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Atomic units (au) form a system of units convenient 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 four fundamental physical constants are all unity by definition:

Not to be confused with Astronomical Units (also abbreviated to 'au').

[edit] Fundamental units

Fundamental Atomic Units
Quantity	Name	Symbol	SI value	Planck unit scale
mass	electron rest mass	$\!m_e$	9.1093826(16)×10⁻³¹ kg	10⁻⁸ kg
length	Bohr radius	$a_0 = \hbar / (m_e c \alpha)$	5.291772108(18)×10⁻¹¹ m	10⁻³⁵ m
charge	elementary charge	$\!e$	1.60217653(14)×10⁻¹⁹ C	10⁻¹⁸ C
angular momentum	Reduced Planck's constant	$\hbar = h/(2 \pi)$	1.05457168(18)×10⁻³⁴ J·s	(same)
energy	Hartree energy	$\!E_h = m_\mathrm{e} c^2\alpha^2$	4.35974417(75)×10⁻¹⁸ J	10⁹ J

These quantities are not all independent; to normalize the first five quantities to 1, it suffices to normalize any four of them to 1.

[edit] Some derived units

Derived Atomic Units
Quantity	Expression	SI value	Planck unit scale
time	$\hbar / E_\mathrm{h}$	2.418884326505(16)×10⁻¹⁷ s	10⁻⁴³ s
velocity	$a_0 E_\mathrm{h} / \hbar$	2.1876912633(73)×10⁶ m·s⁻¹	10⁸ m·s⁻¹
force	$\! E_\mathrm{h} / a_0$	8.2387225(14)×10⁻⁸ N	10⁴⁴ N
temperature	$\! E_\mathrm{h} / k_\mathrm{B}$	3.1577464(55)×10⁵ K	10³² K
pressure	$E_\mathrm{h} / {a_0}^3$	2.9421912(19)×10¹³ N·m⁻²	10¹¹⁴ Pa
electric field	$E h / e$	5.142×10¹¹ V·m⁻¹	10⁶² V·m⁻¹

[edit] Bohr model simplified

Atomic units are chosen to reflect the properties of electrons in atoms. This is particularly clear from the classical Bohr model of the hydrogen atom in its ground state. The ground state electron orbiting the hydrogen nucleus has:

Orbital velocity = 1
Orbital radius = 1
Angular momentum = 1
Orbital period = 2π
Ionization energy = ¹⁄₂
Electric field (due to nucleus) = 1
Electrical attractive force (due to nucleus) = 1

[edit] Comparison with Planck units

Both Planck units and au are derived from certain fundamental properties of the physical world, and are free of anthropocentric considerations. To facilitate comparing the two systems of units, the above tables show the order of magnitude, in SI units, of the Planck 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 for quantum gravity and early-universe cosmology.

Both au and Planck units normalize the reduced Planck constant. Beyond this, Planck units normalize to 1 the two fundamental constants of general relativity and cosmology: the gravitational constant G and the speed of light in 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 a₀, the Bohr radius of the hydrogen atom. Normalizing a₀ to 1 amounts to normalizing the Rydberg constant, R_∞, to 4π/α = 4πc. Given au, the Bohr magneton is μ_B = ¹⁄₂. The corresponding Planck value is e/2m_e. Finally, au normalize the Hartree energy to 1, while Planck units normalize to 1 Boltzmann's constant k_B, which relates energy and temperature.

[edit] Quantum mechanics and electrodynamics simplified

The (non-relativistic) Schrödinger equation for an electron in SI units is

$- \frac{\hbar^2}{2m_e} \nabla^2 \psi(\mathbf{r}, t) + V(\mathbf{r}) \psi(\mathbf{r}, t) = i \hbar \frac{\partial \psi}{\partial t} (\mathbf{r}, t)$ .

The same equation in au is

$- \frac{1}{2} \nabla^2 \psi(\mathbf{r}, t) + V(\mathbf{r}) \psi(\mathbf{r}, t) = i \frac{\partial \psi}{\partial t} (\mathbf{r}, t)$ .

For the special case of the electron around a hydrogen atom, the Hamiltonian in SI units is:

$\hat H = - {{{\hbar^2} \over {2 m_e}}\nabla^2} - {1 \over {4 \pi \epsilon_0}}{{e^2} \over {r}}$ ,

while atomic units transform the preceding equation into

$\hat H = - {{{1} \over {2}}\nabla^2} - {{1} \over {r}}$ .

Finally, Maxwell's equations take the following elegant form in au:

$\nabla \cdot \mathbf{E} = 4\pi\rho$

$\nabla \cdot \mathbf{B} = 0$

$\nabla \times \mathbf{E} = -\alpha\frac{\partial \mathbf{B}} {\partial t}$

$\nabla \times \mathbf{B} = \alpha \left( \frac{\partial \mathbf{E}} {\partial t} + 4\pi \mathbf{J} \right)$

Extending the atomic units to electromagnetism, there is some ambiguity in defining the atomic unit of magnetic field, electric current, etc. The above Maxwell equations use the cgs-Gaussian unit convention, in which a plane wave has electric and magnetic fields of equal magnitude.

[edit] See also

[edit] References

H. Shull, G.G. Hall (1959). "Atomic Units". Nature 184 (4698): 1559. doi:10.1038/1841559a0.
G.W.F. Drake (ed.) (2006). Springer Handbook of Atomic, Molecular, and Optical Physics (2nd ed.). Springer. ISBN 978-0-387-20802-2.

[edit] External links

CODATA Internationally recommended values of the Fundamental Physical Constants.

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