- Atomic units
Atomic units (au) form a
system of unitsconvenient for atomic physics, electromagnetism, and quantum electrodynamics, especially when the focus is on the properties of electrons. There are two different kinds of atomic units, which one might name Hartree atomic units and Rydberg atomic units, which differ in the choice of the unit of mass and charge. This article deals with Hartree atomic units. In au, the numerical values of the following six physical constantsare all unity by definition:
*Two properties of the electron, its mass and charge;
*Two properties of the
hydrogen atom, its Bohr radiusand the absolute valueof its electric potential energyin the ground state;
Planck's constantand that for Coulomb's Law.
These six quantities are not independent; to normalize all six quantities to 1, it suffices to normalize any four of them to 1. The normalizations of the
Hartree energyand Coulomb's constant, for example, are only an incidental consequence of normalizing the other four quantities.
ome derived units
Comparison with Planck units
Planck unitsand au are derived from certain fundamental properties of the physical world, and are free of anthropocentricconsiderations. To facilitate comparing the two systems of units, the above tables show the order of magnitude, in SIunits, of the Planck unitcorresponding to each atomic unit. Generally, when an atomic unit is "large" in SI terms, the corresponding Planck unit is "small", and vice versa. It should be kept in mind that au were designed for atomic-scale calculations in the present-day Universe, while Planck units are more suitable for quantum gravityand early-Universe cosmology.
Both au and Planck units normalize the
Reduced Planck constantand the Coulomb force constantto 1. Beyond this, Planck units normalize to 1 the two fundamental constants of general relativityand cosmology: the gravitational constant"G" and the speed of lightin a vacuum, "c". Letting α denote the fine structure constant, the au value of "c" is α-1 ≈ 137.036.
Atomic units, by contrast, normalize to 1 the mass and charge of the electron, and "a"0, the
Bohr radiusof the hydrogen atom. Normalizing "a"0 to 1 amounts to normalizing the Rydberg constant, "R"∞, to 4π/α = 4π"c". Given au, the Bohr magnetonμB=1/2. The corresponding Planck value is "e"/2"m"e. Finally, au normalize a unit of atomic energy to 1, while Planck units normalize to 1 Boltzmann's constant"k", which relates energy and temperature.
Quantum mechanics and electrodynamics simplified
Schrödinger equationfor an electron in SI units is:.The same equation in au is:.For the special case of the electron around a hydrogen atom, the Hamiltonian in SI units is::,while atomic units transform the preceding equation into:.Finally, Maxwell's equationstake the following elegant form in au:::::(There is actually some ambiguity in defining the atomic unit of magnetic field. The above Maxwell equations use the "Gaussian" convention, in which a plane wave has electric and magnetic fields of equal magnitude. In the "Lorentz force" convention, a factor of α is absorbed into B.)
*H. Shull and G. G. Hall, Atomic Units, Nature, volume 184, no. 4698, page 1559 (Nov. 14, 1959)
* [http://physics.nist.gov/cuu/Constants/index.html CODATA Internationally recommended values of the Fundamental Physical Constants.]
Wikimedia Foundation. 2010.