# atomic units

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 six physical constants are all unity by definition:

## Fundamental units

Fundamental Atomic Units
QuantityNameSymbolSI valuePlanck unit scale
lengthBohr radiusa05.291 772 108(18)Ã—10-11 m10-35 m
masselectron rest massme9.109 3826(16)Ã—10-31 kg10-8 kg
chargeelementary chargee1.602 176 53(14)Ã—10-19 C10-18 C
angular momentumPlanck's constant1.054 571 68(18)Ã—10-34 J s(same)
energyHartree energyEh4.359 744 17(75)Ã—10-18 J109 J
electrostatic force constantCoulomb's constant1/(4πε0)8.9875516Ã—109 C-2 N m2(same)

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 and Coulomb's constant, for example, are only an incidental consequence of normalizing the other four quantities.

## Some derived units

Derived Atomic Units
QuantityExpressionSI valuePlanck unit scale
time2.418 884 326 505(16)Ã—10-17 s10-43 s
velocity2.187 691 2633(73)Ã—106 m s-1108 m s-1
force8.238 7225(14)Ã—10-8 N1044 N
current6.623 617 82(57)Ã—10-3 A1026 A
temperature3.157 7464(55)Ã—105 K1032 K
pressure2.942 1912(19)Ã—1013 N m-210114 Pa

## 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 Dirac constant and the Coulomb force constant to 1. 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 μB=1/2. The corresponding Planck value is e/2me. 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

The (non-relativistic) SchrÃ¶dinger equation for 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 equations take 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.)

Planck units

## References

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

common units, but have now been mostly replaced by the metric system in commercial, scientific, and industrial applications.

Contrarily, however, U.S. customary units are still the main system of measurement in the United States.
Atomic physics (or atom physics) is the field of physics that studies atoms as isolated systems comprised of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and the processes by which these arrangements change.
Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles.
Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. QED was developed by a number of physicists, beginning in the late 1920s.[1]
Electron

Theoretical estimates of the electron density for the first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density
Composition: Elementary particle
Family: Fermion
Group: Lepton
Generation: First
In physics, a physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. It can be contrasted with a mathematical constant, which is a fixed numerical value but does not directly involve any physical measurement.
Electron

Theoretical estimates of the electron density for the first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density
Composition: Elementary particle
Family: Fermion
Group: Lepton
Generation: First
invariant mass or intrinsic mass or proper mass or just mass is a measurement or calculation of the mass of an object that is the same for all frames of reference.
The elementary charge (symbol e or sometimes q) is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron.
hydrogen atom is an atom of the chemical element hydrogen. It is composed of a single negatively-charged electron circling a single positively-charged nucleus of the hydrogen atom.
SI units
010−12 m 010−3 nm
Natural units
01024 lP 0103 le
US customary / Imperial units
010−12 ft 010
In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3.
The electric potential energy of a system is the potential energy associated with the conservative Coulomb forces between charged particles in a system, where the reference potential energy is usually chosen to be zero for particles at infinite separation.
stationary state is an eigenstate of a Hamiltonian, or in other words, a state of definite energy. It is called stationary because the corresponding probability density has no time dependence.
In physics, a physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. It can be contrasted with a mathematical constant, which is a fixed numerical value but does not directly involve any physical measurement.
Planck constant (denoted ) is a physical constant that is used to describe the sizes of quanta. It plays a central role in the theory of quantum mechanics, and is named after Max Planck, one of the founders of quantum theory.
Coulomb's law, developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated as follows:

The magnitude of the electrostatic force between two points electric charges is directly proportional to the product of the magnitudes of each

Si, si, or SI may refer to (all SI unless otherwise stated):

In language:
• One of two Italian words:
• sÃ¬ (accented) for "yes"
• si

In physics, Planck units are physical units of measurement defined exclusively in terms of the five universal physical constants shown in the table below in such a manner that all of these physical constants take on the numerical value of one when expressed in terms of these units.
Length is the long dimension of any object. The length of a thing is the distance between its ends, its linear extent as measured from end to end. This may be distinguished from height, which is vertical extent, and width or breadth
SI units
010−12 m 010−3 nm
Natural units
01024 lP 0103 le
US customary / Imperial units
010−12 ft 010
Mass is a fundamental concept in physics, roughly corresponding to the intuitive idea of "how much matter there is in an object". Mass is a central concept of classical mechanics and related subjects, and there are several definitions of mass within the framework of relativistic
Electron

Theoretical estimates of the electron density for the first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density
Composition: Elementary particle
Family: Fermion
Group: Lepton
Generation: First
invariant mass or intrinsic mass or proper mass or just mass is a measurement or calculation of the mass of an object that is the same for all frames of reference.
Flavour in particle physics

The elementary charge (symbol e or sometimes q) is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron.
angular momentum of an object rotating about some reference point is the measure of the extent to which the object will continue to rotate about that point unless acted upon by an external torque.
Planck constant (denoted ) is a physical constant that is used to describe the sizes of quanta. It plays a central role in the theory of quantum mechanics, and is named after Max Planck, one of the founders of quantum theory.
energy (from the Greek ενεργός, energos, "active, working")[1] is a scalar physical quantity that is a property of objects and systems of objects which is conserved by nature.
A Hartree (symbol Eh) is the atomic unit of energy and is named after physicist Douglas Hartree.

The Hartree energy is equal to the absolute value of the electric potential energy of the hydrogen atom in its ground state.