In physics, natural unit systems are measurement systems for which selected physical constants have been set to 1 through nondimensionalization of physical units. For example, the speed of light 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 equation . A purely natural system of units 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). Dimensional analysis in the collapsed system is uninformative as most quantities have the same dimensions.
| Quantity ! Planck | Stoney ! Atomic | Particle and atomic physics ! Strong | Schrödinger | |||
|---|---|---|---|---|---|---|
| Defining constants | c G \hbar kB | c G e ke | e me \hbar ke | c me \hbar \varepsilon0 | c mp \hbar | \hbar G e ke |
| Speed of light c | 1 | 1 | {1}/{\alpha} | 1 | 1 | {1}/{\alpha} |
| Reduced Planck constant \hbar | 1 | {1}/{\alpha} | 1 | 1 | 1 | 1 |
| Elementary charge e | — | 1 | 1 | \sqrt{4\pi\alpha} | — | 1 |
| Vacuum permittivity \varepsilon0 | — | {1}/{4\pi} | {1}/{4\pi} | 1 | — | {1}/{4\pi} |
| Gravitational constant G | 1 | 1 | {ηe | ηe | ηp | 1 |
See main article: Stoney units.
| Quantity ! Expression | Approx. metric value | |
|---|---|---|
\sqrt{{Gkee2}/{c4}} | ||
\sqrt{{Gkee2}/{c6}} | ||
e | ||
The Stoney unit system uses the following defining constants:
,,,,where is the speed of light, is the gravitational constant, is the Coulomb constant, and is the elementary charge.
George Johnstone Stoney'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.[1] Stoney units did not consider the Planck constant, which was discovered only after Stoney's proposal.
See main article: Planck units.
| Quantity ! Expression | Approx. metric value | |
|---|---|---|
\sqrt{{\hbarG}/{c3}} | ||
\sqrt{{\hbarc}/{G}} | ||
\sqrt{{\hbarG}/{c5}} | ||
\sqrt{{\hbarc5}/{G{kB | ||
The Planck unit system uses the following defining constants:
,,,,where is the speed of light, is the reduced Planck constant, is the gravitational constant, and is the Boltzmann constant.
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. They only refer to the basic structure of the laws of physics: and are part of the structure of spacetime in general relativity, and is at the foundation of quantum mechanics. This makes Planck units particularly convenient and common in theories of quantum gravity, including string theory.
Planck considered only the units based on the universal constants,,, and B to arrive at natural units for length, time, mass, and temperature, but no electromagnetic units.[2] The Planck system of units is now understood to use the reduced Planck constant,, in place of the Planck constant, .[3]
| Quantity ! Expression | Approx. metric value | |||||||
|---|---|---|---|---|---|---|---|---|
\sqrt{{\hbar4G(4\pi
/{e6}} | ||||||||
\sqrt{{e2}/{4\pi\varepsilon0G}} | ||||||||
\sqrt{{\hbar6G(4\pi
/{e10 | ||||||||
e | ||||||||
The Schrödinger system of units (named after Austrian physicist Erwin Schrödinger) is seldom mentioned in literature. Its defining constants are:[4] [5]
,,, .
See main article: Geometrized unit system.
Defining constants:
, .
The geometrized unit system,[6] used in general relativity, the base physical units are chosen so that the speed of light,, and the gravitational constant,, are set to one.
See main article: Atomic units.
| Quantity ! Expression | Metric value | |||||||
|---|---|---|---|---|---|---|---|---|
{(4\pi\epsilon0)\hbar2}/{mee2} | [7] | |||||||
me | [8] | |||||||
{(4\pi
\hbar3}/{mee4} | [9] | |||||||
e | [10] | |||||||
The atomic unit system[11] uses the following defining constants:[12]
,,, .
The atomic units were first proposed by Douglas Hartree and are designed to simplify atomic and molecular physics and chemistry, especially the hydrogen atom.[13] For example, in atomic units, in the Bohr model of the hydrogen atom an electron in the ground state has orbital radius, orbital velocity and so on with particularly simple numeric values.
| Quantity ! Expression | Metric value | |
|---|---|---|
{\hbar}/{mec} | [14] | |
me | [15] | |
{\hbar}/{mec2} | [16] | |
\sqrt{\varepsilon0\hbarc} | ||
This natural unit system, used only in the fields of particle and atomic physics, uses the following defining constants:[17]
,,,,where is the speed of light, e is the electron mass, is the reduced Planck constant, 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 fine-structure constant, written, which may be compared to the correspoding expression in SI: .
| Quantity ! Expression | Metric value | |
|---|---|---|
{\hbar}/{mpc} | ||
mp | ||
{\hbar}/{mpc2} | ||
Defining constants:
,, .
Here, is the proton rest mass. Strong units are "convenient for work in QCD and nuclear physics, where quantum mechanics and relativity are omnipresent and the proton is an object of central interest".[18]
In this system of units the speed of light changes in inverse proportion to the fine-structure constant, therefore it has gained some interest recent years in the niche hypothesis of time-variation of fundamental constants.[19]