## Pi and the Quantum RealmNovember 30, 2015 As Physics Nobel Laureate, Eugene Wigner, wrote in 1960,"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning."[1]In this sentiment, Wigner was reframing in a modern context the 2,500 year old statements by the Greek philosopher, Pythagoras, that "Mathematics is the way to understand the universe," and "Number is the measure of all things."[2] Mathematics is a good friend of most scientists, who, as Ernest Rutherford said, would otherwise be doing "stamp collecting."[3] I wrote about science and mathematics in a previous article (Science and Math, July 16, 2012).
"Philosophy is written in this grand book — I mean the universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other Geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth."[4]One mysterious connection between mathematics and the quantum realm was discovered by Wigner, who found that random matrices closely describe the nuclear excitations of heavy atoms.[5] He found that the spacing between the energy levels of highly excited states of heavy nuclei resemble the spacings between the eigenvalues of a random matrix. It appears that God does play dice with the world. A University of Rochester (Rochester, New York) physicist and mathematician were recently surprised to find the mathematical constant, pi, hidden in a quantum mechanical calculation for the energy states of the hydrogen atom.[6-8] Says mathematician, Tamar Friedmann, coauthor of a paper on this discovery in the journal, Mathematical Physics, "We didn't just find pi... We found the classic seventeenth century Wallis formula for pi, making us the first to derive it from physics, in general, and quantum mechanics, in particular."[7]The Wallis formula is quite unlike the most important formulas for calculating pi. Those formulas are rapidly convergent series expansions, the most notable being one developed by the brothers, David Chudnovsky and Gregory Chudnovsky, that delivers three digits of pi per calculated term, Arithmetica Infinitorum, as shown below,
"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."[8]The quantum behavior of electrons in hydrogen has been known for a century, but its connection to pi has been unknown for all that time. This discovery was notable enough that Friedmann and Hagen's paper was accepted in Mathematical Physics in less than 24 hours.[7] ## References:- E.P. Wigner, "The unreasonable effectiveness of mathematics in the natural sciences," Communications on Pure and Applied Mathematics, vol. 13, no. 1 (February, 1960). pp. 1-14. A PDF file is available, here.
- R. W. Hamming, "The Unreasonable Effectiveness of Mathematics," The American Mathematical Monthly, vol. 87, no. 2 (February, 1980), via the Center for Cancer Research Nanobiology Program.
- Ernest Rutherford quotation page on Wikiquotes.
- Selections of a translation of The Assayer; Stillman Drake, "Discoveries and Opinions of Galileo," Doubleday & Co.(New York, 1957), pp. 231-280.
- E. Wigner, "Characteristic vectors of bordered matrices with infinite dimensions," Annals of Mathematics, vol. 62, no. 3 (November, 1955), pp. 548-564, doi:10.2307/1970079.
- Tamar Friedmann and C. R. Hagen, "Quantum mechanical derivation of the Wallis formula for π," J. Math. Phys., vol. 56 (In Press, 2015), Document No. 112101, doi:10.1063/1.4930800. This is an open access paper with a PDF file available at the same URL.
- Discovery of classic pi formula a ‘cunning piece of magic’, University of Rochester Press Release, November 10, 2015.
- New derivation of pi links quantum physics and pure math, American Institute of Physics Press Release, November 10, 2015.
- Derivation of the Wallis formula for pi on Wikipedia.
Linked Keywords: Nobel Prize in Physics; Physics Nobel Laureate; Eugene Wigner; language; mathematics; physical law; laws of physics; gift; research; learning; Ancient Greek philosophy; Greek philosopher; Pythagoras; universe; number; scientist; Ernest Rutherford; stamp collecting; science; Nuremberg Chronicle; woodcut; Wikimedia Commons; the proof of the pudding is in the eating; Galileo Galilei; physicist; experiment; The Assayer; philosophy; triangle; circle; geometry; geometric; human; labyrinth; quantum mechanics; quantum realm; random matrices; nuclear isomer; nuclear excitation; atom; energy level; excited state; atomic nucleus; nuclei; eigenvalue; random matrix; God; Albert Einstein; play dice with the world; University of Rochester (Rochester, New York); mathematician; mathematical constant; pi; quantum mechanical; calculation; hydrogen atom; Tamar Friedmann; coauthor; academic publishing; paper; scientific journal; Mathematical Physics; 17th century; seventeenth century; Wallis product; Wallis formula; formula; calculating pi; convergent series expansion; David Chudnovsky; Gregory Chudnovsky; numerical digit; algorithm; genius; prodigy; Srinivasa Ramanujan; John Wallis (1616-1703); English; infinity; multiplication; product; Arithmetica Infinitorum; scholar; Latin; multiplicative inverse; Google Books; electron; variational approach; graduate school; variational principle; least time principle; Pierre de Fermat; ground state; Carl Hagen; orbit; uncertainty principle; uncertainty; radius; ratio; error. |
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