In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge e may be used as a natural unit of electric charge, and the speed of light c may be used as a natural unit of speed. A purely natural system of units has all of its units defined such that each of these can be expressed as a product of powers of defining physical constants.
Through nondimensionalization, physical quantities may then be redefined so that the defining constants can be omitted from mathematical expressions of physical laws, and while this has the apparent advantage of simplicity, it may entail a loss of clarity due to the loss of information for dimensional analysis. It precludes the interpretation of an expression in terms of constants, such as e and c, unless it is known which units (in dimensionful units) the expression is supposed to have. In this case, the reinsertion of the correct powers of e, c, etc., can be uniquely determined.
Quantity | Expression | Metric value |
---|---|---|
Length (L) | 1.616×10^{−35} m^{[1]} | |
Mass (M) | 2.176×10^{−8} kg^{[2]} | |
Time (T) | 5.391×10^{−44} s^{[3]} | |
Temperature (Θ) | 1.417×10^{32} K^{[4]} |
The Planck unit system uses the following defining constants:
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.^{[citation needed]}
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]}
Quantity | Expression | Metric value |
---|---|---|
Length (L) | 1.38068×10^{−36} m | |
Mass (M) | 1.85921×10^{−9} kg | |
Time (T) | 4.60544×10^{−45} s | |
Electric charge (Q) | 1.60218×10^{−19} C |
The Stoney unit system uses the following defining constants:
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. He presented the idea in a lecture entitled "On the Physical Units of Nature" delivered to the British Association in 1874.^{[7]} Stoney units did not consider the Planck constant, which was discovered only after Stoney's proposal.
Stoney units are rarely used in modern physics for calculations, but they are of historical interest.
Quantity | Expression | Metric value |
---|---|---|
Length (L) | 5.292×10^{−11} m | |
Mass (M) | 9.109×10^{−31} kg | |
Time (T) | 2.419×10^{−17} s | |
Electric charge (Q) | 1.602×10^{−19} C |
The Hartree atomic unit system uses the following defining constants:
The Coulomb constant, k_{e}, is generally expressed as 1/4πε_{0} when working with this system.
These units are designed to simplify atomic and molecular physics and chemistry, especially the hydrogen atom, and are widely used in these fields. The Hartree units were first proposed by Douglas Hartree.
The units are designed especially to characterize the behavior of an electron in the ground state of a hydrogen atom. For example, in Hartree atomic units, in the Bohr model of the hydrogen atom an electron in the ground state has orbital radius (the Bohr radius) a_{0} = 1 l_{A}, orbital velocity = 1 l_{A}⋅t_{A}^{−1}, angular momentum = 1 m_{A}⋅l_{A}⋅t_{A}^{−1}, ionization energy = 1/2 m_{A}⋅l_{A}^{2}⋅t_{A}^{−2}, etc.
The unit of energy is called the Hartree energy in the Hartree system. The speed of light is relatively large in Hartree atomic units (c = 1/α l_{A}⋅t_{A}^{−1} ≈ 137 l_{A}⋅t_{A}^{−1}) since an electron in hydrogen tends to move much more slowly than the speed of light. The gravitational constant is extremely small in atomic units (G ≈ 10^{−45} m_{A}^{−1}⋅l_{A}^{3}⋅t_{A}^{−2}), which is due to the gravitational force between two electrons being far weaker than the Coulomb force between them.
A less commonly used closely related system is the system of Rydberg atomic units, in which e^{2}/2, 2m_{e}, ħ, k_{e} are used as the defining constants, with resulting units l_{R} = a_{0} = (4πε_{0})ħ^{2}/m_{e}e^{2}, t_{R} = 2(4πε_{0})^{2}ħ^{3}/m_{e}e^{4}, m_{R} = 2m_{e}, q_{R} = e⁄√2.^{[8]}
Quantity | Expression | Metric value |
---|---|---|
Length (L) | 3.862×10^{−13} m^{[9]} | |
Mass (M) | 9.109×10^{−31} kg^{[10]} | |
Time (T) | 1.288×10^{−21} s^{[11]} | |
Electric charge (Q) | 5.291×10^{−19} C |
This natural unit system, used only in the fields of particle and atomic physics, uses the following defining constants:^{[12]}^{: 509 }
where c is the speed of light, m_{e} is the electron mass, ħ is the reduced Planck constant, and ε_{0} is the vacuum permittivity.
The vacuum permittivity ε_{0} is implicitly used as a nondimensionalization constant, as is evident from the physicists' expression for the fine-structure constant, written α = e^{2}/(4π),^{[13]}^{[14]} which may be compared to the same expression in SI: α = e^{2}/(4πε_{0}ħc).^{[15]}^{: 128 }
Quantity | Expression | Metric value |
---|---|---|
Length (L) | 2.103×10^{−16} m | |
Mass (M) | 1.673×10^{−27} kg | |
Time (T) | 7.015×10^{−25} s | |
Electric charge (Q) | 5.291×10^{−19} C |
Defining constants:
Here, m_{p} is the proton rest mass. Strong units, also called quantum chromodynamics (QCD) 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".^{[16]}
Quantity | Expression | Metric value |
---|---|---|
Length (L) | 2.593×10^{−32} m^{[17]} | |
Mass (M) | 1.859×10^{−9} kg^{[17]} | |
Time (T) | 1.185×10^{−38} s^{[17]} | |
Electric charge (Q) | 1.602×10^{−19} C^{[18]} | |
Temperature (Θ) | 6.445×10^{26} K^{[17]} |
A system of units in which and is seldom mentioned in literature as Schrödinger's system of units (after Austrian physicist Erwin Schrödinger).^{[19]}^{[20]}^{[17]}
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.^{[21]}
Defining constants:
The geometrized unit system, used in general relativity, is an incompletely defined system. In this system, the base physical units are chosen so that the speed of light and the gravitational constant are coherent units and often used for nondimensionalization. Other units may be treated however desired. Planck units and Stoney units are examples of geometrized unit systems.
Quantity / Symbol | Planck | Stoney | Hartree | particle and atomic physics | QCD | Schrödinger |
---|---|---|---|---|---|---|
Defining constants | , , , | , , , | , , , | , , , | , , , | , , , |
Speed of light |
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Reduced Planck constant |
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Elementary charge |
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Vacuum permittivity |
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Gravitational constant |
where:
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, c_{ψ} changes in proportion to c but the speed of light in Planck units, c_{P} 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, c_{P} and c_{ψ}). Whether or not c changes is unambiguous because the measuring system has been defined.