Last time: light used to be faster → This time: 1/137, the star of that footnote, takes center stage
Last time, we rewrote cosmic expansion as "the speed of light slowly drops." In the closing footnote, one number that never moves made an appearance — the fine-structure constant \(\alpha\approx 1/137\). The speed of light, lengths, energies — their values all shift depending on how you measure. But \(\alpha\) alone comes out to \(1/137\) no matter who measures it, in any units. This time we'll verify, by hand, how this strange number shows up as a "speed ratio" inside the atom.
A hydrogen atom is the simplest atom: one electron circling a single proton at the center. In the simple model built by the physicist Bohr, the speed \(v\) of the innermost orbiting electron can be written using the speed of light \(c\) like this:
So the electron orbits at exactly one-137th the speed of light. In everyday terms, about 2,200 km/s — blazingly fast, but only 1/137 of light's speed (300,000 km/s). The name of this "one-137th" ratio is the fine-structure constant \(\alpha\). Written out, it's built from a combination of several fundamental constants:
$$\alpha = \frac{e^2}{4\pi\varepsilon_0\,\hbar\, c}$$Here \(e\) is the electron's charge, \(\varepsilon_0\) a constant describing the vacuum, \(\hbar\) the Planck constant (the basic unit of the quantum world), and \(c\) the speed of light. It looks scary at first glance, but the key point is this: multiply and divide all of it together and the units vanish, leaving just a number. We'll check that in the next step.
In physics, "a quantity with units" and "a number with the units gone" are in different leagues. A number with the units gone (a dimensionless quantity) has the same value whether you use meters or inches — the same for anyone, anywhere in the universe. Let's confirm by hand that \(\alpha\) belongs to that club.
Values we'll use (SI units)
\(e = 1.602\times10^{-19}\ \mathrm{C}\), \(\varepsilon_0 = 8.854\times10^{-12}\ \mathrm{F/m}\), \(\hbar = 1.055\times10^{-34}\ \mathrm{J\cdot s}\), \(c = 2.998\times10^{8}\ \mathrm{m/s}\).
Numerator and denominator separately
$$\text{numerator}=e^2=(1.602\times10^{-19})^2 \approx 2.566\times10^{-38}$$ $$\text{denominator}=4\pi\varepsilon_0\hbar c \approx 4\pi(8.854\times10^{-12})(1.055\times10^{-34})(2.998\times10^{8})$$Working the denominator out step by step gives \(\approx 3.517\times10^{-36}\). Therefore
$$\alpha = \frac{2.566\times10^{-38}}{3.517\times10^{-36}} \approx 7.30\times10^{-3} = \frac{1}{137}.$$What happened to the units?
The numerator is \(\mathrm{C^2}\); the denominator is \(\mathrm{(F/m)(J\cdot s)(m/s)}\). Recalling \(\mathrm{F=C^2/J}\) and canceling, every unit cleanly cancels out. What remains is the pure number \(1/137\). Switch from meters to feet and this value doesn't move a hair.
There's a second way to read \(\alpha=v/c\). When something approaches the speed of light, Einstein's relativistic effects (like time dilation) kick in. Far below the speed of light, relativity can be ignored. The gauge of "how much relativity matters" is precisely \(v/c\).
For the hydrogen electron, \(v/c=\alpha=1/137\) — so it's only slightly relativistic. That "slightly" splits the color of the light the atom emits by a tiny amount (fine structure). The size of the split scales roughly as \(\alpha^2\). Let's work it out.
Scale of the relativistic correction
$$\left(\frac{v}{c}\right)^2 = \alpha^2 = \left(\frac{1}{137}\right)^2 = \frac{1}{18769} \approx 5.3\times10^{-5}$$About one part in twenty thousand. That's roughly the fraction of an atom's energy that the tiny relativistic shift accounts for. Small — but a precise spectrometer sees it clearly. This is the origin of "fine structure," the tiny splitting of spectral lines. The ratio 1/137 is carved right into the fine detail of the light's color.
This is the most important part of the episode — and the heart of last time's footnote. The atom's "size" and the electron's "speed" themselves are, in fact, quantities that can shift depending on how you measure (which units you pick). But take their ratio, and the moving parts cancel cleanly, leaving only the immovable number \(\alpha\).
For example, take the gauge of atomic size, the "Bohr radius \(a_0\)," and another fundamental length of the electron, the "Compton wavelength \(\lambda_C\)." Their ratio is
$$\frac{\lambda_C}{a_0} = \alpha \approx \frac{1}{137}.$$Each length shifts depending on the speed of light and the choice of units. But the instant you make it a ratio, the units and the moving parts vanish, and only \(1/137\) shows its face. When we said last time "even if light slows, \(\alpha\) is invariant," this is exactly it — even if \(c\) moves, the other constants move along with it, and the ratio \(\alpha\) does not.
Quantities with units (speed of light, lengths, energies) shift with the measurer's "ledger." The unit-free ratio \(\alpha\) is "a property of the universe itself," coming out \(1/137\) for anyone. Only the latter is something physics can truly ask about.
Why \(\alpha\) is exactly 1/137 is, in truth, something nobody yet knows. It's one of the greatest mysteries in physics. Long ago the great physicist Eddington tried to derive this number from theory alone, and failed spectacularly. For now, \(\alpha\) is an "input value fixed by measurement," not something computable from anything more fundamental.
To be even more honest: \(\alpha\) actually changes a little depending on "how finely you look" (at high energy it's about \(1/128\), not \(1/137\)). So strictly, "1/137 is the value of \(\alpha\) when you look at the coarsest scale." We'll cover this "running" in a later episode.
If we're going to claim "\(\alpha\) doesn't move," there ought to be an experiment that checked. There is. It's a precision experiment: line up atomic clocks of different types and, over a long time, compare the ratio of their ticking rates. If \(\alpha\) moved even a little, the clocks' rhythms would drift apart.
The answer from the latest experiments: \(\alpha\)'s change is less than \(10^{-19}\) per year — a staggering precision, smaller than "one trillionth of one hundred-millionth per year." Recall last time's universe (light changing by a fraction \(7.2\times10^{-11}\) per year): if only \(c\) changed while the others stayed fixed, \(\alpha\) would move at that same rate — instantly contradicting this experiment. So last time's universe must be one where "if \(c\) moves, the other constants move along and \(\alpha\) stays invariant." This episode is the very justification for last time's footnote.
The speed of light, lengths, energies — all "ledger numbers" that shift with how you measure. But \(\alpha=v/c\approx 1/137\), as their ratio, is the same for anyone, anytime — a number of the universe itself, carved inside the atom. The star of the footnote from Episode 1, "even if light slows, only \(\alpha\) stays put," we found this time inside the atom.
And the greatest mystery remains: why exactly one-137th? Nobody yet knows. Maybe you'll be the one to solve it — physics leaves quite a few open doors like this, on purpose.
Print / save as PDF: ⌘+P (Ctrl+P on Windows). Printing hides the answers, turning it into a problem set. On screen, tap "Show answer" to open them.