Skip to main content

Mathematics for Physics

96 Bohr Radius

A physical constant that is used to set the length scale of the hydrogen atom, where a length scale is the typical size that appears in the ground-state wavefunction.

The definition of the Bohr radius. The Bohr radius is the most probable distance of the electron from the proton in the hydrogen ground state. This principle is used to quote the size of the hydrogen atom.

The Bohr radius is

\[ a_{0} = \dfrac{4\pi\epsilon_{0}\hbar^{2}}{m_{e}e^{2}} \]

where

  • \(a_{0}\) is the Bohr radius.
  • \(\epsilon_{0}\) is the permittivity of free space.
  • \(\hbar\) is the reduced Planck constant.
  • \(m_{e}\) is the electron mass.
  • \(e\) is the elementary charge.

Its numerical size. The numerical value is about \(0.529\,\mathrm{\AA}\). This principle is used to convert atomic calculations into laboratory lengths.

Its role as the hydrogen length unit. The hydrogen radial wavefunctions are written in units of \(a_{0}\). This principle is used to scale all hydrogen lengths by one constant.

The ground-state radial scale is

\[ \psi_{100}(r) \propto e^{-r/a_{0}} \]

where

  • \(\psi_{100}\) is the ground-state wavefunction.
  • \(r\) is the electron-proton distance.
  • \(a_{0}\) is the Bohr radius.

96.1 References

  1. Griffiths, D. J. Introduction to Quantum Mechanics. Cambridge University Press, 2018. §4.2 — Bohr radius.
  2. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — source for the heading explanation.