- Atomic units
**Atomic units**(**au**) form asystem of units convenient foratomic physics ,electromagnetism , andquantum electrodynamics , especially when the focus is on the properties ofelectron s. 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 sixphysical constants are all unity by definition:

*Two properties of the electron, its mass and charge;

*Two properties of thehydrogen atom , itsBohr radius and theabsolute value of itselectric potential energy in theground state ;

*Two constants,Planck's constant and that forCoulomb's Law .**Fundamental units**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 energy andCoulomb's constant , for example, are only an incidental consequence of normalizing the other four quantities.**ome derived units****Comparison with Planck units**Both

Planck units and**au**are derived from certain fundamental properties of the physical world, and are free ofanthropocentric considerations. To facilitate comparing the two systems of units, the above tables show theorder of magnitude , inSI units, of thePlanck unit corresponding 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 forquantum gravity and early-Universe cosmology.Both

**au**and Planck units normalize theReduced Planck constant and theCoulomb force constant to 1. Beyond this, Planck units normalize to 1 the two fundamental constants ofgeneral relativity and cosmology: thegravitational constant "G" and thespeed of light in a vacuum, "c". Letting α denote thefine 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}, theBohr radius of thehydrogen atom . Normalizing "a"_{0}to 1 amounts to normalizing theRydberg constant , "R"_{∞}, to 4π/α = 4π"c". Given**au**, theBohr magneton μ_{B}=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 1Boltzmann's constant "k", which relates energy and temperature.**Quantum mechanics and electrodynamics simplified**The (non-relativistic)

Schrödinger equation for an electron in SI units is:$-\; frac\{hbar^2\}\{2m\_e\}\; abla^2\; psi(mathbf\{r\},\; t)\; +\; V(mathbf\{r\})\; psi(mathbf\{r\},\; t)\; =\; i\; hbar\; frac\{partial\; psi\}\{partial\; t\}\; (mathbf\{r\},\; t)$.The same equation in**au**is:$-\; frac\{1\}\{2\}\; abla^2\; psi(mathbf\{r\},\; t)\; +\; V(mathbf\{r\})\; psi(mathbf\{r\},\; t)\; =\; i\; frac\{partial\; psi\}\{partial\; t\}\; (mathbf\{r\},\; t)$.For the special case of the electron around a hydrogen atom, the Hamiltonian in SI units is::$hat\; H\; =\; -${hbar^2} over {2 m_e abla^2} - {1 over {4 pi epsilon_0e^2} over {r,while**atomic units**transform the preceding equation into:$hat\; H\; =\; -${1} over {2 abla^2} - 1} over {r.Finally,Maxwell's equations take the following elegant form in**au**::$abla\; cdot\; mathbf\{E\}\; =\; 4pi\; ho$:$abla\; cdot\; mathbf\{B\}\; =\; 0$:$abla\; imes\; mathbf\{E\}\; =\; -alpha\; frac\{partial\; mathbf\{B\; \{partial\; t\}$:$abla\; imes\; mathbf\{B\}\; =\; alpha\; left(\; frac\{partial\; mathbf\{E\; \{partial\; t\}\; +\; 4pi\; mathbf\{J\}\; ight)$(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**.)**ee also***

Planck units

*Natural units **References***H. Shull and G. G. Hall, Atomic Units, Nature, volume 184, no. 4698, page 1559 (Nov. 14, 1959)

**External links*** [

*http://physics.nist.gov/cuu/Constants/index.html CODATA Internationally recommended values of the Fundamental Physical Constants.*]

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