Uncertainty in Physical Constants

Analyze the very different uncertainties of physical constants

Consider the gravitational constant

appearing in the law of universal gravitation. Get its best current value together with its associated uncertainty:

In[1]:=

TextGridTransposeThroughCommonName,

gravitational constant

PHYSICAL CONSTANT



primary source

value

standard uncertainty

relative uncertainty

,



Out[1]=

primary source	CODATA2018
value	6.674× -11 10 3 m /(kg 2 s )
standard uncertainty	1.5× -15 10 3 m /(kg 2 s )
relative uncertainty	0.0000224743

Show the constant value in different forms, using

together with

and

In[2]:=

TextGridTableunc,EntityValue

gravitational constant

PHYSICAL CONSTANT

,EntityProperty["PhysicalConstant","Value",{"Uncertainty"unc}],{unc,{"Around","Interval","Precision"}},



Out[2]=

Around	(6.67430 ± 0.00015 )× -11 10 3 m /(kg 2 s )
Interval	(6.67415× -11 10 to 6.67445× -11 10 ) 3 m /(kg 2 s )
Precision	6.674× -11 10 3 m /(kg 2 s )

To examine the relative uncertainties of physical constants, first retrieve the values for all constants, then remove constants with exact values, unknown uncertainties and the constants corresponding to atomic and natural units:

In[3]:=

TextGridTakeuncertaintydata=DeleteCasesSortBy[EntityValue["PhysicalConstant",{"Entity","Symbol","RelativeUncertainty"}],Last],

Alternatives[

]

,20,



Out[3]=

electron g factor	g e	1.74797× -13 10
Hartree energy	E h	1.9× -12 10
Rydberg constant	R ∞	1.9× -12 10
electron relative atomic mass	m e	2.91662× -11 10
proton relative atomic mass	m e	5.26171× -11 10
proton-electron mass ratio	m p / m e	5.99079× -11 10
electron-proton mass ratio	m e / m p	6.0593× -11 10
inverse square root electric constant	κ	7.5× -11 10
electric constant	ε 0	1.46823× -10 10
Coulomb constant	k e	1.5× -10 10
vacuum admittance	Y 0	1.5× -10 10
vacuum impedance	Z 0	1.5× -10 10
fine structure constant	α	1.5074× -10 10
Bohr radius	a 0	1.51178× -10 10
magnetic constant	μ 0	1.51197× -10 10
reduced Bohr radius	* a 0	1.519× -10 10
inverse fine structure constant	-1 α	1.53244× -10 10
electric Bohr dipole moment	e a 0	1.53332× -10 10
electron magnetic moment anomaly	a e	1.55219× -10 10
proton g factor	g p	2.86446× -10 10

Construct a new list of data in which relative uncertainties are converted to precisions. Also single out a few important constants to ensure they are labeled in the resulting plot using the

LabelVisibility

option:

In[4]:=

labels=

fine structure constant

PHYSICAL CONSTANT

gravitational constant

PHYSICAL CONSTANT

;

In[5]:=

precisiondata=Callout[-Log10[2#3],#1,LabelVisibilityIf[MemberQ[labels,#1],All,Automatic]]&@@@uncertaintydata;

Now plot the sorted precisions of tabulated constants:

In[6]:=

ListPlotprecisiondata,



Out[6]=

Find the precision of the least well known of the core set of fundamental constants of nature, the gravitational constant (to only 4.3 decimal digits), and the precision of the most well known constant, the electron

factor, known to 12.4 digits:

In[7]:=

TextGridJoin{Thread@Style[{"constant","prec."},Italic]},{#,-Log10[2#["RelativeUncertainty"]]}&/@

gravitational constant

PHYSICAL CONSTANT

electron g factor

PHYSICAL CONSTANT

,



Out[7]=

constant	prec.
gravitational constant	4.34728
electron g factor	12.4564

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Visualize the precisions to which physical constants are known as a

WordCloud

, where symbols are shown with size weighted by the precision to which they are known (and where the precision has been scaled by cubing it to better emphasize the variations):

In[1]:=

WordCloud-Log10[#3]^3TooltipStyle#2/.

RuleDelayed[

]

,#1&@@@uncertaintydata,MaxItems∞

Out[1]=

Publisher Information

Contributed by: Wolfram Staff

Uncertainty in Physical Constants

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