# 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 *a*_{0}.

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

where ε_{0} is the vacuum permittivity (electric constant),
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, *a*_{0} 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, *a*_{0} is the position of the maximum in the radial distribution 4π*r*^{ 2} |Ψ_{1s}(*r*) |^{2} and in that sense *a*_{0} is a measure for the "size" of the hydrogen atom.