Natural units

 ( Metrology ) 

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 may be set to 1, and it may then be omitted, equating mass and energy directly rather than using 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).

! Quantity
! Planck
! Stoney
! Atomic
! Particle and atomic physics
! Strong
! Schrödinger
     

! Defining constants

! Speed of light $c$

! Reduced Planck constant $\backslash hbar$

! Elementary charge $e$

! Vacuum permittivity $\backslash varepsilon\_0$

! Gravitational constant $G$

• is the fine-structure constant ( ≈ 0.007297)
• ≈
• ≈
• A dash (—) indicates where the system is not sufficient to express the quantity.

! Quantity
! Expression
! Approx.
metric value
     

The Stoney unit system uses the following defining constants:

, , , ,

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.

! Quantity
! Expression
! Approx.
metric value
     

The Planck unit system uses the following defining constants:

, , , ,

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: and are part of the structure of spacetime in general relativity, and 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 , , , and _B to arrive at natural units for length, time, mass, and temperature, but no electromagnetic units.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 would be implicit in the list of defining constants, giving a charge unit . The Planck system of units is now understood to use the reduced Planck constant, , in place of the Planck constant, .Tomilin, K. A., 1999, " Natural Systems of Units: To the Centenary Anniversary of the Planck System ", 287–296.

! Quantity
! Expression
! Approx.
metric value
     

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

, , , .

, .

The geometrized unit system,

! Quantity
! Expression
! Metric value
     

The atomic unit system uses the following defining constants:

, , , .

The atomic units were first proposed by Douglas Hartree and are designed to simplify atomic and molecular physics and chemistry, especially the hydrogen atom.

! Quantity
! Expression
! Metric value
     

This natural unit system, used only in the fields of particle and atomic physics, uses the following defining constants:

, , , ,

The vacuum permittivity ₀ is implicitly used as a nondimensionalization constant, as is evident from the physicists' expression for the fine-structure constant, written , which may be compared to the corresponding expression in SI: .

! Quantity
! Expression
! Metric value
     

Defining constants:

, , .

Here, 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". . Further see.

• Anthropic units
• Astronomical system of units
• Dimensionless physical constant
• International System of Units
• N-body units
• Outline of metrology and measurement
• Unit of measurement

• The NIST website (NIST) is a convenient source of data on the commonly recognized constants.
• K.A. Tomilin: NATURAL SYSTEMS OF UNITS; To the Centenary Anniversary of the Planck System A comparative overview/tutorial of various systems of natural units having historical use.
• Pedagogic Aides to Quantum Field Theory Click on the link for Chap. 2 to find an extensive, simplified introduction to natural units.
• Natural System Of Units In General Relativity (PDF), by Alan L. Myers (University of Pennsylvania). Equations for conversions from natural to SI units.