Physical constants
Fundamental constants, length scales, and precision tests
Fundamental constants
| Quantity | Symbol | Value |
|---|---|---|
| Fine structure constant | $\alpha$ | $1/137.035\,999\,177(21)$ |
| Electron mass | $m_e$ | $0.510\,998\,950\,69(16)$ MeV/$c^2$ |
| Electron charge | $e$ | $1.602\,176\,634 \times 10^{-19}$ C (exact) |
| Speed of light | $c$ | $299\,792\,458$ m/s (exact) |
| Planck's constant | $\hbar$ | $1.054\,571\,817 \times 10^{-34}$ J s (exact) |
| Bohr radius | $a_0 = 1/(m_e\alpha)$ | $0.529\,177 \times 10^{-10}$ m |
| Classical electron radius | $r_e = \alpha/m_e$ | $2.818 \times 10^{-15}$ m |
| Reduced Compton wavelength | $\lambdabar_C = 1/m_e$ | $3.862 \times 10^{-13}$ m |
| Thomson cross section | $\sigma_T = 8\pi\alpha^2/(3m_e^2)$ | $0.665 \times 10^{-24}$ cm$^2$ |
| Schwinger critical field | $E_c = m_e^2/e$ | $1.32 \times 10^{18}$ V/m |
Length scales
Three length scales characterize electron-only QED, each associated with a power of $\alpha$:
| Scale | Expression | Value | Physics |
|---|---|---|---|
| Bohr radius | $a_0 = 1/(m\alpha)$ | $0.53$ Å | Size of hydrogen, atomic physics |
| Reduced Compton wavelength | $\lambdabar_C = 1/m$ | $3.9 \times 10^{-3}$ Å | Pair creation threshold, QFT effects |
| Classical electron radius | $r_e = \alpha/m$ | $2.8 \times 10^{-5}$ Å | Thomson scattering, classical EM |
They are related by $a_0 : \lambda_C : r_e = 1/\alpha : 1 : \alpha = 137 : 1 : 1/137$. The Bohr radius is the largest (atomic scale). The Compton wavelength is where quantum field effects become important. The classical electron radius is the smallest and is where the classical theory of point charges becomes inconsistent.
QED precision tests
| Observable | Theory | Experiment | Precision |
|---|---|---|---|
| Electron $g-2$ | $0.001\,159\,652\,181\,643(764)$ | $0.001\,159\,652\,180\,73(28)$ | $\sim 10^{-12}$ |
| Hydrogen Lamb shift ($2s-2p$) | $1\,057.845(9)$ MHz | $1\,057.845(3)$ MHz | $\sim 10^{-6}$ |
| Hydrogen $1s$ hyperfine | $1\,420.405\,751\,768(1)$ MHz | $1\,420.405\,751\,768(1)$ MHz | $\sim 10^{-12}$ |