The Hartree atomic units are a
system of
natural units
In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
of measurement which is especially convenient for
atomic physics and
computational chemistry calculations. They are named after the physicist
Douglas Hartree
Douglas Rayner Hartree (27 March 1897 – 12 February 1958) was an English mathematician and physicist most famous for the development of numerical analysis and its application to the Hartree–Fock equations of atomic physics and the ...
.
By definition, the following four fundamental
physical constants
A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant, ...
may each be expressed as the numeric value 1 multiplied by a
coherent unit
A coherent system of units is a system of units of measurement used to express physical quantities that are defined in such a way that the equations relating the numerical values expressed in the units of the system have exactly the same form, inc ...
of this system:
*
Reduced Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
:
, also known as the atomic unit of action
*
Elementary charge:
, also known as the atomic unit of charge
*
Bohr radius
The Bohr radius (''a''0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an ...
:
, also known as the atomic unit of length
*
Electron mass
The electron mass (symbol: ''m''e) is the mass of a stationary electron, also known as the invariant mass of the electron. It is one of the fundamental constants of physics. It has a value of about or about , which has an energy-equivalent of ...
:
, also known as the atomic unit of mass
Atomic units are often abbreviated "a.u." or "au", not to be confused with the same abbreviation used also for
astronomical unit
The astronomical unit (symbol: au, or or AU) is a unit of length, roughly the distance from Earth to the Sun and approximately equal to or 8.3 light-minutes. The actual distance from Earth to the Sun varies by about 3% as Earth orbits ...
s,
arbitrary unit
In science and technology, an arbitrary unit (abbreviated arb. unit, '' see below'')
or procedure defined unit (p.d.u.)
is a relative unit of measurement to show the ratio of amount of substance, intensity, or other quantities, to a predetermined ...
s, and
absorbance units in other contexts.
Defining constants
Each unit in this system can be expressed as a product of powers of four physical constants without a multiplying constant. This makes it a
coherent system of units
A coherent system of units is a system of units of measurement used to express physical quantities that are defined in such a way that the equations relating the numerical values expressed in the units of the system have exactly the same form, in ...
, as well as making the numerical values of the defining constants in atomic units equal to unity.
As of the
2019 redefinition of the SI base units
In 2019, four of the seven SI base units specified in the International System of Quantities were redefined in terms of natural physical constants, rather than human artifacts such as the standard kilogram.
Effective 20 May 2019, the 144th ...
, the elementary charge
and the Planck constant
(and consequently also the reduced Planck constant
) are defined as having an exact numerical values in
SI units.
Five symbols are commonly used as units in this system, only four of them being independent:
Units
Below are listed units that can be derived in the system. A few are given names, as indicated in the table.
Here,
*
is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
*
is the
vacuum permittivity
*
is the
Rydberg constant
In spectroscopy, the Rydberg constant, symbol R_\infty for
heavy atoms or R_\text for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to the electromagnetic spectra of an atom. The constant first aro ...
*
is the
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
*
is the
fine-structure constant
In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between el ...
*
is the
Bohr magneton
In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum.
The Bohr magneton, in SI units is defined as
\mu_\mat ...
* denotes ''correspondence'' between quantities since equality does not apply.
Use and notation
Atomic units, like
SI units, have a unit of mass, a unit of length, and so on. However, the use and notation is somewhat different from SI.
Suppose a particle with a mass of ''m'' has 3.4 times the mass of electron. The value of ''m'' can be written in three ways:
* "
". This is the clearest notation (but least common), where the atomic unit is included explicitly as a symbol.
* "
" ("a.u." means "expressed in atomic units"). This notation is ambiguous: Here, it means that the mass ''m'' is 3.4 times the atomic unit of mass. But if a length ''L'' were 3.4 times the atomic unit of length, the equation would look the same, "
" The dimension must be inferred from context.
* "
". This notation is similar to the previous one, and has the same dimensional ambiguity. It comes from formally setting the atomic units to 1, in this case
, so
.
Physical constants
Dimensionless physical constant
In physics, a dimensionless physical constant is a physical constant that is dimensionless, i.e. a pure number having no units attached and having a numerical value that is independent of whatever system of units may be used.
For example, if one c ...
s retain their values in any system of units. Of note is the
fine-structure constant
In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between el ...
, which appears in expressions as a consequence of the choice of units. For example, the numeric value of the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
, expressed in atomic units, has a value related to the fine-structure constant.
Bohr model in atomic units
Atomic units are chosen to reflect the properties of electrons in atoms, which is particularly clear in the classical
Bohr model
In atomic physics, the Bohr model or Rutherford–Bohr model, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar Syst ...
of the
hydrogen atom for the bound electron in its
ground state:
* Mass = 1 a.u. of mass
* Orbital radius = 1 a.u. of length
* Orbital velocity = 1 a.u. of velocity
* Orbital period = 2''π'' a.u. of time
* Orbital
angular velocity = 1 radian per a.u. of time
* Orbital
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
= 1 a.u. of momentum
*
Ionization energy = a.u. of energy
* Electric field (due to nucleus) = 1 a.u. of electric field
* Electrical attractive force (due to nucleus) = 1 a.u. of force
Non-relativistic quantum mechanics in atomic units
In the context of atomic physics,
nondimensionalization
Nondimensionalization is the partial or full removal of dimensional analysis, physical dimensions from an mathematical equation, equation involving physical quantity, physical quantities by a suitable substitution of variables. This technique can ...
using the defining constants of the Hartree atomic system can be a convenient shortcut, since it can be thought of as eliminating these constants wherever they occur. Nondimesionalization involves a substitution of variables that results in equations in which these constants (
,
,
and
) "have been set to 1".
Though the variables are no longer the original variables, the same symbols and names are typically used.
For example, the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
for an electron with quantities that use SI units is
:
The same equation with corresponding nondimensionalized quantity definitions is
:
For the special case of the electron around a hydrogen atom, the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
with SI quantities is:
:
while the corresponding nondimensionalized equation is
:
Comparison with Planck units
Both
Planck units and atomic units are derived from certain fundamental properties of the physical world, and have little
anthropocentric
Anthropocentrism (; ) is the belief that human beings are the central or most important entity in the universe. The term can be used interchangeably with humanocentrism, and some refer to the concept as human supremacy or human exceptionalism. ...
arbitrariness, but do still involve some arbitrary choices in terms of the defining constants. Atomic units were designed for atomic-scale calculations in the present-day universe, while Planck units are more suitable for
quantum gravity and early-universe
cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', and in 1731 taken up in Latin by German philosopher ...
. Both atomic units and Planck units use the
reduced Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
. Beyond this, Planck units use the two fundamental constants of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
and cosmology: the
gravitational constant and the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
in vacuum,
. Atomic units, by contrast, use the mass and charge of the electron, and, as a result, the speed of light in atomic units is
The orbital velocity of an electron around a small atom is of the order of 1 atomic unit, so the discrepancy between the velocity units in the two systems reflects the fact that electrons orbit small atoms by around 2 orders of magnitude more slowly than the speed of light.
There are much larger differences for some other units. For example, the unit of mass in atomic units is the mass of an electron, while the unit of mass in Planck units is the
Planck mass, which is 22 orders of magnitude larger than the atomic unit of mass. Similarly, there are many orders of magnitude separating the Planck units of energy and length from the corresponding atomic units.
See also
*
Natural units
In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
*
Planck units
*
Various extensions of the CGS system to electromagnetism
Notes and references
*
*
External links
{{Systems of measurement
Systems of units
Natural units
Atomic physics