ATM-04

Fine-structure constant

α = e² / (4π ε₀ ħ c) ≈ 1/137. The dimensionless electromagnetic coupling.

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StructureFine structure

Governing equation

\alpha=e^2/(4\pi\varepsilon_0\hbar c)

$s \in \left[0.5, 2\right]$

where

s: Scale vs SI (—)
\alpha: α (—)
1/\alpha: 1/α (—)

The figure, part by part

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①

Governing law — red

\alpha=e^2/(4\pi\varepsilon_0\hbar c)

$s \in \left[0.5, 2\right]$

$s = 0.5$ $\alpha = 274.1$

Along the curve

②

Domain of the variable — blue

s_{\min} = 0.5,\quad s_{\max} = 2

$s \in \left[0.5, 2\right]$

$s = 0.5$ $s = 0.5$

Along the curve

③

Response curve — green

\alpha=e^2/(4\pi\varepsilon_0\hbar c)

$s \in \left[0.5, 2\right]$

$s = 0.5$ $\alpha = 274.1$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\alpha}{\mathrm{d}s}

$s \in \left[0.5, 2\right]$

$s = 0.509$ $\alpha^{\prime} = -528.2$

Along the curve

Lecture brief

Historical brief

Balmer, Rydberg, Bohr (1913), fine structure, Zeeman and Stern–Gerlach built the spectrum of one atom. The lab computes levels, selection and magnetic splitting. This sheet (ATM-04 — Fine-structure constant) is the form associated with Fine structure. Working symbols: $s$ $\rightarrow$ $\alpha$ , $1/\alpha$ . Fine-structure splits ~ α². This sheet just evaluates α from the SI constants you set.

Purpose

Purpose: compute

\alpha

1/\alpha

from

s

in Atomic physics via

\alpha=e^2/(4\pi\varepsilon_0\hbar c)

α = e² / (4π ε₀ ħ c) ≈ 1/137. The dimensionless electromagnetic coupling. Use it when a real atomic physics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

s = 1.000\,\mathrm{—}

, the governing relation

\alpha=e^2/(4\pi\varepsilon_0\hbar c)

yields

\alpha = 0.007297\,\mathrm{—}

1/\alpha = 137.0360\,\mathrm{—}

. Four constants combining into a pure number. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Scale vs SI s1 —

Outputs

α \alpha0.007297 —
1/α 1/\alpha137.0360 —

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Narration of this film

Four constants combining into a pure number.

Fine-structure splits ~ α². This sheet just evaluates α from the SI constants you set.

Reading speed

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