:Natural units

| $\sqrt{{G k_\text{e} e^2} / {c^4}}$

| {{val|1.380|e=-36|u=m}}{{citation |last=Barrow |first=John D. |url=https://articles.adsabs.harvard.edu/full/1983QJRAS..24...24B |title=Natural units before Planck |journal=Quarterly Journal of the Royal Astronomical Society |volume=24 |pages=24–26 |year=1983 |bibcode=1983QJRAS..24...24B }}

| $\sqrt{{k_\text{e} e^2} / {G}}$

| {{val|1.859|e=-9|u=kg}}

| $\sqrt{{G k_\text{e} e^2} / {c^6}}$

| {{val|4.605|e=-45|u=s}}

| $e$

| {{val|1.602|e=-19|u=C}}

The Stoney unit system uses the following defining constants:

: {{math|1=c}}, {{math|G}}, {{math|k{{sub|e}}}}, {{math|e}},

where {{math|c}} is the speed of light, {{math|G}} is the gravitational constant, {{math|k{{sub|e}}}} is the Coulomb constant, and {{math|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.

{{cite journal

| last=Ray | first = T.P.

| year=1981

| title=Stoney's Fundamental Units

| journal=Irish Astronomical Journal

| volume=15

| page=152

|bibcode = 1981IrAJ...15..152R

}}

Stoney units did not consider the Planck constant, which was discovered only after Stoney's proposal.

= Planck units =

class="wikitable" align="right" style="margin-left: 1em;" \|+ Planck dimensions in SI units
Quantity ! Expression ! Approx. metric value
align="left" \| Length \| $\sqrt{{\hbar G} / {c^3}}$ \| {{val\|1.616\|e=-35\|u=m}}{{physconst\|lP\|ref=only}}
Mass \| $\sqrt{{\hbar c} / {G}}$ \| {{val\|2.176\|e=-8\|u=kg}}{{physconst\|mP\|ref=only}}
Time \| $\sqrt{{\hbar G} / {c^5}}$ \| {{val\|5.391\|e=-44\|u=s}}{{physconst\|tP\|ref=only}}
Temperature \| $\sqrt{{\hbar c^5} / {G {k_\text{B}}^2}}$ \| {{val\|1.417\|e=32\|u=K}}{{physconst\|TP\|ref=only}}

class="wikitable" align="right" style="margin-left: 1em;"

|+ Planck dimensions in SI units

Quantity

! Expression

! Approx.
metric value

align="left"

| $\sqrt{{\hbar G} / {c^3}}$

| {{val|1.616|e=-35|u=m}}{{physconst|lP|ref=only}}

| $\sqrt{{\hbar c} / {G}}$

| {{val|2.176|e=-8|u=kg}}{{physconst|mP|ref=only}}

| $\sqrt{{\hbar G} / {c^5}}$

| {{val|5.391|e=-44|u=s}}{{physconst|tP|ref=only}}

Temperature

| $\sqrt{{\hbar c^5} / {G {k_\text{B}}^2}}$

| {{val|1.417|e=32|u=K}}{{physconst|TP|ref=only}}

The Planck unit system uses the following defining constants:

: {{math|c}}, {{math|ħ}}, {{math|G}}, {{math|k{{sub|B}}}},

where {{math|c}} is the speed of light, {{math|ħ}} is the reduced Planck constant, {{math|G}} is the gravitational constant, and {{math|k{{sub|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: {{math|c}} and {{math|G}} are part of the structure of spacetime in general relativity, and {{math|ħ}} is at the foundation of quantum mechanics. This makes Planck units particularly convenient and common in theories of quantum gravity, including string theory.{{citation needed|date=September 2020}}

Planck considered only the units based on the universal constants {{math|G}}, {{math|h}}, {{math|c}}, and {{math|k}}_B to arrive at natural units for length, time, mass, and temperature, but no electromagnetic units.However, if it is assumed that at the time the Gaussian definition of electric charge was used and hence not regarded as an independent quantity, 4{{math|πε}}{{sub|0}} would be implicitly in the list of defining constants, giving a charge unit {{math|{{radic|4πε{{sub|0}}ħc}}}}. The Planck system of units is now understood to use the reduced Planck constant, {{math|ħ}}, in place of the Planck constant, {{math|h}}.Tomilin, K. A., 1999, "[http://old.ihst.ru/personal/tomilin/papers/tomil.pdf Natural Systems of Units: To the Centenary Anniversary of the Planck System] {{Webarchive|url=https://web.archive.org/web/20201212041222/http://old.ihst.ru/personal/tomilin/papers/tomil.pdf |date=2020-12-12 }}", 287–296.

= Schrödinger units =

class="wikitable" align="right" style="margin-left: 1em;" \|+Schrödinger system dimensions in SI units
Quantity ! Expression ! Approx. metric value
align="left" \| Length \| $\sqrt{{\hbar^4 G (4 \pi \varepsilon_0)^3} / {e^6}}$ \| {{val\|2.593\|e=-32\|u=m}}
Mass \| $\sqrt{{e^2} / {4 \pi \varepsilon_0 G}}$ \| {{val\|1.859\|e=-9\|u=kg}}
Time \| $\sqrt{{\hbar^6 G (4 \pi \varepsilon_0)^5} / {e^{10}}}$ \| {{val\|1.185\|e=-38\|u=s}}
Electric charge \| $e$ \| {{val\|1.602\|e=-19\|u=C}}{{physconst\|e\|ref=only}}

class="wikitable" align="right" style="margin-left: 1em;"

|+Schrödinger system dimensions in SI units

Quantity

! Expression

! Approx.
metric value

align="left"

| $\sqrt{{\hbar^4 G (4 \pi \varepsilon_0)^3} / {e^6}}$

| {{val|2.593|e=-32|u=m}}

| $\sqrt{{e^2} / {4 \pi \varepsilon_0 G}}$

| {{val|1.859|e=-9|u=kg}}

| $\sqrt{{\hbar^6 G (4 \pi \varepsilon_0)^5} / {e^{10}}}$

| {{val|1.185|e=-38|u=s}}

| $e$

| {{val|1.602|e=-19|u=C}}{{physconst|e|ref=only}}

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

{{cite book

| last1 = Stohner | first1 = Jürgen

| last2 = Quack | first2 = Martin

| year = 2011

| title = Handbook of High-resolution Spectroscopy

| chapter = Conventions, Symbols, Quantities, Units and Constants for High-Resolution Molecular Spectroscopy

| url = https://www.ir.ethz.ch/handbook/MQ333_Handbook_Stohner_Quack_bearbeitet.pdf

| page = 304

| access-date = 19 March 2023

| doi = 10.1002/9780470749593.hrs005 | isbn = 9780470749593

}}

{{cite arXiv

| last = Duff | first = Michael James | author-link = Michael James Duff

| date = 11 July 2004

| title = Comment on time-variation of fundamental constants

| page = 3

| eprint = hep-th/0208093

}}

: {{math|e}}, {{math|ħ}}, {{math|G}}, {{math|k{{sub|e}}}}.

= Geometrized units =

Defining constants:

: {{math|c}}, {{math|G}}.

The geometrized unit system,

{{cite book

|last1=Misner |first1=Charles W.

|last2=Thorne |first2=Kip S.

|last3=Wheeler |first3=John Archibald

|date=2008

|title=Gravitation

|edition=27. printing

|location=New York, NY

|publisher=Freeman

|isbn=978-0-7167-0344-0

}}{{rp|36}} used in general relativity, the base physical units are chosen so that the speed of light, {{math|c}}, and the gravitational constant, {{math|G}}, are set to one.

= Atomic units =

class="wikitable" align="right" style="margin-left: 1em;" \|+ Atomic-unit dimensions in SI units
Quantity ! Expression ! Metric value
align="left" \| Length \| ${(4 \pi \epsilon_0) \hbar^2} / {m_\text{e} e^2}$ \| {{val\|5.292\|e=-11\|u=m}}{{cite web \|title=2018 CODATA Value: atomic unit of length \|work=The NIST Reference on Constants, Units, and Uncertainty \|url=http://physics.nist.gov/cgi-bin/cuu/Value?Abohrrada0 \|publisher=NIST \|access-date=2023-12-31 }}
Mass \| $m_\text{e}$ \| {{val\|9.109\|e=-31\|u=kg}}{{cite web \|title=2018 CODATA Value: atomic unit of mass \|work=The NIST Reference on Constants, Units, and Uncertainty \|url=http://physics.nist.gov/cgi-bin/cuu/Value?Ame \|publisher=NIST \|access-date=2023-12-31 }}
Time \| ${(4 \pi \epsilon_0)^2 \hbar^3} / {m_\text{e} e^4}$ \| {{val\|2.419\|e=-17\|u=s}}{{cite web \|title=2018 CODATA Value: atomic unit of time \|work=The NIST Reference on Constants, Units, and Uncertainty \|url=http://physics.nist.gov/cgi-bin/cuu/Value?aut \|publisher=NIST \|access-date=2023-12-31 }}
Electric charge \| $e$ \| {{val\|1.602\|e=-19\|u=C}}{{cite web \|title=2018 CODATA Value: atomic unit of charge \|work=The NIST Reference on Constants, Units, and Uncertainty \|url=http://physics.nist.gov/cgi-bin/cuu/Value?Ae \|publisher=NIST \|access-date=2023-12-31 }}

class="wikitable" align="right" style="margin-left: 1em;"

|+ Atomic-unit dimensions in SI units

Quantity

! Expression

! Metric value

align="left"

| ${(4 \pi \epsilon_0) \hbar^2} / {m_\text{e} e^2}$

| {{val|5.292|e=-11|u=m}}{{cite web |title=2018 CODATA Value: atomic unit of length |work=The NIST Reference on Constants, Units, and Uncertainty |url=http://physics.nist.gov/cgi-bin/cuu/Value?Abohrrada0 |publisher=NIST |access-date=2023-12-31 }}

| $m_\text{e}$

| {{val|9.109|e=-31|u=kg}}{{cite web |title=2018 CODATA Value: atomic unit of mass |work=The NIST Reference on Constants, Units, and Uncertainty |url=http://physics.nist.gov/cgi-bin/cuu/Value?Ame |publisher=NIST |access-date=2023-12-31 }}

| ${(4 \pi \epsilon_0)^2 \hbar^3} / {m_\text{e} e^4}$

| {{val|2.419|e=-17|u=s}}{{cite web |title=2018 CODATA Value: atomic unit of time |work=The NIST Reference on Constants, Units, and Uncertainty |url=http://physics.nist.gov/cgi-bin/cuu/Value?aut |publisher=NIST |access-date=2023-12-31 }}

| $e$

| {{val|1.602|e=-19|u=C}}{{cite web |title=2018 CODATA Value: atomic unit of charge |work=The NIST Reference on Constants, Units, and Uncertainty |url=http://physics.nist.gov/cgi-bin/cuu/Value?Ae |publisher=NIST |access-date=2023-12-31 }}

The atomic unit system

{{cite journal

|last1=Shull | first1=H.

|last2=Hall | first2=G. G.

|year=1959

|title=Atomic Units

|journal=Nature

|volume=184 |issue=4698 |page=1559

|doi=10.1038/1841559a0 |bibcode = 1959Natur.184.1559S | s2cid=23692353

}} uses the following defining constants:{{rp|349}}

{{cite journal

|last=McWeeny |first=R.

|date=May 1973

|title=Natural Units in Atomic and Molecular Physics

|url=https://www.nature.com/articles/243196a0

|journal=Nature

|language=en

|volume=243 |issue=5404 |pages=196–198

|doi=10.1038/243196a0 |bibcode=1973Natur.243..196M |s2cid=4164851 |issn=0028-0836

}}

: {{math|m{{sub|e}}}}, {{math|e}}, {{math|ħ}}, {{math|4πε₀}}.

The atomic units were first proposed by Douglas Hartree and are designed to simplify atomic and molecular physics and chemistry, especially the hydrogen atom.

{{cite book

|last=Levine |first=Ira N.

|date=1991

|title=Quantum chemistry

|edition=4

|series=Pearson advanced chemistry series

|location=Englewood Cliffs, NJ

|publisher=Prentice-Hall International

|isbn=978-0-205-12770-2

}}{{rp|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) =

class="wikitable" align="right" style="margin-left: 1em;"
Quantity ! Expression ! Metric value
align="left" \| Length \| ${\hbar} / {m_\text{e} c}$ \| {{val\|3.862\|e=-13\|u=m}}{{cite web \|url=http://physics.nist.gov/cgi-bin/cuu/Value?eqNecomwlbar \|title=2018 CODATA Value: natural unit of length \|work=The NIST Reference on Constants, Units, and Uncertainty \|publisher=NIST \|access-date=2020-05-31}}
Mass \| $m_\text{e}$ \| {{val\|9.109\|e=-31\|u=kg}}{{cite web \|url=http://physics.nist.gov/cgi-bin/cuu/Value?Nme \|title=2018 CODATA Value: natural unit of mass \|work=The NIST Reference on Constants, Units, and Uncertainty \|publisher=NIST \|access-date=2020-05-31}}
Time \| ${\hbar} / {m_\text{e} c^2}$ \| {{val\|1.288\|e=-21\|u=s}}{{cite web \|url=http://physics.nist.gov/cgi-bin/cuu/Value?nut \|title=2018 CODATA Value: natural unit of time \|work=The NIST Reference on Constants, Units, and Uncertainty \|publisher=NIST \|access-date=2020-05-31}}
Electric charge \| $\sqrt{\varepsilon_0 \hbar c}$ \| {{val\|5.291\|e=-19\|u=C}}

class="wikitable" align="right" style="margin-left: 1em;"

Quantity

! Expression

! Metric value

align="left"

| ${\hbar} / {m_\text{e} c}$

| {{val|3.862|e=-13|u=m}}{{cite web |url=http://physics.nist.gov/cgi-bin/cuu/Value?eqNecomwlbar |title=2018 CODATA Value: natural unit of length |work=The NIST Reference on Constants, Units, and Uncertainty |publisher=NIST |access-date=2020-05-31}}

| $m_\text{e}$

| {{val|9.109|e=-31|u=kg}}{{cite web |url=http://physics.nist.gov/cgi-bin/cuu/Value?Nme |title=2018 CODATA Value: natural unit of mass |work=The NIST Reference on Constants, Units, and Uncertainty |publisher=NIST |access-date=2020-05-31}}

| ${\hbar} / {m_\text{e} c^2}$

| {{val|1.288|e=-21|u=s}}{{cite web |url=http://physics.nist.gov/cgi-bin/cuu/Value?nut |title=2018 CODATA Value: natural unit of time |work=The NIST Reference on Constants, Units, and Uncertainty |publisher=NIST |access-date=2020-05-31}}

| $\sqrt{\varepsilon_0 \hbar c}$

| {{val|5.291|e=-19|u=C}}

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

{{cite book

|first=Mike |last=Guidry

|date=1991

|title=Gauge Field Theories

|chapter=Appendix A: Natural Units

|pages=509–514

|publication-place=Weinheim, Germany

|publisher=Wiley-VCH Verlag

|doi=10.1002/9783527617357.app1

|isbn=978-0-471-63117-0

}}{{rp|509}}

: {{math|c}}, {{math|m{{sub|e}}}}, {{math|ħ}}, {{math|ε₀}},

where {{math|c}} is the speed of light, {{math|m}}_e is the electron mass, {{math|ħ}} is the reduced Planck constant, and {{math|ε}}₀ is the vacuum permittivity.

The vacuum permittivity {{math|ε}}₀ is implicitly used as a nondimensionalization constant, as is evident from the physicists' expression for the fine-structure constant, written {{math|1=α = e{{i sup|2}}/(4π)}},

{{citation

|author=Frank Wilczek

|year=2005

|title=On Absolute Units, I: Choices

|journal=Physics Today |volume=58 |issue=10 |page=12

|doi=10.1063/1.2138392 |bibcode=2005PhT....58j..12W

|access-date=2020-05-31

|url=http://ctpweb.lns.mit.edu/physics_today/phystoday/Abs_limits388.pdf |url-status=dead

|archive-url=https://web.archive.org/web/20200613120809/http://ctpweb.lns.mit.edu/physics_today/phystoday/Abs_limits388.pdf

|archive-date=2020-06-13

}}

{{citation

|author=Frank Wilczek

|year=2006

|title=On Absolute Units, II: Challenges and Responses

|journal=Physics Today |volume=59 |issue=1 |page=10

|doi=10.1063/1.2180151 |bibcode=2006PhT....59a..10W

|access-date=2020-05-31

|url=http://ctpweb.lns.mit.edu/physics_today/phystoday/Abs_limits393.pdf |url-status=dead

|archive-url=https://web.archive.org/web/20170812231026/http://ctpweb.lns.mit.edu/physics_today/phystoday/Abs_limits393.pdf

|archive-date=2017-08-12

}} which may be compared to the corresponding expression in SI: {{math|1=α = e{{i sup|2}}/(4πε₀ħc)}}.{{SIbrochure9th}}{{rp|128}}

= Strong units =

class="wikitable" align="right" style="margin-left: 1em;" \|+ Strong-unit dimensions in SI units
Quantity ! Expression ! Metric value
align="left" \| Length \| ${\hbar} / {m_\text{p} c}$ \| {{val\|2.103\|e=-16\|u=m}}
Mass \| $m_\text{p}$ \| {{val\|1.673\|e=-27\|u=kg}}
Time \| ${\hbar} / {m_\text{p} c^2}$ \| {{val\|7.015\|e=-25\|u=s}}

class="wikitable" align="right" style="margin-left: 1em;"

|+ Strong-unit dimensions in SI units

Quantity

! Expression

! Metric value

align="left"

| ${\hbar} / {m_\text{p} c}$

| {{val|2.103|e=-16|u=m}}

| $m_\text{p}$

| {{val|1.673|e=-27|u=kg}}