Quantum constants

19 values, each with its units, its uncertainty, and where it came from.

Compton wavelength of the electron measured

\lambda_{\mathrm{C}} = 2.42631023538 \times 10^{-12}\ \text{m}

mλ_C = h/(m_e c) = 2.426 310 235 × 10⁻¹² m — the wavelength shift of a photon scattered through 90° by a free electron.

G_{0} = 0.00007748091729863649\ \text{S}

SThe conductance 2e²/h of a single ballistic quantum channel, about 77.5 μS — the step size in nanoscale wires and atomic point contacts.

g_{\mathrm{e}} = -2.00231930436092

dimensionlessThe electron's magnetic moment in Bohr magnetons, −2.002 319 304 360 92 — the most precisely tested prediction in all of science.

\alpha = 0.0072973525643

dimensionlessThe dimensionless strength of the electromagnetic interaction, 0.007 297 352 564 — roughly 1/137, and pure number with no units at all.

E_{\mathrm{h}} = 4.359744722206 \times 10^{-18}\ \text{J}

JThe atomic unit of energy, 4.359 744 722 × 10⁻¹⁸ J or 27.211 386 eV — twice the Rydberg and the currency of quantum chemistry.

G_{0}^{-1} = 12,906.40372965225\ \Omega

ΩHalf the von Klitzing constant, h/2e² ≈ 12.906 kΩ — the resistance of one perfect ballistic channel and the floor for any nanoscale wire.

\alpha^{-1} = 137.035999177

dimensionlessThe reciprocal of the fine-structure constant, 137.035 999 177 — famously near 137, and definitively not equal to it.

K_{\mathrm{J}} = 4.835978484169836 \times 10^{14}\ \text{Hz/V}

Hz/VThe frequency-to-voltage ratio 2e/h of a Josephson junction — 483.6 THz per volt, and the modern practical realisation of the volt.

\Phi_{0} = 2.067833848461929 \times 10^{-15}\ \text{Wb}

WbThe smallest unit of magnetic flux that can thread a superconducting loop, h/2e — exact since the 2019 SI redefinition fixed h and e.

h = 6.62607015 \times 10^{-34}\ \text{J}{\cdot}\text{s}

J·sThe quantum of action, exactly 6.626 070 15 × 10⁻³⁴ J·s — the constant that has defined the kilogram since 2019.

E_{\mathrm{P}} = 1,956,081,637\ \text{J}

Jm_P c² = 1.956 × 10⁹ J, or 1.221 × 10¹⁹ GeV — the energy scale of quantum gravity, and about the kinetic energy of a car on the motorway.

\ell_{\mathrm{P}} = 1.616255 \times 10^{-35}\ \text{m}

m√(ħG/c³) = 1.616 255 × 10⁻³⁵ m — the length scale where quantum mechanics and gravity must both apply, and neither alone works.

m_{\mathrm{P}} = 2.176434 \times 10^{-8}\ \text{kg}

kg√(ħc/G) = 2.176 434 × 10⁻⁸ kg — about 22 micrograms, the only Planck unit on a human scale, and the mass where gravity meets quantum.

T_{\mathrm{P}} = 1.416784 \times 10^{32}\ \text{K}

K√(ħc⁵/G)/k = 1.416 784 × 10³² K — the temperature at which thermal photons carry the Planck energy and gravity becomes quantum.

t_{\mathrm{P}} = 5.391247 \times 10^{-44}\ \text{s}

s√(ħG/c⁵) = 5.391 247 × 10⁻⁴⁴ s — the time light takes to cross a Planck length, and the earliest instant physics can describe.

\frac{h}{2m_{\mathrm{e}}} = 0.00036369475467\ \text{m}^{2}\text{/s}

m²/sh/(2m_e) = 3.636 947 547 × 10⁻⁴ m²/s — the ratio of Planck's constant to mass that atom interferometers measure directly.

\bar{\lambda}_{\mathrm{C}} = 3.8615926744 \times 10^{-13}\ \text{m}

mħ/(m_e c) = 3.861 592 674 × 10⁻¹³ m — the Compton wavelength divided by 2π, and the natural length scale of the Dirac equation.

\hbar = 1.054571817 \times 10^{-34}\ \text{J}{\cdot}\text{s}

J·sPlanck's constant divided by 2π, 1.054 571 817 × 10⁻³⁴ J·s — the natural quantum of angular momentum and spin.

R_{\mathrm{K}} = 25,812.8074593045\ \Omega

ΩThe quantum Hall resistance h/e² ≈ 25.813 kΩ, exact since 2019 and the reference by which the ohm is now realised worldwide.