematicians and computer scientists discovered new approaches that, when combined with increasing computational power, extended the decimal representation of pi to, as of 2015, over 13.3 trillion digits.[3]
Practically all scientific applications require no more than a few hundred digits of pi, and many substantially fewer, so the primary motivation for these computations is the human desire to break records.[4][5]
However, the extensive calculations involved have been used to test supercomputers and high-precision multiplication algorithms.
Because its definition relates to the circle, pi is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses or spheres.
It is also found in formulae used in other branches of science such as cosmology, number theory, statistics, fractals, thermodynamics, mechanics and electromagnetism. The ubiquity of pi makes it one of the most widely known mathematical constants both inside and outside the scientific community: Several books devoted to it have been published, the number is celebrated on Pi Day and record-setting calculations of the digits of pi often result in news headlines.
Attempts to memorize the value of pi with increasing precision have led to records of over 67,000 digits.
Sounds like a suburb of Washington DC.
Pie are round. Cornbread are squared.
Cowboy logic
Pie are round....
OK That is just too freakin’ cool!!!!!!!!!!!!!!!
Bump
Here is one use of pi, e (another transcendental number, namely the base of the natural logarithm aka Euler’s number) and i (the square root of -1) that I still find to be incredible:
e^(i*pi) = -1
/yes, geek mode here
Don’t know why everyone is getting so excited as the whole thing was clear from the start. Like, duuuuuh!
I was blown away when I first learned that
d/dx (e^x) = e^x
Yup. That one took me a bit to really digest, too!
“Pi is also found in formulae used in other branches of science such as cosmology,”
I will have to ask my cosmetician about that.
Sure you can. The genius is in the simplicity. Truth just spills itself out like light.
God created the world through the Hebrew letters. That is to say, God's *word*, which is Truth.
1. In the beginning God created the heaven and the earth:
2. And the earth was without form, and void; and darkness was upon the face of the deep; And a *wind* from God moved upon the face of the waters:
3. And God *said*, Let there be light; and there was light:
Yep, there is an order to the universe that I refuse to believe is accidental or was left to random entropy, but was created by design IMHO.
When I happened upon this FR thread, I had just finished reading the article at this link (still had the page open):
by Reb Gutman Locks
***
Might as well add Ecc 12:
10. Kohelet sought to find out acceptable words; and *words of truth written in proper form*:
11. The words of the wise are like goads, and *like nails firmly fixed are the collected sayings*, which are given by one shepherd:
12. And furthermore, by these, my son, be admonished; of making many books there is no end; and much study is a weariness of the flesh:
13. The end of the matter, all has been heard; Fear God, and keep his commandments; for this is the whole duty of man:
14. For God shall bring every deed into judgment, with every secret thing, whether it is good, or whether it is evil:
Gen 1, cont'd...
4. And God saw the light, that it was good; and God divided the light from the darkness:
5. And God called the light Day, and the darkness he called Night; And there was evening and there was morning, one day:
Hashem gave us His mitzvahs so we could understand His ways and what He wants from us. He wants us to emulate Him. For instance, Hashem is charitable so He commands us to give charity. Hashem is loving so He commands us to love one another. Hashem is holy so He gave us His holy mitzvahs to fulfill so we can become holy too.
There you go. Everything comes around full circle..
And some people claim there is no God.
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