Bohr radius

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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 4πr² |Ψ_1s(r) |² and in that sense a₀ is a measure for the "size" of the hydrogen atom.