Bohr radius

Bohr's theory of the hydrogen atom (1913) predicts the existence of a smallest orbit for the electron circulating the hydrogen nucleus. Today the radius of this orbit is called the Bohr radius. It is usually indicated by a₀.

In the old quantum theory of Bohr and Arnold Sommerfeld, as well as in the new quantum theory of Werner Heisenberg and Erwin Schrödinger the radius is given by

$a_{0}={\frac {4\pi \epsilon _{0}\hbar ^{2}}{\mu e^{2}}}\approx 0.529\,177\,208\,59\cdot 10^{-10}\;\mathrm {m}$

where ε₀ is the vacuum permittivity (electric constant), $\hbar$ is Planck's reduced constant, μ is the reduced mass of the hydrogen atom (is equal to the electron mass when the proton mass may supposed to be infinite; for the numerical value given this assumption is made) and e is the charge of the electron.

In quantum mechanics, a₀ appears as the maximum in the radial distribution associated with the electronic wave function Ψ_1s(r) of lowest energy of the hydrogen atom, the 1s atomic orbital. That is, a₀ is the position of the maximum in the radial distribution 4πr ² |Ψ_1s(r) |² and in that sense a₀ is a measure for the "size" of the hydrogen atom.