Planck units

Planck dimensions in SI units

Quantity	Expression	Approx. metric value
Length	${\displaystyle {\sqrt {{\hbar G}/{c^{3}}}}}$	1.616×10−35 m[1]
Mass	${\displaystyle {\sqrt {{\hbar c}/{G}}}}$	2.176×10−8 kg[2]
Time	${\displaystyle {\sqrt {{\hbar G}/{c^{5}}}}}$	5.391×10−44 s[3]
Temperature	${\displaystyle {\sqrt {{\hbar c^{5}}/{G{k_{\text{B}}}^{2}}}}}$	1.417×1032 K[4]

The Planck unit system uses the following defining constants:

c, ħ, G, k_B,

where c is the speed of light, ħ is the reduced Planck constant, G is the gravitational constant, and k_B 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: c and G are part of the structure of spacetime in general relativity, and ħ 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 G, h, c, and k_B to arrive at natural units for length, time, mass, and temperature, but no electromagnetic units.^[5] The Planck system of units is now understood to use the reduced Planck constant, ħ, in place of the Planck constant, h.^[6]

Stoney units

Stoney system dimensions in SI units

Quantity	Expression	Approx. metric value
Length	${\displaystyle {\sqrt {{Gk_{\text{e}}e^{2}}/{c^{4}}}}}$	1.380×10−36 m[7]
Mass	${\sqrt {{k_{\text{e}}e^{2}}/{G}}}$	1.859×10−9 kg[7]
Time	${\displaystyle {\sqrt {{Gk_{\text{e}}e^{2}}/{c^{6}}}}}$	4.605×10−45 s[7]
Electric charge	${\displaystyle e}$	1.602×10−19 C

The Stoney unit system uses the following defining constants:

c, G, k_e, e,

where c is the speed of light, G is the gravitational constant, k_e is the Coulomb constant, and e 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.^[8] Stoney units did not consider the Planck constant, which was discovered only after Stoney's proposal. The Stoney units and the Planck units differ by the constant factor ${\displaystyle {\sqrt {e^{2}/hc}}}$ ^[7]

Atomic units

Atomic-unit dimensions in SI units

Quantity	Expression	Metric value
Length	${\displaystyle {(4\pi \epsilon _{0})\hbar ^{2}}/{m_{\text{e}}e^{2}}}$	5.292×10−11 m[9]
Mass	${\displaystyle m_{\text{e}}}$	9.109×10−31 kg[10]
Time	${\displaystyle {(4\pi \epsilon _{0})^{2}\hbar ^{3}}/{m_{\text{e}}e^{4}}}$	2.419×10−17 s[11]
Electric charge	${\displaystyle e}$	1.602×10−19 C[12]

The atomic unit system^[13] uses the following defining constants: the electron mass, m_e, the charge on the proton, e, and the unit of angular momentum, ħ.^[14]: 349

The atomic units were first proposed by Douglas Hartree and are designed to simplify atomic and molecular physics and chemistry, especially the hydrogen atom.^[14]: 349 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.

Natural units (particle and atomic physics)

Quantity	Expression	Metric value
Length	${\displaystyle {\hbar }/{m_{\text{e}}c}}$	3.862×10−13 m[15]
Mass	${\displaystyle m_{\text{e}}}$	9.109×10−31 kg[16]
Time	${\displaystyle {\hbar }/{m_{\text{e}}c^{2}}}$	1.288×10−21 s[17]
Electric charge	${\displaystyle {\sqrt {\varepsilon _{0}\hbar c}}}$	5.291×10−19 C

This natural unit system, used only in the fields of particle and atomic physics, uses the following defining constants:^[18]: 509

c, m_e, ħ, ε₀,

where c is the speed of light, m_e is the electron mass, ħ is the reduced Planck constant, and ε₀ is the vacuum permittivity.

The vacuum permittivity ε₀ is implicitly used as a nondimensionalization constant, as is evident from the physicists' expression for the fine-structure constant, written α = e²/(4π),^[19][20] which may be compared to the same expression in SI: α = e²/(4πε₀ħc).^[21]: 128

Strong units

Strong-unit dimensions in SI units

Quantity	Expression	Metric value
Length	${\displaystyle {\hbar }/{m_{\text{p}}c}}$	2.103×10−16 m
Mass	${\displaystyle m_{\text{p}}}$	1.673×10−27 kg
Time	${\displaystyle {\hbar }/{m_{\text{p}}c^{2}}}$	7.015×10−25 s

Defining constants:

c, m_p, ħ.

Here, m_p 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".^[22]

Schrödinger units

Schrödinger system dimensions in SI units

Quantity	Expression	Approx. metric value
Length	${\displaystyle {\sqrt {{\hbar ^{4}G(4\pi \varepsilon _{0})^{3}}/{e^{6}}}}}$	2.593×10−32 m
Mass	${\displaystyle {\sqrt {{e^{2}}/{4\pi \varepsilon _{0}G}}}}$	1.859×10−9 kg
Time	${\displaystyle {\sqrt {{\hbar ^{6}G(4\pi \varepsilon _{0})^{5}}/{e^{10}}}}}$	1.185×10−38 s
Electric charge	${\displaystyle e}$	1.602×10−19 C[23]

The Schrödinger system of units (named after Austrian physicist Erwin Schrödinger) is seldom mentioned in literature. Its defining constants are:^[24][25]

e, ħ, G, k_e.

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.^[26]

Geometrized units

Defining constants:

c, G.

The geometrized unit system,^[27]: 36 used in general relativity, the base physical units are chosen so that the speed of light, c, and the gravitational constant, G, are set to one.

Summary table

Quantity	Planck	Stoney	Hartree	Particle and atomic physics	Strong	Schrödinger
Defining constants	${\displaystyle c}$, ${\displaystyle G}$, ${\displaystyle \hbar }$, ${\displaystyle k_{\text{B}}}$	${\displaystyle c}$, ${\displaystyle G}$, ${\displaystyle e}$, ${\displaystyle k_{\text{e}}}$	${\displaystyle e}$, ${\displaystyle m_{\text{e}}}$, ${\displaystyle \hbar }$, ${\displaystyle k_{\text{e}}}$	${\displaystyle c}$, ${\displaystyle m_{\text{e}}}$, ${\displaystyle \hbar }$, ${\displaystyle \varepsilon _{0}}$	${\displaystyle c}$, ${\displaystyle m_{\text{p}}}$, ${\displaystyle \hbar }$	${\displaystyle \hbar }$, ${\displaystyle G}$, ${\displaystyle e}$, ${\displaystyle k_{\text{e}}}$
Speed of light ${\displaystyle c}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle {1}/{\alpha }}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle {1}/{\alpha }}$
Reduced Planck constant ${\displaystyle \hbar }$	${\displaystyle 1}$	${\displaystyle {1}/{\alpha }}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$
Elementary charge ${\displaystyle e}$	—	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle {\sqrt {4\pi \alpha }}}$	—	${\displaystyle 1}$
Vacuum permittivity ${\displaystyle \varepsilon _{0}}$	—	${\displaystyle {1}/{4\pi }}$	${\displaystyle {1}/{4\pi }}$	${\displaystyle 1}$	—	${\displaystyle {1}/{4\pi }}$
Gravitational constant ${\displaystyle G}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle {\eta _{\mathrm {e} }}/{\alpha }}$	${\displaystyle \eta _{\mathrm {e} }}$	${\displaystyle \eta _{\mathrm {p} }}$	${\displaystyle 1}$

where:

• α is the fine-structure constant (α = e² / 4πε₀ħc ≈ 0.007297)
• η_e = Gm_e² / ħc ≈ 1.7518×10⁻⁴⁵
• η_p = Gm_p² / ħc ≈ 5.9061×10⁻³⁹
• A dash (—) indicates where the system is not sufficient to express the quantity.