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Encyclopedia > Natural units

In physics, natural units are physical units of measurement defined in terms of universal physical constants in such a manner that some chosen physical constants take on the numerical value of one when expressed in terms of a particular set of natural units. Natural units are intended to elegantly simplify particular algebraic expressions appearing in physical law or to normalize some chosen physical quantities that are properties of universal elementary particles and that may be reasonably believed to be constant. However, what may be believed and forced to be constant in one system of natural units can very well be allowed or even assumed to vary in another natural unit system. Natural units are natural because the origin of their definition comes only from properties of nature and not from any human construct. Planck units are often, without qualification, called "natural units" but are only one system of natural units among other systems. Planck units might be considered unique in that the set of units are not based on properties of any prototype, object, or particle but are based only on properties of free space. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... The former Weights and Measures office in Middlesex, England. ... Various meters Measurement is an observation that reduces an uncertainty expressed as a quantity. ... In science, a physical constant is a physical quantity whose numerical value does not change. ... Nondimensionalization refers to the partial or full removal of units from a mathematical equation by a suitable substitution of variables. ... An expression is a combination of numbers, operators, grouping symbols (such as brackets and parentheses) and/or free variables and bound variables arranged in a meaningful way which can be evaluated. ... For the novel, see The Elementary Particles. ... â€œNaturalâ€ redirects here. ... In physics, Planck units are physical units of measurement defined exclusively in terms of the five universal physical constants shown in the table below in such a manner that all of these physical constants take on the numerical value of one when expressed in terms of these units. ... For other uses, see Prototype (disambiguation). ... Helium atom (schematic) Showing two protons (red), two neutrons (green) and two electrons (yellow). ... In physics, free space is a concept of electromagnetic theory, corresponding roughly to the vacuum, the baseline state of the electromagnetic field, or the replacement for the electromagnetic aether. ...

As with any set of base units or fundamental units the base units of a set of natural units will include definitions and values for length, mass, time, temperature, and electric charge. Some physicists have not recognized temperature as a fundamental dimension of physical quantity since it simply expresses the energy per degree of freedom of a particle which can be expressed in terms of energy (or mass, length, and time). Virtually every system of natural units normalizes the Boltzmann constant to k=1, which can be thought of as simply another expression of the definition of the unit temperature. In addition, some physicists recognize electric charge as a separate fundamental dimension of physical quantity, even if it has been expressed in terms of mass, length, and time in unit systems such as the electrostatic cgs system. Virtually every system of natural units normalize the permittivity of free space to ε0=(4π)-1, which can be thought of as an expression of the definition of the unit charge. In the language of measurement, quantities are quantifiable aspects of the world, such as time, distance, velocity, mass, and weight, and units measure them. ... For other uses of this word, see Length (disambiguation). ... This article or section is in need of attention from an expert on the subject. ... Look up time in Wiktionary, the free dictionary. ... For other uses, see Temperature (disambiguation). ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... This article or section is in need of attention from an expert on the subject. ... This article is in need of attention. ...

## Candidate physical constants used in natural unit systems GA_googleFillSlot("encyclopedia_square");

The candidate physical constants to be normalized are chosen from those in the following table. Note that only a smaller subset of the following can be normalized in any one system of units without contradiction in definition (e.g., me and mp cannot both be defined as the unit mass in a single system).

Constant Symbol Dimension
speed of light in vacuum ${ c }$ L T-1
Gravitational constant ${ G }$ M-1L3T-2
Dirac's constant or "reduced Planck's constant" $hbar=frac{h}{2 pi}$ where ${h}$ is Planck's constant ML2T-1
Coulomb force constant $frac{1}{4 pi epsilon_0}$ where ${ epsilon_0 }$ is the permittivity of free space Q-2 M L3 T-2
Elementary charge $e$ Q
Electron mass $m_e$ M
Proton mass $m_p$ M
Boltzmann constant ${ k }$ ML2T-2Θ-1

Dimensionless physical constants such as the fine-structure constant The speed of light in vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... For other uses of this word, see Length (disambiguation). ... Look up time in Wiktionary, the free dictionary. ... According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... This article or section is in need of attention from an expert on the subject. ... Plancks constant, denoted h, is a physical constant that is used to describe the sizes of quanta. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... Coulombs torsion balance In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrostatic force that one stationary, electrically charged object of small dimensions (ideally, a point source) exerts on another. ... This article is in need of attention. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... The elementary charge (symbol e or sometimes q) is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron. ... Properties The electron is a fundamental subatomic particle which carries a negative electric charge. ... // For alternative meanings see proton (disambiguation). ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... For other uses, see Temperature (disambiguation). ... Fundamental physical constant redirects here. ... The fine-structure constant or Sommerfeld fine-structure constant, usually denoted , is the fundamental physical constant characterizing the strength of the electromagnetic interaction. ...

$alpha stackrel{mathrm{def}}{=} frac{e^2}{hbar c (4 pi epsilon_0)} = frac{1}{137.03599911}$

cannot take on a different numerical value no matter what system of units are used. Judiciously choosing units can only normalize physical constants that have dimension. Since α is a fixed dimensionless number not equal to 1, it is not possible to define a system of natural units that will normalize all of the physical constants that comprise α. Any 3 of the 4 constants: c, $hbar$, e, or 4πε0, can be normalized (leaving the remaining physical constant to take on a value that is a simple function of α, alluding to the fundamental nature of the fine-structure constant) but not all 4.

## Planck units

Main article: Planck units
Quantity Expression Metric value
Length (L) $l_P = sqrt{frac{hbar G}{c^3}}$ 1.61609735×10-35 m
Mass (M) $m_P = sqrt{frac{hbar c}{G}}$ 21.7664598 μg
Time (T) $t_P = sqrt{frac{hbar G}{c^5}}$ 5.3907205×10-44 s
Electric charge (Q) $q_P = sqrt{hbar c (4 pi epsilon_0)}$ 1.87554573×10-18 C
Temperature (Θ) $T_P = sqrt{frac{hbar c^5}{G k^2}}$ 1.4169206×1032 K
$c = 1$
$G = 1$
$hbar = 1$
$frac{1}{4 pi epsilon_0} = 1$
$k = 1$
$e = sqrt{alpha}$

The physical constants that Planck units normalize are properties of free space and not properties (such as charge, mass, size or radius) of any object or elementary particle (that would have to be arbitrarily chosen). Being so, the Planck units are defined independently of the elementary charge which comes out to be the square root of the fine-structure constant, √α if measured in terms of Planck units. In Planck units a conceivable variation in the value of the dimensionless α would be considered to be due to a variation in the elementary charge.
In physics, Planck units are physical units of measurement defined exclusively in terms of the five universal physical constants shown in the table below in such a manner that all of these physical constants take on the numerical value of one when expressed in terms of these units. ... For other uses of this word, see Length (disambiguation). ... This article or section is in need of attention from an expert on the subject. ... Look up time in Wiktionary, the free dictionary. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... For other uses, see Temperature (disambiguation). ... In physics, free space is a concept of electromagnetic theory, corresponding roughly to the vacuum, the baseline state of the electromagnetic field, or the replacement for the electromagnetic aether. ... For the novel, see The Elementary Particles. ... The elementary charge (symbol e or sometimes q) is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron. ... The fine-structure constant or Sommerfeld fine-structure constant, usually denoted , is the fundamental physical constant characterizing the strength of the electromagnetic interaction. ...

## Stoney units

Quantity Expression
Length (L) $l_S = sqrt{frac{G e^2}{c^4 (4 pi epsilon_0)}}$
Mass (M) $m_S = sqrt{frac{e^2}{G (4 pi epsilon_0)}}$
Time (T) $t_S = sqrt{frac{G e^2}{c^6 (4 pi epsilon_0)}}$
Electric charge (Q) $q_S = e$
Temperature (Θ) $T_S = sqrt{frac{c^4 e^2}{G (4 pi epsilon_0) k^2}}$
$c = 1$
$G = 1$
$e = 1$
$frac{1}{4 pi epsilon_0} = 1$
$k = 1$
$hbar = frac{1}{alpha}$

Proposed by George Stoney in 1881. Stoney units fix the elementary charge and allow Planck's constant to float. They can be obtained from Planck units with the substitution: For other uses of this word, see Length (disambiguation). ... This article or section is in need of attention from an expert on the subject. ... Look up time in Wiktionary, the free dictionary. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... For other uses, see Temperature (disambiguation). ... George Johnstone Stoney (1826-1911) was an Irish physicist. ... The elementary charge (symbol e or sometimes q) is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... In physics, Planck units are physical units of measurement defined exclusively in terms of the five universal physical constants shown in the table below in such a manner that all of these physical constants take on the numerical value of one when expressed in terms of these units. ...

$hbar leftarrow alpha hbar = frac{e^2}{c (4 pi epsilon_0)}$.

This removes Planck's constant from the definitions and the value it takes on in Stoney units is the reciprocal of the fine-structure constant, 1/α. In Stoney units a conceivable variation in the value of the dimensionless α would be considered to be due to a variation in Planck's constant. The fine-structure constant or Sommerfeld fine-structure constant, usually denoted , is the fundamental physical constant characterizing the strength of the electromagnetic interaction. ...

## "Schrödinger" units

Quantity Expression
Length (L) $l_{psi} = sqrt{frac{hbar^4 G (4 pi epsilon_0)^3}{e^6}}$
Mass (M) $m_{psi} = sqrt{frac{e^2}{G (4 pi epsilon_0)}}$
Time (T) $t_{psi} = sqrt{frac{hbar^6 G (4 pi epsilon_0)^5}{e^{10}}}$
Electric charge (Q) $q_{psi} = e$
Temperature (Θ) $T_{psi} = sqrt{frac{e^{10}}{hbar^4 (4 pi epsilon_0)^5 G k^2}}$
$e = 1$
$G = 1$
$hbar = 1$
$frac{1}{4 pi epsilon_0} = 1$
$k = 1$
$c = frac{1}{alpha}$

The name coined by Michael Duff[1]. They can be obtained from Planck units with the substitution: For other uses of this word, see Length (disambiguation). ... This article or section is in need of attention from an expert on the subject. ... Look up time in Wiktionary, the free dictionary. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... For other uses, see Temperature (disambiguation). ... Sir Michael Duff, the bon vivant and society figure, was the son of Sir Robin Duff, 2nd Bt, of Vaynol, and his wife Lady Juliet Lowther, only child of the 4th Earl of Lonsdale and his wife Lady Gwladys Herbert (later Marchioness of Ripon). ... In physics, Planck units are physical units of measurement defined exclusively in terms of the five universal physical constants shown in the table below in such a manner that all of these physical constants take on the numerical value of one when expressed in terms of these units. ...

$c leftarrow alpha c = frac{e^2}{hbar (4 pi epsilon_0)}$.

This removes the speed of light from the definitions and the value it takes on in Schrödinger units is the reciprocal of the fine-structure constant, 1/α. In Schrödinger units a conceivable variation in the value of the dimensionless α would be considered to be due to a variation in the speed of light. The speed of light in vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... The fine-structure constant or Sommerfeld fine-structure constant, usually denoted , is the fundamental physical constant characterizing the strength of the electromagnetic interaction. ...

## Atomic units (Hartree)

Main article: Atomic units
Quantity Expression
Length (L) $l_A = frac{hbar^2 (4 pi epsilon_0)}{m_e e^2}$
Mass (M) $m_A = m_e$
Time (T) $t_A = frac{hbar^3 (4 pi epsilon_0)^2}{m_e e^4}$
Electric charge (Q) $q_A = e$
Temperature (Θ) $T_A = frac{m_e e^4}{hbar^2 (4 pi epsilon_0)^2 k}$
$e = 1$
$m_e = 1$
$hbar = 1$
$frac{1}{4 pi epsilon_0} = 1$
$k = 1$
$c = frac{1}{alpha}$

First proposed by Douglas Hartree to simplify the physics of the Hydrogen atom. Michael Duff[2] calls these "Bohr units". The unit energy in this system is the total energy of the electron in the first circular orbit of the Bohr atom and called the Hartree energy, Eh. The unit velocity is the velocity of that electron, the unit mass is the electron mass, me, and the unit length is the Bohr radius, $a_0 = 4 pi epsilon_0hbar^2/m_e e^2$. They can be obtained from "Schrödinger" units with the substitution: Atomic units (au) form a system of units convenient for electromagnetism, atomic physics, and quantum electrodynamics, especially when the focus is on the properties of electrons. ... For other uses of this word, see Length (disambiguation). ... This article or section is in need of attention from an expert on the subject. ... Look up time in Wiktionary, the free dictionary. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... For other uses, see Temperature (disambiguation). ... Douglas Rayner Hartree (March 27, 1897 - February 12, 1958) was an English mathematician and physicist most famous for the development of numerical analysis and its application to atomic physics. ... Depiction of a hydrogen atom showing the diameter as about twice the Bohr model radius. ... Sir Michael Duff, the bon vivant and society figure, was the son of Sir Robin Duff, 2nd Bt, of Vaynol, and his wife Lady Juliet Lowther, only child of the 4th Earl of Lonsdale and his wife Lady Gwladys Herbert (later Marchioness of Ripon). ... For other uses, see Electron (disambiguation). ... The Bohr model of the atom The Bohr Model is a physical model that depicts the atom as a small positively charged nucleus with electrons in orbit at different levels, similar in structure to the solar system. ... A Hartree (symbol Eh) is the atomic unit of energy and is named after physicist Douglas Hartree. ... Properties The electron is a fundamental subatomic particle which carries a negative electric charge. ... In the Bohr model of the structure of an atom, put forward by Niels Bohr in 1913, electrons orbit a central nucleus. ...

$G leftarrow alpha G left( frac{m_P}{m_e} right)^2 = frac{e^2}{4 pi epsilon_0 m_e^2}$.

This removes the speed of light (as well as the gravitational constant) from the definitions and the value it takes on in atomic units is the reciprocal of the fine-structure constant, 1/α. In atomic units a conceivable variation in the value of the dimensionless α would be considered to be due to a variation in the speed of light. The speed of light in vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... The fine-structure constant or Sommerfeld fine-structure constant, usually denoted , is the fundamental physical constant characterizing the strength of the electromagnetic interaction. ...

## Electronic system of units

Quantity Expression
Length (L) $l_e = frac{e^2}{c^2 m_e (4 pi epsilon_0)}$
Mass (M) $m_e = m_e$
Time (T) $t_e = frac{e^2}{c^3 m_e (4 pi epsilon_0)}$
Electric charge (Q) $q_e = e$
Temperature (Θ) $T_e = frac{m_e c^2}{k}$
$c = 1$
$e = 1$
$m_e = 1$
$frac{1}{4 pi epsilon_0} = 1$
$k = 1$
$hbar = frac{1}{alpha}$

Michael Duff[3] calls these "Dirac units". They can be obtained from Stoney units with the substitution: For other uses of this word, see Length (disambiguation). ... This article or section is in need of attention from an expert on the subject. ... Look up time in Wiktionary, the free dictionary. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... For other uses, see Temperature (disambiguation). ... Sir Michael Duff, the bon vivant and society figure, was the son of Sir Robin Duff, 2nd Bt, of Vaynol, and his wife Lady Juliet Lowther, only child of the 4th Earl of Lonsdale and his wife Lady Gwladys Herbert (later Marchioness of Ripon). ...

$G leftarrow alpha G left( frac{m_P}{m_e} right)^2 = frac{e^2}{4 pi epsilon_0 m_e^2}$.

They can be also obtained from Atomic units with the substitution: Atomic units (au) form a system of units convenient for electromagnetism, atomic physics, and quantum electrodynamics, especially when the focus is on the properties of electrons. ...

$hbar leftarrow alpha hbar = frac{e^2}{c (4 pi epsilon_0)}$.

Similarly to Stoney units, a conceivable variation in the value of α would be considered to be due to a variation in Planck's constant.

## Quantum electrodynamical system of units (Stille)

Quantity Expression
Length (L) $l_{mathrm{QED}} = frac{e^2}{c^2 m_p (4 pi epsilon_0)}$
Mass (M) $m_{mathrm{QED}} = m_p$
Time (T) $t_{mathrm{QED}} = frac{e^2}{c^3 m_p (4 pi epsilon_0)}$
Electric charge (Q) $q_{mathrm{QED}} = e$
Temperature (Θ) $T_{mathrm{QED}} = frac{m_p c^2}{k}$
$c = 1$
$e = 1$
$m_p = 1$
$frac{1}{4 pi epsilon_0} = 1$
$k = 1$
$hbar = frac{1}{alpha}$

Similar to the electronic system of units except that the proton mass is normalized rather that the electron mass. Also a conceivable variation in the value of α would be considered to be due to a variation in Planck's constant. For other uses of this word, see Length (disambiguation). ... This article or section is in need of attention from an expert on the subject. ... Look up time in Wiktionary, the free dictionary. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... For other uses, see Temperature (disambiguation). ... // For alternative meanings see proton (disambiguation). ... Properties The electron is a fundamental subatomic particle which carries a negative electric charge. ...

## Geometrized units

$c = 1$
$G = 1$

The geometrized unit system is not a completely defined or unique system. In this system, the base physical units are chosen so that the speed of light and the gravitational constant are set equal to unity leaving latitude to also set some other constant such as the Boltzmann constant and Coulomb force constant equal to unity: In physics, especially in the general theory of relativity, geometrized units or geometric units constitute a physical unit system in which all physical quantities are identified with geometric quantities such as areas, lengths, dimensionless numbers, path curvatures, or sectional curvatures. ... The speed of light in vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... Coulombs torsion balance In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrostatic force that one stationary, electrically charged object of small dimensions (ideally, a point source) exerts on another. ...

$k = 1$
$frac{1}{4 pi epsilon_0} = 1$

If Dirac's constant (also called the "reduced Planck's constant") is also set equal to unity, Plancks constant, denoted h, is a physical constant that is used to describe the sizes of quanta. ...

$hbar = 1$

then geometrized units are identical to Planck units. In physics, Planck units are physical units of measurement defined exclusively in terms of the five universal physical constants shown in the table below in such a manner that all of these physical constants take on the numerical value of one when expressed in terms of these units. ...

## N-body units

Quantity Expression
Length (R) $frac{1}{R} = frac{1}{N(N-1)} sum_{i=1}^{N} sum_{j=1}^{N} frac{1}{r_j-r_i}$
Mass (M) $M = sum_{i=1}^{N} m_i$
$M = 1$
$G = 1$
$R = 1$

N-body units are a completely self-contained system of units used for N-body simulations of self gravitating systems in astrophysics. In this system, the base physical units are chosen so that the total mass (M), the gravitational constant (G) and the virial radius (R) are set equal to unity. The underlying assumption is that the system of N objects (stars) satisfies the virial theorem. The consequence of standard N-body units is that the velocity dispersion of the system is $v = 1/sqrt{2}$ and that the dynamical -crossing- time scales as $t = 2sqrt{2}$. The first mention of standard N-body units was by Michel Hénon (1971) [4]. They were taken up by Haldan Cohn (1979) [5] and later widely advertised and generalized by Douglas Heggie and Robert Mathieu (1986) [6]. For other uses of this word, see Length (disambiguation). ... This article or section is in need of attention from an expert on the subject. ... The n-body problem is the problem of finding, given the initial positions, masses, and velocities of n bodies, their subsequent motions as determined by classical mechanics, i. ... According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... In mechanics, the virial theorem provides a general equation relating the average total kinetic energy of a system with its average total potential energy , where angle brackets represent the average of the enclosed quantity. ...

A set of fundamental units is a set of units for physical quantities from which every other unit can be generated. ... Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ... In physics, a physical constant is a physical quantity of a value that is generally believed to be both universal in nature and not believed to change in time. ...

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