| Bohr radius | |
|---|---|
| Name | Bohr radius |
| Quantity | length |
| Value | 5.29177210903×10^−11 m (exactly by definition of constants?) |
| Units | metre (m) |
Bohr radius
The Bohr radius is a physical constant that characterizes the typical scale of the hydrogen atom's ground-state electron orbital in early atomic models and in nonrelativistic quantum mechanics. It provides a natural length unit for describing atomic and molecular sizes, atomic units, and scaling relations in spectroscopy and scattering theory.
The Bohr radius, conventionally denoted a_0 or a_0, is the radius of the lowest-energy circular orbit in the Bohr model of the hydrogen atom and the characteristic length scale of the ground-state wavefunction of hydrogen in the Schrödinger solution. It sets the scale for many atomic phenomena, appearing in expressions for the Rydberg constant, atomic unit system, and the expectation values of radial operators. As a natural unit of length it links fundamental constants: the reduced Planck constant (ℏ), the elementary charge (e), the electron mass (m_e), and the vacuum permittivity (ε_0).
In the Bohr model introduced by Niels Bohr (1913), the electron orbits a proton under a classical Coulomb force with quantized angular momentum L = nℏ. Equating centripetal force and Coulomb attraction and imposing the angular momentum quantization for principal quantum number n = 1 yields the radius a_0 = 4πε_0ℏ^2/(m_e e^2). This derivation links the Bohr radius to early semiclassical ideas and to observed spectral lines explained by the Rydberg formula and Rydberg constant for hydrogen emission governed historically by Balmer and Rydberg.
In the nonrelativistic treatment using the time-independent Schrödinger equation for the hydrogenic Coulomb potential, the ground-state wavefunction ψ_100(r) has radial dependence proportional to exp(−r/a_0), making a_0 the exponential decay length and the expectation value ⟨r⟩ = 3a_0/2. The Bohr radius arises when expressing the Hamiltonian in atomic units where ℏ = m_e = e = 4πε_0 = 1, simplifying many-body electronic structure calculations employed in quantum chemistry and density functional theory. Corrections from relativistic quantum mechanics (via the Dirac equation) and quantum electrodynamics modify energies and effective radii by small factors involving the fine-structure constant α.
Numerically, the Bohr radius is approximately 5.29177210903×10^−11 metres. It is commonly used with SI units and in atomic units (a_0 = 1 a.u. of length). The value is obtained from well-established constants: ℏ, e, m_e, and ε_0, whose values are maintained by organizations such as the International Bureau of Weights and Measures (BIPM) and reflected in the CODATA recommended values. Experimental spectroscopy of hydrogen and precision tests of quantum electrodynamics contribute to refinement of related constants like the Rydberg constant and the electron mass.
The Bohr radius defines typical magnitudes for atomic radii, bond lengths, and the spatial extent of atomic orbitals used in molecular orbital theory and valence bond theory. For hydrogen-like ions with nuclear charge Z, characteristic radii scale as a_0/Z. In condensed matter physics, the effective Bohr radius (a*_0) incorporating an effective mass and dielectric constant describes donor-bound electrons in semiconductors such as silicon and gallium arsenide. In quantum scattering and low-energy collision theory, a_0 often appears in cross-section estimates and in the parameterization of van der Waals interactions and Rydberg states of atoms and molecules.
For a two-body Coulomb system the Bohr radius generalizes by replacing the electron mass m_e with the reduced mass μ of the electron–nucleus system, giving a_μ = 4πε_0ℏ^2/(μ e^2). This correction is essential for high-precision spectroscopy of isotopes (e.g., deuterium, tritium) and for exotic atoms such as muonic hydrogen, where the muon mass dramatically reduces the effective radius. The Bohr radius relates to the Rydberg constant R_∞ through R_∞ = α^2 m_e c/(4πℏ) and to energy scales via the Rydberg energy. These relations underpin precise determinations of fundamental constants and tests against quantum electrodynamics predictions.
The Bohr radius emerged from Bohr's 1913 model, which reconciled discrete spectral lines observed by Johann Balmer and formalized quantization principles that influenced the development of matrix mechanics and the Schrödinger wave mechanics. While the Bohr model was superseded by modern quantum mechanics, the Bohr radius remains a fundamental derived constant and pedagogical bridge linking early atomic theory to rigorous quantum treatments by scientists such as Erwin Schrödinger, Werner Heisenberg, and Paul Dirac. Its presence in atomic units and spectroscopy continues to shape research in atomic physics, quantum chemistry, precision measurement programs (e.g., at institutions like National Institute of Standards and Technology NIST), and technological applications spanning semiconductor design to spectroscopy of cold and ultracold gases.
Category:Atomic physics Category:Physical constants Category:Quantum mechanics