Physical constant

A physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. It can be contrasted with a mathematical constant, which is a fixed numerical value but does not directly involve any physical measurement.

There are many physical constants in science, some of the most widely recognized being the speed of light in vacuum c, the gravitational constant G, Planck's constant h, the electric constant ε₀, and the elementary charge e. Physical constants can take many dimensional forms: the speed of light signifies a maximum speed limit of the universe and is expressed dimensionally as length divided by time; while the fine-structure constant α, which characterizes the strength of the electromagnetic interaction, is dimensionless.

1 Dimensionful and dimensionless physical constants
2 How constant are the physical constants?
3 Anthropic principle
4 Table of universal constants
5 Table of electromagnetic constants
6 Table of atomic and nuclear constants
7 Table of physico-chemical constants
8 Table of adopted values
9 Natural units
10 Notes
11 References
12 See also
13 External links

Dimensionful and dimensionless physical constants

Whereas the values of physical constants do not depend on the unit system used, the numerical values of dimensionful physical constants do depend on the unit used. Therefore, these numerical values (such as 299,792,458 for the constant speed of light c expressed in units of meters per second) are not values that a theory of physics can be expected to predict.

Ratios of like-dimensioned physical constants do not depend on unit systems in this way (the units cancel), so they are pure (dimensionless) numbers whose values a future theory of physics could conceivably hope to predict. Additionally, all equations describing laws of physics can be expressed without dimensional physical constants via a process known as nondimensionalisation, but the dimensionless constants will remain. Thus, theoretical physicists tend to regard these dimensionless quantities as fundamental physical constants.

However, the phrase fundamental physical constant is also used in other ways. For example, the National Institute of Standards and Technology^[1] uses it to refer to any universal physical quantity believed to be constant, such as the speed of light, c, and the gravitational constant G.

The fine-structure constant α is probably the best known dimensionless fundamental physical constant. Many attempts have been made to derive its value (currently measured at about 1/137.035999) from theory, but so far none have succeeded. The same holds for the dimensionless ratios of masses of fundamental particles (the most apparent is m_p/m_e, approximately 1836.152673). With the development of quantum chemistry in the 20th century, however, a vast number of previously inexplicable dimensionless physical constants were successfully computed from theory. As such, some theoretical physicists still hope for continued progress in explaining the values of dimensionless physical constants.

It is known that the universe would be very different if these constants took values significantly different from those we observe. For example, a few percent change in the value of the fine structure constant would be enough to eliminate stars like our Sun. This has prompted attempts at anthropic explanations of the dimensionless physical constants.

How constant are the physical constants?

Beginning with Paul Dirac in 1937, some scientists have speculated that physical constants may actually decrease in proportion to the age of the universe. Scientific experiments have not yet pinpointed any definite evidence that this is the case, although they have placed upper bounds on the maximum possible relative change per year at very small amounts (roughly 10⁻⁵ per year for the fine structure constant α and 10⁻¹¹ for the gravitational constant G).

It is currently disputed^[2]^[3] whether any changes in dimensional physical constants such as G, c, ħ, or ε₀ are operationally meaningful;^[4] however, a sufficient change in a dimensionless constant such as α is generally agreed to be something that would definitely be noticed. If a measurement indicated that a dimensional physical constant had changed, this would be the result or interpretation of a more fundamental dimensionless constant changing, which is the salient metric. From John D. Barrow 2002:

[An] important lesson we learn from the way that pure numbers like α define the world is what it really means for worlds to be different. The pure number we call the fine structure constant and denote by α is a combination of the electron charge, e, the speed of light, c, and Planck's constant, h. At first we might be tempted to think that a world in which the speed of light was slower would be a different world. But this would be a mistake. If c, h, and e were all changed so that the values they have in metric (or any other) units were different when we looked them up in our tables of physical constants, but the value of α remained the same, this new world would be observationally indistinguishable from our world. The only thing that counts in the definition of worlds are the values of the dimensionless constants of Nature. If all masses were doubled in value you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged.

Anthropic principle

Main article: Anthropic principle

Some physicists have explored the notion that if the dimensionless physical constants had sufficiently different values, our universe would be so radically different that intelligent life would probably not have emerged, and that our universe therefore seems to be fine-tuned for intelligent life. The anthropic principle states a logical truism: the fact of our existence as intelligent beings who can measure physical constants requires those constants to be such that beings like us can exist. There are a variety of interpretations of the apparent fine-tuning, including a divine creator (the apparent fine-tuning is actual and intentional), or that ours is one universe of many in a multiverse (e.g. the Many-worlds interpretation of quantum mechanics), or even that a universe without the capacity for conscious beings cannot exist.

Table of universal constants

	Symbol	Value^[5]	Relative Standard Uncertainty
speed of light in vacuum	$c \,$	299 792 458 m·s⁻¹	defined
Newtonian constant of gravitation	$G \,$	6.67428(67)×10⁻¹¹ m³·kg⁻¹·s⁻²	1.0 × 10⁻⁴
Planck constant	$h \,$	6.626 068 96(33) × 10⁻³⁴ J·s	5.0 × 10⁻⁸
reduced Planck constant	$\hbar = h / (2 \pi)$	1.054 571 628(53) × 10⁻³⁴ J·s	5.0 × 10⁻⁸

Table of electromagnetic constants

	Symbol	Value^[5] (SI units)	Relative Standard Uncertainty
magnetic constant (vacuum permeability)	$\mu_0 \,$	4π × 10⁻⁷ N·A⁻² = 1.256 637 061... × 10⁻⁶ N·A⁻²	defined
electric constant (vacuum permittivity)	$\varepsilon_0 = 1/(\mu_0 c^2) \,$	8.854 187 817... × 10⁻¹² F·m⁻¹	defined
characteristic impedance of vacuum	$Z_0 = \mu_0 c \,$	376.730 313 461... Ω	defined
Coulomb's constant	$k_e = 1 / 4\pi\varepsilon_0 \,$	8.987 551 787... × 10⁹ N·m²·C⁻²	defined
elementary charge	$e \,$	1.602 176 487(40) × 10⁻¹⁹ C	2.5 × 10⁻⁸
Bohr magneton	$\mu_B = e \hbar / 2 m_e$	9.274 009 15(23) × 10⁻²⁴ J·T⁻¹	2.5 × 10⁻⁸
conductance quantum	$G_0 = 2 e^2 / h \,$	7.748 091 700 4(53) × 10⁻⁵ S	6.8 × 10⁻¹⁰
inverse conductance quantum	$G_0^{-1} = h / 2 e^2 \,$	12 906.403 778 7(88) Ω	6.8 × 10⁻¹⁰
Josephson constant	$K_J = 2 e / h \,$	4.835 978 91(12) × 10¹⁴ Hz·V⁻¹	2.5 × 10⁻⁸
magnetic flux quantum	$\phi_0 = h / 2 e \,$	2.067 833 667(52) × 10⁻¹⁵ Wb	2.5 × 10⁻⁸
nuclear magneton	$\mu_N = e \hbar / 2 m_p$	5.050 783 43(43) × 10⁻²⁷ J·T⁻¹	8.6 × 10⁻⁸
von Klitzing constant	$R_K = h / e^2 \,$	25 812.807 557(18) Ω	6.8 × 10⁻¹⁰

Table of atomic and nuclear constants

	Symbol	Value^[5] (SI units)	Relative Standard Uncertainty
Bohr radius	$a_0 = \alpha / 4 \pi R_\infin \,$	5.291 772 108(18) × 10⁻¹¹ m	3.3 × 10⁻⁹
classical electron radius	$r_e = e^2 / 4\pi\varepsilon_0 m_e c^2\,$	2.817 940 289 4(58) × 10⁻¹⁵ m	2.1 × 10⁻⁹
electron mass	$m_e \,$	9.109 382 15(45) × 10⁻³¹ kg	5.0 × 10⁻⁸
Fermi coupling constant	$G_F / (\hbar c)^3$	1.166 39(1) × 10⁻⁵ GeV⁻²	8.6 × 10⁻⁶
fine-structure constant	$\alpha = \mu_0 e^2 c / (2 h) = e^2 / (4 \pi \varepsilon_0 \hbar c) \,$	7.297 352 537 6(50) × 10⁻³	6.8 × 10⁻¹⁰
Hartree energy	$E_h = 2 R_\infin h c \,$	4.359 744 17(75) × 10⁻¹⁸ J	1.7 × 10⁻⁷
proton mass	$m_p \,$	1.672 621 637(83) × 10⁻²⁷ kg	5.0 × 10⁻⁸
quantum of circulation	$h / 2 m_e \,$	3.636 947 550(24) × 10⁻⁴ m² s⁻¹	6.7 × 10⁻⁹
Rydberg constant	$R_\infin = \alpha^2 m_e c / 2 h \,$	10 973 731.568 525(73) m⁻¹	6.6 × 10⁻¹²
Thomson cross section	$(8 \pi / 3)r_e^2$	6.652 458 73(13) × 10⁻²⁹ m²	2.0 × 10⁻⁸
weak mixing angle	$\sin^2 \theta_W = 1 - (m_W / m_Z)^2 \,$	0.222 15(76)	3.4 × 10⁻³

Table of physico-chemical constants

		Symbol	Value^[5] (SI units)	Relative Standard Uncertainty
atomic mass unit (unified atomic mass unit)		$m_u = 1 \ u \,$	1.660 538 86(28) × 10⁻²⁷ kg	1.7 × 10⁻⁷
Avogadro's number		$N_A, L \,$	6.022 141 5(10) × 10²³ mol⁻¹	1.7 × 10⁻⁷
Boltzmann constant		$k = k_B = R / N_A \,$	1.380 650 4(24) × 10⁻²³ J·K⁻¹	1.8 × 10⁻⁶
Faraday constant		$F = N_A e \,$	96 485.338 3(83)C·mol⁻¹	8.6 × 10⁻⁸
first radiation constant		$c_1 = 2 \pi h c^2 \,$	3.741 771 18(19) × 10⁻¹⁶ W·m²	5.0 × 10⁻⁸
first radiation constant	for spectral radiance	$c_{1L} \,$	1.191 042 82(20) × 10⁻¹⁶ W·m² sr⁻¹	1.7 × 10⁻⁷
Loschmidt constant	at $T$ =273.15 K and $p$ =101.325 kPa	$n_0 = N_A / V_m \,$	2.686 777 3(47) × 10²⁵ m⁻³	1.8 × 10⁻⁶
gas constant		$R \,$	8.314 472(15) J·K⁻¹·mol⁻¹	1.7 × 10⁻⁶
molar Planck constant		$N_A h \,$	3.990 312 716(27) × 10⁻¹⁰ J·s·mol⁻¹	6.7 × 10⁻⁹
molar volume of an ideal gas	at $T$ =273.15 K and $p$ =100 kPa	$V_m = R T / p \,$	2.271 098 1(40) × 10⁻² m³·mol⁻¹	1.7 × 10⁻⁶
molar volume of an ideal gas	at $T$ =273.15 K and $p$ =101.325 kPa	$V_m = R T / p \,$	2.241 399 6(39) × 10⁻² m³·mol⁻¹	1.7 × 10⁻⁶
Sackur-Tetrode constant	at $T$ =1 K and $p$ =100 kPa	$S_0 / R = \frac{5}{2}$ $+ \ln\left[ (2\pi m_u k T / h^2)^{3/2} k T / p \right]$	−1.151 704 7(44)	3.8 × 10⁻⁶
Sackur-Tetrode constant	at $T$ =1 K and $p$ =101.325 kPa		−1.164 867 7(44)	3.8 × 10⁻⁶
second radiation constant		$c_2 = h c / k \,$	1.438 775 2(25) × 10⁻² m·K	1.7 × 10⁻⁶
Stefan–Boltzmann constant		$\sigma = (\pi^2 / 60) k^4 / \hbar^3 c^2$	5.670 400(40) × 10⁻⁸ W·m⁻²·K⁻⁴	7.0 × 10⁻⁶
Wien displacement law constant		$b = (h c / k) / \,$ 4.965 114 231...	2.897 768 5(51) × 10⁻³ m·K	1.7 × 10⁻⁶

Table of adopted values

Quantity		Symbol	Value (SI units)	Relative Standard Uncertainty
conventional value of Josephson constant^[6]		$K_{J-90} \,$	4.835 979 × 10¹⁴ Hz·V⁻¹	defined
conventional value of von Klitzing constant^[7]		$R_{K-90} \,$	25 812.807 Ω	defined
molar mass	constant	$M_u = M(\,^{12}\mbox{C}) / 12$	1 × 10⁻³ kg·mol⁻¹	defined
molar mass	of carbon-12	$M(\,^{12}\mbox{C}) = N_A m(\,^{12}\mbox{C})$	1.2 × 10⁻² kg·mol⁻¹	defined
standard acceleration of gravity (gee, free-fall on Earth)		$g_n \,\!$	9.806 65 m·s⁻²	defined
standard atmosphere		$\mbox{atm} \,$	101 325 Pa	defined

Natural units

Using dimensional analysis, it is possible combine fundamental physical constants to produce basic units of measurement. Depending on the choice and arrangement of constants used, the resulting natural units may have physical meaning. For example, Planck units, presented below, use $c, G, \hbar , \varepsilon_0$ and $k$ to derive constants relevant to unified theories, including quantum gravity.

	Dimension	Expression	Value^[8] (SI units)
Planck length	Length (L)	$l_\text{P} = \sqrt{\frac{\hbar G}{c^3}}$	1.616 252(81) × 10⁻³⁵ m
Planck mass	Mass (M)	$m_\text{P} = \sqrt{\frac{\hbar c}{G}}$	2.176 44(11) × 10⁻⁸ kg
Planck time	Time (T)	$t_\text{P} = \frac{l_\text{P}}{c} = \frac{\hbar}{m_\text{P}c^2} = \sqrt{\frac{\hbar G}{c^5}}$	5.391 24(27) × 10⁻⁴⁴ s
Planck charge	Electric charge (Q)	$q_\text{P} = \sqrt{4 \pi \varepsilon_0 \hbar c}$	1.875 545 870(47) × 10⁻¹⁸ C
Planck temperature	Temperature (Θ)	$T_\text{P} = \frac{m_\text{P} c^2}{k_B} = \sqrt{\frac{\hbar c^5}{G k_B^2}}$	1.416 785(71) × 10³² K

Notes

↑ Latest (2006) Values of the Constants; NIST, 2006.
↑ Duff, Michael J. "Comment on time-variation of fundamental constants." High Energy Physics - Theory, 2004.
↑ Duff, M. J.; Okun, L. B.; Veneziano, G. "Trialogue on the number of fundamental constants." Classical Physics, 2002.
↑ c, and ε₀ now are defined numerical values, independent of experiment, so observations now are trained elsewhere, for example, upon a changing value of the meter.
↑ ^5.0 ^5.1 ^5.2 ^5.3 The values are given in the so-called concise form; the number in brackets is the standard uncertainty, which is the value multiplied by the relative standard uncertainty.
↑ This is the value adopted internationally for realizing representations of the volt using the Josephson effect.
↑ This is the value adopted internationally for realizing representations of the ohm using the quantum Hall effect.
↑ Fundamental Physical Constants from NIST.

References

Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). "CODATA Recommended Values of the Fundamental Physical Constants: 2006". Rev. Mod. Phys. 80: 633–730. doi:10.1103/RevModPhys.80.633. http://physics.nist.gov/cuu/Constants/codata.pdf.
Barrow, John D., The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe. Pantheon Books, 2002. ISBN 0-375-42221-8.

External links

Sixty Symbols, University of Nottingham
IUPAC - Gold Book