Atomic units

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Atomic units

Atomic units (au) form a system of units convenient for atomic physics

Atomic physics

Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nuclei. It is primarily concerned with the Electron configuration and...
, electromagnetism

Electromagnetism

Electromagnetism is the physics of the electromagnetic field, a field which exerts a force on Elementary particles with the property of electric charge and which is reciprocally affected by the presence and motion of such particles....
, and quantum electrodynamics

Quantum electrodynamics

Quantum electrodynamics is a relativity theory quantum field theory of electrodynamics. QED was developed by a number of physicists, beginning in the late 1920s....
, especially when the focus is on the properties of electron

Electron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
s. There are two different kinds of atomic units, which one might name Hartree atomic units and Rydberg
Rydberg

Rydberg can refer to:* Rydberg , a crater on the moon*In physics,** Rydberg constant**Rydberg formula**Rydberg atom **a unit of energy, derived from the Rydberg constant, equal to half the Hartree...
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

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

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Encyclopedia

Atomic units (au) form a system of units convenient for atomic physics

Atomic physics

Electromagnetism

Quantum electrodynamics

Electron

Two properties of the electron, its mass and charge
Elementary charge

The elementary charge, usually denoted e, is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron....
;
Two properties of the hydrogen atom
Hydrogen atom

A hydrogen atom is an atom of the chemical element hydrogen. The Electric charge neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force....
, its Bohr radius
Bohr radius

In the Bohr model of the structure of an atom, put forward by Niels Bohr in 1913, electrons orbit a central atomic nucleus. The model says that the electrons orbit only at certain distances from the nucleus, depending on their energy....
and the absolute value
Absolute value

In mathematics, the absolute value of a real number is its numerical value without regard to its Negative and non-negative numbers. So, for example, 3 is the absolute value of both 3 and -3....
of its electric potential energy
Electric potential energy

Electric energy is the potential energy associated with the conservative force Coulomb forces between charged particles contained within a physical system, where the reference potential energy is usually chosen to be zero for particles at infinite separation....
in the ground state;
Two constants
Physical constant

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's constant and that for Coulomb's Law
Coulomb's law

Coulomb's law, sometimes called the Coulomb law, is an equation describing the electrostatic force between electric charges. It was developed in the 1780s by French physicist Charles Augustin de Coulomb and was essential to the development of the classical electromagnetism....
.

Fundamental units

Fundamental Atomic Units
Quantity	Name	Symbol	SI Si Si, si, or SI may refer to :... value	Planck unit scale
mass Mass In physical science, mass refers to the degree of acceleration a body acquires when subject to a force: bodies with greater mass are accelerated less by the same force....	electron Electron The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle.... rest mass		9.109 3826(16)×10^-31 kg	10^-8 kg
length Length 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....	Bohr radius Bohr radius In the Bohr model of the structure of an atom, put forward by Niels Bohr in 1913, electrons orbit a central atomic nucleus. The model says that the electrons orbit only at certain distances from the nucleus, depending on their energy....		5.291 772 108(18)×10^-11 m	10^-35 m
charge Electric charge Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields....	elementary charge Elementary charge The elementary charge, usually denoted e, is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron....		1.602 176 53(14)×10^-19 C	10^-18 C
angular momentum Angular momentum In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....	Reduced Planck's constant		1.054 571 68(18)×10^-34 J s	(same)
energy Energy In physics, energy is a scalar physical quantity that describes the amount of Work_ that can be performed by a force. Energy is an attribute of objects and systems that is subject to a conservation law....	Hartree energy Hartree energy A hartree is the atomic units 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....		4.359 744 17(75)×10^-18 J	10⁹ J
electrostatic force constant	Coulomb's constant		8.9875516×10⁹ 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

Hartree energy

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

Some derived units

Derived Atomic Units
Quantity	Expression	SI Si Si, si, or SI may refer to :... value	Planck unit scale
time Time Time is a component of the measurement used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects....		2.418 884 326 505(16)×10^-17 s	10^-43 s
velocity Velocity In physics, velocity is defined as the Derivative of Position vector. It is a vector physical quantity; both speed and direction are required to define it....		2.187 691 2633(73)×10⁶ m s^-1	10⁸ m s^-1
force Force In physics, a force is that which can cause an object with mass to change its velocity. Force has both Euclidean_vector#Length of a vector and Direction , making it a Vector quantity....		8.238 7225(14)×10^-8 N	10⁴⁴ N
current Current Current may refer to:* Current affairs* Electric current* Current ** Ocean current* Current , geometrical current in differential topology...		6.623 617 82(57)×10^-3 A	10²⁶ A
temperature Temperature In physics, temperature is a physical property of a Physical system that underlies the common notions of hot and cold; something that feels hotter generally has the greater temperature....		3.157 7464(55)×10⁵ K	10³² K
pressure Pressure Pressure is the force per unit area applied to an object in a direction surface normal to the surface. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure....		2.942 1912(19)×10¹³ N m^-2	10¹¹⁴ Pa

Comparison with Planck units

Both Planck units

Planck units

Planck units are units of measurement named after the German physicist Max Planck, who first proposed them in 1899. They are an example of natural units, i.e....
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

Order of magnitude

An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed Geometric progression to the class preceding it....
, in SI

Si, si, or SI may refer to :...
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

Quantum gravity

Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the Fundamental interaction , with general relativity, the theory of the fourth fundamental force: Gravitation....
and early-Universe cosmology

Physical cosmology

Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of our universe and is concerned with fundamental questions about its formation and evolution....
.

Both au and Planck units normalize the Reduced Planck constant and the Coulomb force constant to 1. Beyond this, Planck units normalize to 1 the two fundamental constants of general relativity

General relativity

General relativity or the general theory of relativity is the Geometry Theoretical physics of gravitation published by Albert Einstein in 1916....
and cosmology: the gravitational constant

Gravitational constant

The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitation between objects with mass....
G and the speed of light

Speed of light

The speed of light in an free space is an important physical constant usually written as c, with a value of 299,792,458 metres per second....
in a vacuum, c. Letting a denote the fine structure constant, the au value of c is a^-1 ˜ 137.036.

Atomic units, by contrast, normalize to 1 the mass and charge of the electron, and a₀, the Bohr radius

Bohr radius

In the Bohr model of the structure of an atom, put forward by Niels Bohr in 1913, electrons orbit a central atomic nucleus. The model says that the electrons orbit only at certain distances from the nucleus, depending on their energy....
of the hydrogen atom

Hydrogen atom

A hydrogen atom is an atom of the chemical element hydrogen. The Electric charge neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force....
. Normalizing a₀ to 1 amounts to normalizing the Rydberg constant

Rydberg constant

The Rydberg Physical constant, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to atomic spectrum in the science of spectroscopy....
, R₈, to 4p/a = 4pc. Given au, the Bohr magneton

Bohr magneton

In atomic physics, the Bohr magneton is a physical constant of magnetic moment of electrons. It was discovered in 1913 by Romanian physicist Stefan Procopiu and rediscovered independently two years later by Denmark physicist Niels Bohr....
µ_B=1/2. The corresponding Planck value is e/2m_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

The (non-relativistic) Schrödinger equation

Schrödinger equation

In physics, especially quantum mechanics, the Schr?dinger equation is an equation that describes how the quantum state of a physical system changes in time....
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

Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H is the observable corresponding to the total energy of the system. As with all observables, the Spectrum of the Hamiltonian is the set of possible outcomes when one measures the total energy of a system....
in SI units is: , while atomic units transform the preceding equation into . Finally, Maxwell's equations

Maxwell's equations

In electromagnetism, James Clerk Maxwell equations are a set of four partial differential equations that describe the properties of the electric field and magnetic field fields and relate them to their sources, charge density and current density....
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 a is absorbed into B.)

Atomic units

Fundamental units

Some derived units

Comparison with Planck units

Quantum mechanics and electrodynamics simplified

See also

External links