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Natural units
Units of measurement based on universal physical constants
Units of measurement based on universal physical constants
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 c may be set to 1, and it may then be omitted, equating mass and energy directly rather than using c 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).
Systems of natural units
Summary table
| Quantity | Planck | Stoney | Atomic | Particle and atomic physics | Strong | Schrödinger | Defining constants | c | ℏ | e | ε0 | G |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| c, G, ℏ, kB | c, G, ℏ, ke, kB | c, G, ℏ, ε0, kB | c, G, e, ke | e, me, ℏ, ke | c, me, ℏ, ε0 | c, m_\text{p}, ℏ | ℏ, G, e, ke | |||||
| 1 | 1 | 1 | 1 | 1 | 1 | |||||||
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||
| — | \sqrt{\alpha} | 2\sqrt{\pi\alpha} | 1 | 1 | 2\sqrt{\pi\alpha} | — | 1 | |||||
| — | 1 | 1 | — | \frac{1}{4\pi} | ||||||||
| 1 | 1 | 1 | 1 | ηe | ηp | 1 |
where:
- α is the fine-structure constant, e2 / 4πε0ℏc =
- ke is the Coulomb constant, 1 / 4πε0 ≈ , so assigning it a value also assigns ε0 a value.
- ≈
- me is the mass of an electron,
- ≈
- mp is the mass of a proton,
- — indicates where the system is not sufficient to express the quantity.
- kB, the Boltzmann constant, has no interactions with the other constants - it is used only to redefine temperature.
Stoney units
Main article: Stoney units
| Quantity | Expression | Approx. |
|---|---|---|
| metric value | ||
| Length | \frac{e\sqrt{G k_\text{e}}}{c^2} | |
| Mass | e\sqrt{\frac{k_\text{e}}{G}} | |
| Time | \frac{e\sqrt{G k_\text{e}}}{c^3} | |
| Electric charge | e |
The Stoney unit system uses the following defining constants: : , G, k, e, where c is the speed of light, G is the gravitational constant, k 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. Stoney units did not consider the Planck constant, which was discovered only after Stoney's proposal.
Planck units
Main article: Planck units
| Quantity | Expression | Approx. | ||||||
|---|---|---|---|---|---|---|---|---|
| metric value | Length | Mass | Time | Temperature | Electric charge | |||
| ε0 is also chosen | Electric charge | |||||||
| ke is also chosen | ||||||||
| \sqrt{\frac{\hbar G}{c^3}} | ||||||||
| \sqrt{\frac{\hbar c}{G}} | ||||||||
| \sqrt{\frac{\hbar G}{c^5}} | ||||||||
| \frac{\sqrt{\hbar c^5}}{k_\text{B}\sqrt{G}} | ||||||||
| \sqrt{\varepsilon_0 \hbar c} | ||||||||
| \sqrt{\frac{\hbar c}{k_\mathrm{e}}} |
The Planck unit system uses the following defining constants: : c, ℏ, G, k, where c is the speed of light, ℏ is the reduced Planck constant, G is the gravitational constant, and k 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 kB to arrive at natural units for length, time, mass, and temperature, but no electromagnetic units. The Planck system of units is now understood to use the reduced Planck constant, ℏ, in place of the Planck constant, h.
Schrödinger units
| Quantity | Expression | Approx. metric value | Length | Mass | Time | Electric charge |
|---|---|---|---|---|---|---|
| \frac{\hbar^2}{e^3}\sqrt{\frac{G}{k_\mathrm{e}^3}} | ||||||
| e\sqrt{\frac{k_\mathrm{e}}{G}} | ||||||
| \frac{\hbar^3}{e^5}\sqrt{\frac{G}{k_\mathrm{e}^5}} | ||||||
| e |
The Schrödinger system of units (named after Austrian physicist Erwin Schrödinger) is seldom mentioned in literature. Its defining constants are: | access-date = 19 March 2023 : e, ħ, G, ke.
Geometrized units
Main article: Geometrized unit system
;Defining constants:c, G.
The geometrized unit system,
Atomic units
Main article: Atomic units
| Quantity | Expression | Metric value |
|---|---|---|
| Length | \frac{4 \pi \epsilon_0 \hbar^2}{m_\text{e} e^2} | |
| Mass | m_\text{e} | |
| Time | \frac{16\pi^2\epsilon_0^2 \hbar^3}{m_\text{e} e^4} | |
| Electric charge | e |
The atomic unit system |url-access=subscription}} : m, e, ħ, 4πε0 (this is exactly the same as using ke, except in which constant you use when expressing the conversion).
The atomic units were first proposed by Douglas Hartree and are designed to simplify atomic and molecular physics and chemistry, especially the hydrogen atom.
Natural units (particle and atomic physics)
| Quantity | Expression | Metric value |
|---|---|---|
| Length | \frac{\hbar}{m_\text{e} c} | |
| Mass | m_\text{e} | |
| Time | \frac{\hbar}{m_\text{e} c^2} | |
| Electric charge | \sqrt{\varepsilon_0 \hbar c} |
This natural unit system, used only in the fields of particle and atomic physics, uses the following defining constants: |publication-place=Weinheim, Germany : c, m, 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 , |access-date=2020-05-31 |archive-url=https://web.archive.org/web/20200613120809/http://ctpweb.lns.mit.edu/physics_today/phystoday/Abs_limits388.pdf |archive-date=2020-06-13 |access-date=2020-05-31 |archive-url=https://web.archive.org/web/20170812231026/http://ctpweb.lns.mit.edu/physics_today/phystoday/Abs_limits393.pdf |archive-date=2017-08-12
Strong units
| Quantity | Expression | Metric value |
|---|---|---|
| Length | \frac{\hbar}{m_\text{p} c} | |
| Mass | m_\text{p} | |
| Time | \frac{\hbar}{m_\text{p} c^2} |
Defining constants: : c, m, ħ.
Here, m 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".
References
- Barrow, John D.. (1983). "Natural units before Planck". Quarterly Journal of the Royal Astronomical Society.
- 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 implicit in the list of defining constants, giving a charge unit {{math. {{radic. 4''πε''{{sub. 0ℏ''c''.
- 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. link. (2020-12-12 ", 287–296.)
- "2018 CODATA Value: atomic unit of length". [[National Institute of Standards and Technology.
- "2018 CODATA Value: atomic unit of mass". [[National Institute of Standards and Technology.
- "2018 CODATA Value: atomic unit of time". [[National Institute of Standards and Technology.
- "2018 CODATA Value: atomic unit of charge". [[National Institute of Standards and Technology.
- "2018 CODATA Value: natural unit of length". [[National Institute of Standards and Technology.
- "2018 CODATA Value: natural unit of mass". [[National Institute of Standards and Technology.
- "2018 CODATA Value: natural unit of time". [[National Institute of Standards and Technology.
- {{SIbrochure9th
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