physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

, natural unit systems are

measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared to ...

systems for which selected physical constants have been set to 1 through nondimensionalization of

physical units A unit of measurement, or unit of measure, is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can ...

. For example, the

speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...

may be set to 1, and it may then be omitted, equating mass and energy directly rather than using as a conversion factor in the typical

mass–energy equivalence In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame. The two differ only by a multiplicative constant and the units of measurement. The principle is described by the physicist Albert Einstei ...

equation . A purely natural

system of units A system of units of measurement, also known as a system of units or system of measurement, is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been important, regulated and defi ...

has all of its dimensions collapsed, such that the physical constants completely define the system of units and the relevant physical laws contain no conversion constants. While natural unit systems simplify the form of each equation, it is still necessary to keep track of the non-collapsed dimensions of each quantity or expression in order to reinsert physical constants (such dimensions uniquely determine the full formula).

Systems of natural units

Summary table

where: * is the

fine-structure constant In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Alpha, Greek letter ''alpha''), is a Dimensionless physical constant, fundamental physical constant that quantifies the strength of the el ...

( ≈ 0.007297) * ≈ * ≈ * A dash (—) indicates where the system is not sufficient to express the quantity.

Stoney units

The Stoney unit system uses the following defining constants: : , , , , where is the

, is the gravitational constant, is the

Coulomb constant Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that calculates the amount of force (physics), force between two electric charge, electrically charged particles at rest. This electric for ...

, and is the

elementary charge The elementary charge, usually denoted by , is a fundamental physical constant, defined as the electric charge carried by a single proton (+1 ''e'') or, equivalently, the magnitude of the negative electric charge carried by a single electron, ...

George Johnstone Stoney George Johnstone Stoney (15 February 1826 – 5 July 1911) was an Irish physicist known for introducing the term ''electron'' as the "fundamental unit quantity of electricity". He initially named it ''electrolion'' in 1881, and later named it ...

's unit system preceded that of Planck by 30 years. He presented the idea in a lecture entitled "On the Physical Units of Nature" delivered to the British Association in 1874. Stoney units did not consider the

Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...

, which was discovered only after Stoney's proposal.

Planck units

The Planck unit system uses the following defining constants: : , , , , where is the

, is the

reduced Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...

, is the gravitational constant, and is the

Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...

. Planck units form a system of natural units that is not defined in terms of properties of any prototype, physical object, or even

elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. The Standard Model presently recognizes seventeen distinct particles—twelve fermions and five bosons. As a c ...

. They only refer to the basic structure of the laws of physics: and are part of the structure of

spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...

general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...

, and is at the foundation of

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

. This makes Planck units particularly convenient and common in theories of

quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...

, including string theory. Planck considered only the units based on the universal constants , , , and _B to arrive at natural units for

length Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...

time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...

mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...

, and

temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...

, but no electromagnetic units. The Planck system of units is now understood to use the reduced Planck constant, , in place of the Planck constant, .Tomilin, K. A., 1999,
Natural Systems of Units: To the Centenary Anniversary of the Planck System
", 287–296.

Schrödinger units

The Schrödinger system of units (named after Austrian physicist

Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger ( ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was an Austrian-Irish theoretical physicist who developed fundamental results in quantum field theory, quantum theory. In particul ...

) is seldom mentioned in literature. Its defining constants are: : , , , .

Geometrized units

Defining constants: : , . The geometrized unit system, used in

, the base physical units are chosen so that the

, , and the gravitational constant, , are set to one.

Atomic units

The atomic unit system uses the following defining constants: : , , , . The atomic units were first proposed by Douglas Hartree and are designed to simplify atomic and molecular physics and chemistry, especially the hydrogen atom. For example, in atomic units, in the

Bohr model In atomic physics, the Bohr model or Rutherford–Bohr model was a model of the atom that incorporated some early quantum concepts. Developed from 1911 to 1918 by Niels Bohr and building on Ernest Rutherford's nuclear Rutherford model, model, i ...

of the hydrogen atom an electron in the ground state has orbital radius, orbital velocity and so on with particularly simple numeric values.

Natural units (particle and atomic physics)

This natural unit system, used only in the fields of particle and atomic physics, uses the following defining constants: : , , , , where is the

, _e is the electron mass, is the

, and ₀ is the vacuum permittivity. The vacuum permittivity ₀ is implicitly used as a nondimensionalization constant, as is evident from the physicists' expression for the

, written , which may be compared to the corresponding expression in SI: .

Strong units

Defining constants: : , , . Here, is the

proton A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...

rest mass. ''Strong units'' are "convenient for work in

QCD In theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in ...

and nuclear physics, where quantum mechanics and relativity are omnipresent and the proton is an object of central interest". . Furthe
see

Notes and references

External links

The NIST website
(

National Institute of Standards and Technology The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into Outline of p ...

) is a convenient source of data on the commonly recognized constants.
K.A. Tomilin: ''NATURAL SYSTEMS OF UNITS; To the Centenary Anniversary of the Planck System''
A comparative overview/tutorial of various systems of natural units having historical use.
Pedagogic Aides to Quantum Field Theory
Click on the link for Chap. 2 to find an extensive, simplified introduction to natural units.
Natural System Of Units In General Relativity (PDF)
by Alan L. Myers (University of Pennsylvania). Equations for conversions from natural to SI units. {{DEFAULTSORT:Natural Units Metrology