Bohr radius

In Bohr's 1913 model a₀ is the radius of the first allowed orbit; in real quantum mechanics the electron has no orbit, but a₀ survives as the decay length of the 1s wavefunction and the radius at which the radial probability density peaks. Either way it sets the scale of the atomic world: about 0.53 ångström, so atoms are roughly 1 Å across and a solid packs about 10²⁹ of them per cubic metre.

Written as a₀ = ħ/(m_e c α), it is α times smaller than the reduced Compton wavelength and α⁻² times bigger than the classical electron radius: the same three lengths, separated twice over by 137. Two cautions: a₀ is defined for infinite nuclear mass, and it scales as 1/Z for hydrogen-like ions, so a ground-state electron in U⁹¹⁺ sits 92 times closer in and moves at two-thirds the speed of light, which is why heavy-element chemistry needs relativistic corrections.