In physics, natural units are physical units of measurement based only on universal physical constants. For example, the elementary charge e is a natural unit of electric charge, and the speed of light c is a natural unit of speed. A purely natural system of units has all of its units defined in this way, and usually such that the numerical values of the selected physical constants in terms of these units are exactly dimensionless 1. These constants are then typically 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 fundamental physical constants, such 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. Contents 1 Introduction 1.1 Summary table 2 Notation and use 2.1 Advantages and disadvantages 3 Choosing constants to normalize 4 Electromagnetism units 5 Systems of natural units 5.1 Planck units
5.2 Stoney units
5.3 Atomic units
5.4
6 See also 7 Notes and references 8 External links Introduction[edit]
Quantity / Symbol Planck (with Gauss) Stoney Hartree Rydberg "Natural" (with L-H) "Natural" (with Gauss)
c displaystyle c, 1 displaystyle 1, 1 displaystyle 1, 1 α
displaystyle frac 1 alpha 2 α
displaystyle frac 2 alpha 1 displaystyle 1, 1 displaystyle 1,
ℏ = h 2 π displaystyle hbar = frac h 2pi 1 displaystyle 1, 1 α
displaystyle frac 1 alpha 1 displaystyle 1, 1 displaystyle 1, 1 displaystyle 1, 1 displaystyle 1, Elementary charge e displaystyle e, α displaystyle sqrt alpha , 1 displaystyle 1, 1 displaystyle 1, 2 displaystyle sqrt 2 , 4 π α displaystyle sqrt 4pi alpha α displaystyle sqrt alpha Josephson constant K J = e π ℏ displaystyle K_ text J = frac e pi hbar , α π displaystyle frac sqrt alpha pi , α π displaystyle frac alpha pi , 1 π displaystyle frac 1 pi , 2 π displaystyle frac sqrt 2 pi , 4 α π displaystyle sqrt frac 4alpha pi , α π displaystyle frac sqrt alpha pi , von Klitzing constant R K = h e 2 displaystyle R_ text K = frac h e^ 2 , 2 π α displaystyle frac 2pi alpha , 2 π α displaystyle frac 2pi alpha , 2 π displaystyle 2pi , π displaystyle pi , 1 2 α displaystyle frac 1 2alpha 2 π α displaystyle frac 2pi alpha Gravitational constant G displaystyle G, 1 displaystyle 1, 1 displaystyle 1, α G α displaystyle frac alpha _ text G alpha , 8 α G α displaystyle frac 8alpha _ text G alpha , α G m e 2 displaystyle frac alpha _ text G m_ text e ^ 2 , α G m e 2 displaystyle frac alpha _ text G m_ text e ^ 2 , Boltzmann constant k B displaystyle k_ text B , 1 displaystyle 1, 1 displaystyle 1, 1 displaystyle 1, 1 displaystyle 1, 1 displaystyle 1, 1 displaystyle 1, Electron rest mass m e displaystyle m_ text e , α G displaystyle sqrt alpha _ text G , α G α displaystyle sqrt frac alpha _ text G alpha , 1 displaystyle 1, 1 2 displaystyle frac 1 2 , 511 keV displaystyle 511 text keV 511 keV displaystyle 511 text keV where: α is the fine-structure constant, (e/qPlanck)2 ≈ 0.007297, αG is the gravitational coupling constant, (me/mPlanck)2 ≈ 6955175200000000000♠1.752×10−45. Notation and use[edit]
Simplified equations: By setting constants to 1, equations containing
those constants appear more compact and in some cases may be simpler
to understand. For example, the special relativity equation E2 = p2c2
+ m2c4 appears somewhat complicated, but the natural units version, E2
= p2 + m2, appears simpler.
Physical interpretation: Natural unit systems automatically subsume
dimensional analysis. For example, in Planck units, the units are
defined by properties of quantum mechanics and gravity. Not
coincidentally, the Planck unit of length is approximately the
distance at which quantum gravity effects become important. Likewise,
atomic units are based on the mass and charge of an electron, and not
coincidentally the atomic unit of length is the
Choosing constants to normalize[edit]
Out of the many physical constants, the designer of a system of
natural unit systems must choose a few of these constants to normalize
(set equal to 1). It is not possible to normalize just any set of
constants. For example, the mass of a proton and the mass of an
electron cannot both be normalized: if the mass of an electron is
defined to be 1, then the mass of a proton has to be approximately
1836. In a less trivial example, the fine-structure constant, α ≈
1/137, cannot be set to 1, at least not independently, because it is a
dimensionless number defined in terms of other quantities, some of
which one may want to set to unity as well. The fine-structure
constant is related to other fundamental constants through α =
kee2/ħc, where ke is the
Of these, Lorentz–Heaviside is somewhat more common,[2] mainly
because
e = √4παħc (Lorentz–Heaviside), e = √αħc (Gaussian) where ħ is the reduced Planck constant, c is the speed of light, and
α ≈ 1/137 is the fine-structure constant.
In a natural unit system where c = 1,
Quantity Expression Metric value Name
l P = ℏ G c 3 displaystyle l_ text P = sqrt hbar G over c^ 3 6965161600000000000♠1.616×10−35 m Planck length
m P = ℏ c G displaystyle m_ text P = sqrt hbar c over G 6992217600000000000♠2.176×10−8 kg Planck mass
t P = ℏ G c 5 displaystyle t_ text P = sqrt hbar G over c^ 5 6956539120000000000♠5.3912×10−44 s Planck time T P = ℏ c 5 G k B 2 displaystyle T_ text P = sqrt frac hbar c^ 5 G k_ text B ^ 2 7032141700000000000♠1.417×1032 K Planck temperature
q P = ℏ c k e displaystyle q_ text P = sqrt hbar c over k_ text e 6982187600000000000♠1.876×10−18 C Planck charge
c = ħ = G = ke = kB = 1, where c is the speed of light, ħ is the reduced Planck constant, G is
the gravitational constant, ke is the
Stoney units[edit] Main article: Stoney units Quantity Expression Metric value
l S = G k e e 2 c 4 displaystyle l_ text S = sqrt frac Gk_ text e e^ 2 c^ 4 6964138100000000000♠1.381×10−36 m
m S = k e e 2 G displaystyle m_ text S = sqrt frac k_ text e e^ 2 G 6991185900000000000♠1.859×10−9 kg
t S = G k e e 2 c 6 displaystyle t_ text S = sqrt frac Gk_ text e e^ 2 c^ 6 6955460500000000000♠4.605×10−45 s T S = c 4 k e e 2 G k B 2 displaystyle T_ text S = sqrt frac c^ 4 k_ text e e^ 2 G k_ text B ^ 2 7031121000000000000♠1.210×1031 K
q S = e displaystyle q_ text S =e 6981160200000000000♠1.602×10−19 C
c = G = ke = e = kB = 1, where c is the speed of light, G is the gravitational constant, ke is
the
Atomic units[edit] Main article: Atomic units Quantity Expression (Hartree atomic units) Metric value (Hartree atomic units)
l A = ℏ 2 ( 4 π ϵ 0 ) m e e 2 displaystyle l_ text A = frac hbar ^ 2 (4pi epsilon _ 0 ) m_ text e e^ 2 6989529200000000000♠5.292×10−11 m
m A = m e
displaystyle m_ text A =m_ text e 6969910900000000000♠9.109×10−31 kg
t A = ℏ 3 ( 4 π ϵ 0 ) 2 m e e 4 displaystyle t_ text A = frac hbar ^ 3 (4pi epsilon _ 0 )^ 2 m_ text e e^ 4 6983241900000000000♠2.419×10−17 s T A = m e e 4 ℏ 2 ( 4 π ϵ 0 ) 2 k B displaystyle T_ text A = frac m_ text e e^ 4 hbar ^ 2 (4pi epsilon _ 0 )^ 2 k_ text B 7005315800000000000♠3.158×105 K
q A = e displaystyle q_ text A =e 6981160200000000000♠1.602×10−19 C There are two types of atomic units, closely related. Hartree atomic units: e = me = ħ = ke = kB = 1 c = 1/α Rydberg atomic units:[5] e/√2 = 2me = ħ = ke = kB = 1 c = 2/α Coulomb's constant is generally expressed as ke = 1/4πε0. 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, and
are more common than the Rydberg units.
The units are designed especially to characterize the behavior of an
electron in the ground state of a hydrogen atom. For example, using
the Hartree convention, in the
Quantity Expression Metric value
l Q C D = ℏ m p c displaystyle l_ mathrm QCD = frac hbar m_ text p c 6984210300000000000♠2.103×10−16 m
m Q C D = m p
displaystyle m_ mathrm QCD =m_ text p 6973167300000000000♠1.673×10−27 kg
t Q C D = ℏ m p c 2 displaystyle t_ mathrm QCD = frac hbar m_ text p c^ 2 6975701500000000000♠7.015×10−25 s T Q C D = m p c 2 k B displaystyle T_ mathrm QCD = frac m_ text p c^ 2 k_ text B 7013108900000000000♠1.089×1013 K
q Q C D = e 4 π α displaystyle q_ mathrm QCD = frac e sqrt 4pi alpha (L–H) 6981529100000000000♠5.291×10−19 C q Q C D = e α displaystyle q_ mathrm QCD = frac e sqrt alpha (G) 6982187600000000000♠1.876×10−18 C c = mp = ħ = kB = 1 The
"Natural units" (particle physics and cosmology)[edit] Unit Metric value Derivation 1 eV−1 of length 6993196999999999999♠1.97×10−7 m = ℏ c 1 eV displaystyle = frac hbar c 1, text eV 1 eV of mass 6964178000000000000♠1.78×10−36 kg = 1 eV c 2 displaystyle = frac 1, text eV c^ 2 1 eV−1 of time 6984658000000000000♠6.58×10−16 s = ℏ 1 eV displaystyle = frac hbar 1, text eV 1 eV of temperature 7004116000000000000♠1.16×104 K = 1 eV k B ⋅ 2 f displaystyle = frac 1, text eV k_ text B cdot frac 2 f with f = 2 displaystyle f=2 1 unit of electric charge (L–H) 6981529000000000000♠5.29×10−19 C = e 4 π α displaystyle = frac e sqrt 4pi alpha 1 unit of electric charge (G) 6982188000000000000♠1.88×10−18 C = e α displaystyle = frac e sqrt alpha In particle physics and cosmology, the phrase "natural units" generally means:[9][10] ħ = c = kB = 1. where ħ is the reduced Planck constant, c is the speed of light, and
kB is the Boltzmann constant.
Both
1.0 cm = 1.0 cm/ħc ≈ 51000 eV−1 Geometrized units[edit] Main article: Geometrized unit system c = G = 1 The geometrized unit system, used in general relativity, is not a
completely defined 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. Other units may be treated however desired. Planck
units and
Anthropic units Dimensional analysis Dimensionless physical constant SI base unit N-body units Physical constant Units of measurement Notes and references[edit] ^ Kowalski, Ludwik, 1986, "A Short History of the SI Units in
Electricity, Archived 2009-04-29 at the Wayback Machine." The Physics
Teacher 24(2): 97–99. Alternate web link (subscription required)
^ Walter Greiner; Ludwig Neise; Horst Stöcker (1995). Thermodynamics
and Statistical Mechanics. Springer-Verlag. p. 385.
ISBN 978-0-387-94299-5.
^ See
External links[edit] The
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