Fine-Structure Constant

Equivalent forms

\alpha=\frac{v_1}{c}\;(\text{Bohr ground state})

A pure number near 1/137 that combines e, ℏ, c, and ε₀ — no units, no scale, just the bare strength of the electromagnetic force.

Symbol	Name	SI unit	Dimension	Range
$\alpha$	Fine-structure constantoutput Dimensionless electromagnetic coupling	1	1	—
$e$	Elementary charge Electron charge magnitude	C	I*T	—
$\hbar$	Reduced Planck constant h/2π	J*s	ML^2T^-1	—
$c$	Speed of light Speed of light in vacuum	m/s	L*T^-1	—
$\varepsilon_0$	Vacuum permittivity Electric constant	F/m	M^-1L^-3T^4*I^2	—

Where it holds

\alpha is

the low-energy value; in quantum field theory the effective coupling 'runs', growing

to \sim 1/128

at the electroweak scale

(\sim 90\,\mathrm{GeV})

. Truly constant only as a fixed low-energy reference.

Dimensional analysis

\alpha

→ dimensionless

e^{2}/(4\pi \varepsilon _{0}\hbar c) \to [I^{2}T^{2}]/([M^-1\,\mathrm{L}^-3\,\mathrm{T}^4\,\mathrm{I}^2][M L^2\,\mathrm{T}^-1][L T^-1])

All base dimensions cancel, leaving a pure number.

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Discovery

Arnold Sommerfeld · 1916

Sommerfeld introduced α while extending Bohr's model with relativity to explain the fine splitting of hydrogen's spectral lines. The ratio of the electron's orbital speed to c came out as this constant. Its near-1/137 value has fascinated physicists ever since; Eddington famously and wrongly argued it must be exactly 1/136, then 1/137.

Research status: stable

Real-world applications

Precision tests of $QED (g-2)$
Defining atomic energy and length scales
Searches for time-variation of fundamental constants (quasar spectra)

Common misconceptions

$\alpha is$ not exactly 1/137 — it is 1/137.035999...
It is not truly constant with energy; it runs in QFT
The number is dimensionless, so it cannot depend on your choice of units (unlike $e or \hbar$ individually)

Experimental verification

Measured to 11 significant figures via the electron's anomalous magnetic moment

g-2

(Gabrielse) and atom-recoil experiments; the agreement between independent methods is one of physics' most precise tests.

Derivation

Compare the electron's Coulomb potential energy at the Bohr radius to its rest-related quantum scale.

In Bohr's model the ground-state speed is

v_{1} = e^{2}/(4\pi \varepsilon _{0}\hbar )

; dividing by c gives the dimensionless

\alpha = e^{2}/(4\pi \varepsilon _{0}\hbar c)

.

Equivalently

\alpha

= (Coulomb energy between two electrons one reduced Compton wavelength apart

)/(m_e c^{2})

.

Every combination of units cancels, leaving a pure number.

What if…

What

if \alpha

were much larger?

Electromagnetic forces would dominate; chemistry and stable atoms as we know them, plus stellar fusion balances, would change drastically — part of the anthropic fine-tuning discussion.

What if you probe at high energy?

Vacuum polarization screens less, so the effective $\alpha$ grows — about 1/128 at 90 GeV, measured at LEP.

1

Bohr-orbit speed

Given ·

α:: 1/137.036

Find · electron speed in hydrogen's ground state

Steps

$v_{1} = \alpha c$
$= 2.998e_{8} / 137.036 \approx 2.19e_{6} m/s (0.7$ % of light speed)

Result ·

v_{1} = \alpha c = (1/137.036)\times 2.998e_{8} \approx 2.19e_{6} m/s

bohr modelRydberg Formula→coulomb law