Our World is full of Paradoxes; contradictions that challenge our notion of reality and Understanding. As absurd it may seem but the propositions are always found to be true.
By denying Principles, one may maintain any Paradox!
Galileo Galilei
The Most famous paradoxes of all time include Schrodinger Cat, Dark matter, Quantum Zeno Effect, and many more(not familiar to me). But we are here to talk about “Fine Structure Constant” denoted by the Greek letter α, a constant that has baffled scientists for more than a century. What intrigues physicists is not it’s origin but the value it seems to hold. The Great Explainer aka Richard Feynman called it “A Magic Number” or “One of the Greatest Damn Mysteries of Physics.”
The Bohr Model Failure
In the era of wave-particle duality, Louis de Broglie introduced a formula known as “De-Broglie Wavelength”.
λ=h/p ; p= mev
At the same time, Niels Bohr was constructing a simple model of the hydrogen atom. Using De-Broglie’s hypothesis he stated that electron could only revolve in an orbit whose angular momentum was an integral multiple of De-Broglie wavelengths. In the Bohr model, a single e¯ revolving around the nucleus made a circular orbit. Considering duality the e¯ moved in a wave similar to that of a wave generated in a fixed string. So the circumference covered by e¯ equals the length of the wave.
L = nλ ; circumference= 2πr= L ; 2πr= nλ ; 2πr= nh/p ; mevr= nħ
Quantizing the angular momentum, Bohr was able to calculate the energy difference between different n levels. He was able to explain the origin of different spectral lines of the hydrogen atom. Like the red Hα line was interpreted as a jump of e¯ from n=3 to n=2. Later on, it was discovered that the red line was a doublet which was termed as ‘Fine Structure’ of lines, an anomaly that could not be explained using the Bohr model.
The Orginal Origin
Arnold Sommerfeld thought he could improvise the Bohr model to explain the anomaly. Instead, he said that the orbit could be both circular and elliptical. He introduced a quantum number k(any integer except 0) where n/k= length of major-axis/length of minor-axis. With an increase in the value of k, ellipticity decreases and becomes circular when n=k.
Sommerfeld suggested that orbits are made of sub energy levels, and k was called “Azimuthal quantum number” which determines the orbital angular momentum. According to this model, he also considered the effect of variation of mass with speed. Summing the whole process, the final expression for Total Energy came out to be:
W(n,k)= -Rhc/n2[1 + α2/n2 (n/k – 3/4
) ] (interesting isn’t it?! )
Sommerfeld found that the energy difference between levels E(2,2) and E(2,1) explained the doublet anomaly. After that, he sought the discovery of a constant that was missing in the equation. When e¯ revolves around a nucleus, a centripetal force acts on it which is provided the coulomb’s electrostatic force.
FC = mv2 /r = e2/4πe0r2 ; mvr= nħ ; K.E=
mv2= e2/8 πe0r ; P.E= -FC*r= -e2/4πe0r
So total energy is, En = P.E + K.E = -e2/8πe0r …..①. Now, mevr= nħ, v2= e2/4πe0mr =( nħ/mr)2.
From this, = e2m/4πe0(n2ħ2), substituting in ① we get,
En = –( e2/4πe0nħ )2*m ; En= –
( e2/4πe0nħ )2*mc2/c2 ;
En= –
( e2/4πe0nħc )2*E0
Re-writing, En= –α2E0/n2,where α= ( e2/4πe0ħc)
Following the MKS units and fundamental values , α≈ 1/137 [Dimensionless].
The Alpha Effect
After the contribution of Sommerfeld, α appeared in many calculations and its value remained the same irrespective of the dimension system. It is also known as “Coupling Constant”, which determines the strength of electromagnetic interactions. With the Schrodinger model and Uncertainty Principle, it became clear that atoms do not have fixed orbits nor e¯s have fixed speeds and energies. These discoveries changed the whole Atom game but α remained the same.
Since α determines the electronic energy levels in atoms, scientists were curious if changing the value would change anything. In 1950 Fred Hoyle and others mapped out the detailed process by which Stars produce heavy elements such as carbon, oxygen, etc, and the Formation of the Star itself. They observed that the abundance of carbon in the universe could only be accounted for if α had a value that favored the fusion of helium nuclei to produce carbon than any other element.
Another study also showed that if you change the value of α even by as little as 4% stars would not be able to sustain nuclear reactions happening in their cores. Even Wolfgang Pauli was obsessed with this number and he famously quipped,
When I die, my first question to Devil will be: What is the meaning of Fine Structure Constant?
The Inconstant Constant Paradox
For long there has been speculation as to how constant alpha is? Does its value change over space-time? One may say that this contradicts the definition of constant in the first place. In 1937, Paul Dirac wrote to Astronomer Arthur Eddington’s attempts to derive the Constants from scratch, “How can we be so sure that constants have not evolved over cosmological time?”
Keck Telescope in Hawaii [Image: Science] 
Quasar Absorption Lines [Image: Astrobites]
In 2010, John Webb along with his team observed that α had changed since the beginning of the universe. Webb and his team collected data from the Keck Telescope showing various Absorption spectra of quasars. He said, “Changing α you change the degree of attraction between e¯ and nucleus”.
This changes the wavelength absorbed by the e¯ affecting the Absorption spectra, meaning that Absorption spectra are a kind of barcode unique to the value of α. Analyzing the data, they discovered that it had increased by an average of 6 parts in a million (not quite the change you were expecting). This gave a possible hint that cosmological constants are not so constant after all.
Another study conducted near the biggest black hole at the very heart of our galaxy “The Saggitarus A*” suggested otherwise. Researchers observed five Stars that cruised around the SMBH and collected data of their absorption spectra. Fortunately for physics lovers, the constant showed no sign of variation even near such extreme gravitational conditions. Perhaps this is a number written by GOD itself and is intrinsic to our very existence and creation of the Universe as we know it.
The Biggest Mysteries of Nature hide in the most Plain sight waiting to be nurtured.
RDX
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The renormalization theory calculated Alpha from scratch, deriving it from a mathematical point of view is theoretically possible. But the same is not observed experimentally.
When Sir Arthur Eddington tried to do the same he failed. Constants cannot be derived but are placed according to our convenience in equations, to satisfy the conditions that need to be met. In short, if alpha is considered a function, other related constants can also challenge the notion of its existence.
alpha is not “considered” a function. It “is” a function. That was what I was saying. It is the function of the momentum of interaction and depends on the energy of the interaction.
Considering the main essence, alpha was proposed as “The probability that an electron will emit or absorb a photon.” So what you are saying is one of the first definitions proposed for alpha. Also in QED alpha is directly related to the coupling constant, but its value cannot be predicted. So detailed research is required to predict the true nature of alpha just as that of light.
“Renormalization theory teaches us that α isn’t really a number at all; it’s a function. In particular, it’s a function of the total amount of momentum involved in the interaction you are considering. Essentially, the strength of electromagnetism is slightly different for processes happening at different energies.”
-Sean Caroll
(https://www.preposterousuniverse.com/blog/2018/09/25/atiyah-and-the-fine-structure-constant/)
Any thoughts about this?
Very nice