137 - Arthur I. Miller [127]
It is also the energy of an electron in the lowest orbit (n = 1) of the hydrogen atom (Z = 1).
Sommerfeld decided that the mathematics in Bohr’s original theory needed to be tidied up. His brilliant idea was to include relativity in the new mathematical formulation, making the mass of the electron behave according to E = mc2, in which mass and energy are equivalent. This was the result:
In this new equation, the additional quantum number k indicated the additional possible orbits for the electron and allowed the possibility for an electron to make additional quantum jumps from orbit to orbit. It therefore also allowed the possibility of the atom having additional spectral lines—a fine structure.
The first term in the equation for En,k—outside the brackets—was the same as in Bohr’s original equation. But a whole extra term had appeared inside the large brackets.
Multiplying this term was an extraordinary bundle of symbols that no one had ever seen before: . In this expression, e is the charge of the electron, h is Planck’s constant, and c is the velocity of light. Sommerfeld deduced the number 0.00729 from this bundle of symbols. He realized that this was the number that set the scale of the splitting of spectral lines—that is, of the atom’s fine structure—and called it the fine structure constant. It is there in the equation because it is there in the atom; it is part of the atom’s existence, which includes the fine structure of a spectral line. Physicists knew the fine structure existed. They had measured the fine structure splitting, but they didn’t have an equation for it that agreed with experiment. Now they did:
This extraordinary equation, in which 2e2/hc is replaced by 1/137, perfectly described the fine structure of spectral lines as observed in experiments.
Not just scientists but many others have grappled with 137. For a start, 137 can be expressed in terms of pi. Some complicated ways of doing this, all of which end up with 137, are
We can also write 137 as a series of Lucas numbers, which are connected with Fibonacci numbers and the Golden Ratio.
Fibonacci numbers is a sequence of whole numbers in which each number starting from the third is the sum of the two previous ones. The sequence begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, and so on (0 + 1 = 1, 1 + 1 = 2, etc.).
If we take the ratio of successive numbers in the series—1/1 = 1.000000, 2/1 = 2.000000, 3/2 = 1.500000, 5/3 = 1.666666, and so on to 987/610 = 1.618033—we reach 1.6180339837, the Golden Ratio, which has been a guideline for architecture since the days of ancient Greece. It appears on the pyramids of ancient Egypt, the Parthenon in Athens, and the United Nations building in New York City.
Fibonacci numbers were discovered by the Italian mathematician Leonardo Fibonacci in the twelfth century. It was Kepler who discovered their relation to the Golden Ratio. Then Edouard Lucas, a French mathematician, used them to develop the Lucas numbers in the nineteenth century.
Lucas numbers are like Fibonacci numbers but begin with 2: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, and so on. Like the Fibonacci series, the Lucas series also produces the Golden Ratio.
Fibonacci and Lucas numbers pop up all over the place, from how rabbits reproduce to the shape of mollusk shells, to leaf arrangements that sometimes spiral at angles derivable from the Golden Ratio.
Because all these numbers are related, any formula for 137 in terms of the Golden Ratio can be rewritten in terms of Fibonacci and Lucas numbers, though whether this is anything more than merely abstruse relationships