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Natural units

## Summary

In physics, natural unit systems are measurement systems for which certain physical constants have been set to 1 through nondimensionalization of physical units. For example, the speed of light c may be set to 1, and it may then be omitted, equating mass and energy directly E=m rather than using c as a conversion factor in the typical mass–energy equivalence equation E=mc2. 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.

## Systems of natural units

### Summary table

Quantity Planck Stoney Atomic 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)
• ηe = Gme2 / ħc1.7518×10−45
• ηp = Gmp2 / ħc5.9061×10−39
• A dash (—) indicates where the system is not sufficient to express the quantity.

### 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[1]
Mass ${\sqrt {{k_{\text{e}}e^{2}}/{G}}}$  1.859×10−9 kg[1]
Time ${\displaystyle {\sqrt {{Gk_{\text{e}}e^{2}}/{c^{6}}}}}$  4.605×10−45 s[1]
Electric charge ${\displaystyle e}$  1.602×10−19 C

The Stoney unit system uses the following defining constants:

c, G, ke, e,

where c is the speed of light, G is the gravitational constant, ke 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.[2] Stoney units did not consider the Planck constant, which was discovered only after Stoney's proposal.

### Planck units

Planck dimensions in SI units
Quantity Expression Approx.
metric value
Length ${\displaystyle {\sqrt {{\hbar G}/{c^{3}}}}}$  1.616×10−35 m[3]
Mass ${\displaystyle {\sqrt {{\hbar c}/{G}}}}$  2.176×10−8 kg[4]
Time ${\displaystyle {\sqrt {{\hbar G}/{c^{5}}}}}$  5.391×10−44 s[5]
Temperature ${\displaystyle {\sqrt {{\hbar c^{5}}/{G{k_{\text{B}}}^{2}}}}}$  1.417×1032 K[6]

The Planck unit system uses the following defining constants:

c, ħ, G, kB,

where c is the speed of light, ħ is the reduced Planck constant, G is the gravitational constant, and kB 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.[citation needed]

Planck considered only the units based on the universal constants G, h, c, and kB to arrive at natural units for length, time, mass, and temperature, but no electromagnetic units.[7] The Planck system of units is now understood to use the reduced Planck constant, ħ, in place of the Planck constant, h.[8]

### 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[9]

The Schrödinger system of units (named after Austrian physicist Erwin Schrödinger) is seldom mentioned in literature. Its defining constants are:[10][11]

e, ħ, G, ke.

### Geometrized units

Defining constants:

c, G.

The geometrized unit system,[12]: 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.

### 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[13]
Mass ${\displaystyle m_{\text{e}}}$  9.109×10−31 kg[14]
Time ${\displaystyle {(4\pi \epsilon _{0})^{2}\hbar ^{3}}/{m_{\text{e}}e^{4}}}$  2.419×10−17 s[15]
Electric charge ${\displaystyle e}$  1.602×10−19 C[16]

The atomic unit system[17] uses the following defining constants:[18]: 349 [19]

me, e, ħ, 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.[18]: 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[20]
Mass ${\displaystyle m_{\text{e}}}$  9.109×10−31 kg[21]
Time ${\displaystyle {\hbar }/{m_{\text{e}}c^{2}}}$  1.288×10−21 s[22]
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:[23]: 509

c, me, ħ, ε0,

where c is the speed of light, me 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),[24][25] which may be compared to the correspoding expression in SI: α = e2/(4πε0ħc).[26]: 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, mp, ħ.

Here, mp 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".[27]

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.[28]

## Notes and references

1. ^ a b c Barrow, John D. (1983), "Natural units before Planck", Quarterly Journal of the Royal Astronomical Society, 24: 24–26
2. ^ Ray, T.P. (1981). "Stoney's Fundamental Units". Irish Astronomical Journal. 15: 152. Bibcode:1981IrAJ...15..152R.
3. ^ "2022 CODATA Value: Planck length". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18.
4. ^ "2022 CODATA Value: Planck mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18.
5. ^ "2022 CODATA Value: Planck time". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18.
6. ^ "2022 CODATA Value: Planck temperature". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18.
7. ^ 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, 4πε0 would be implicitly in the list of defining constants, giving a charge unit 4πε0ħc.
8. ^ Tomilin, K. A., 1999, "Natural Systems of Units: To the Centenary Anniversary of the Planck System Archived 2020-12-12 at the Wayback Machine", 287–296.
9. ^ "2022 CODATA Value: elementary charge". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18.
10. ^ Stohner, Jürgen; Quack, Martin (2011). "Conventions, Symbols, Quantities, Units and Constants for High-Resolution Molecular Spectroscopy". Handbook of High‐resolution Spectroscopy (PDF). p. 304. doi:10.1002/9780470749593.hrs005. ISBN 9780470749593. Retrieved 19 March 2023.
11. ^ Duff, Michael James (11 July 2004). "Comment on time-variation of fundamental constants". p. 3. arXiv:hep-th/0208093.
12. ^ Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (2008). Gravitation (27. printing ed.). New York, NY: Freeman. ISBN 978-0-7167-0344-0.
13. ^ "2018 CODATA Value: atomic unit of length". The NIST Reference on Constants, Units, and Uncertainty. NIST. Retrieved 2023-12-31.
14. ^ "2018 CODATA Value: atomic unit of mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. Retrieved 2023-12-31.
15. ^ "2018 CODATA Value: atomic unit of time". The NIST Reference on Constants, Units, and Uncertainty. NIST. Retrieved 2023-12-31.
16. ^ "2018 CODATA Value: atomic unit of charge". The NIST Reference on Constants, Units, and Uncertainty. NIST. Retrieved 2023-12-31.
17. ^ Shull, H.; Hall, G. G. (1959). "Atomic Units". Nature. 184 (4698): 1559. Bibcode:1959Natur.184.1559S. doi:10.1038/1841559a0. S2CID 23692353.
18. ^ a b Levine, Ira N. (1991). Quantum chemistry. Pearson advanced chemistry series (4 ed.). Englewood Cliffs, NJ: Prentice-Hall International. ISBN 978-0-205-12770-2.
19. ^ McWeeny, R. (May 1973). "Natural Units in Atomic and Molecular Physics". Nature. 243 (5404): 196–198. Bibcode:1973Natur.243..196M. doi:10.1038/243196a0. ISSN 0028-0836. S2CID 4164851.
20. ^ "2018 CODATA Value: natural unit of length". The NIST Reference on Constants, Units, and Uncertainty. NIST. Retrieved 2020-05-31.
21. ^ "2018 CODATA Value: natural unit of mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. Retrieved 2020-05-31.
22. ^ "2018 CODATA Value: natural unit of time". The NIST Reference on Constants, Units, and Uncertainty. NIST. Retrieved 2020-05-31.
23. ^ Guidry, Mike (1991). "Appendix A: Natural Units". Gauge Field Theories. Weinheim, Germany: Wiley-VCH Verlag. pp. 509–514. doi:10.1002/9783527617357.app1.
24. ^ Frank Wilczek (2005), "On Absolute Units, I: Choices" (PDF), Physics Today, 58 (10): 12, Bibcode:2005PhT....58j..12W, doi:10.1063/1.2138392, archived from the original (PDF) on 2020-06-13, retrieved 2020-05-31
25. ^ Frank Wilczek (2006), "On Absolute Units, II: Challenges and Responses" (PDF), Physics Today, 59 (1): 10, Bibcode:2006PhT....59a..10W, doi:10.1063/1.2180151, archived from the original (PDF) on 2017-08-12, retrieved 2020-05-31
26. ^ Le Système international d’unités [The International System of Units] (PDF) (in French and English) (9th ed.), International Bureau of Weights and Measures, 2019, ISBN 978-92-822-2272-0
27. ^ Wilczek, Frank (2007). "Fundamental Constants". arXiv:0708.4361 [hep-ph].. Further see.
28. ^ Davis, Tamara Maree (12 February 2004). "Fundamental Aspects of the Expansion of the Universe and Cosmic Horizons". p. 103. arXiv:astro-ph/0402278. In this set of units the speed of light changes in inverse proportion to the fine structure constant. From this we can conclude that if c changes but e and ℏ remain constant then the speed of light in Schrödinger units, cψ changes in proportion to c but the speed of light in Planck units, cP stays the same. Whether or not the "speed of light" changes depends on our measuring system (three possible definitions of the "speed of light" are c, cP and cψ). Whether or not c changes is unambiguous because the measuring system has been defined.