Editing Fine-structure constant (section)

== Physical interpretations ==
The fine-structure constant, {{mvar|α}}, has several physical interpretations. {{mvar|α}} is:{{unordered list
| The ratio of two energies:{{ordered list |type=lower-roman
  | the energy needed to overcome the [[electrostatic repulsion]] between two electrons a distance of {{mvar|d}} apart, and
  | the energy of a single [[photon]] of wavelength {{math|''λ'' {{=}} 2''πd''}} (or of [[wavelength#Angular wavelength|angular wavelength]] {{mvar|d}}; see ''[[Planck relation]]''): <math display="block">\alpha = \left. { \left( \frac{e^2}{4\pi \varepsilon_0 d} \right) }\right/ { \left( \frac{hc}{\lambda} \right) } = \frac{e^2}{4\pi\varepsilon_0 d } \times {\frac{ 2 \pi d }{hc}} = \frac{e^2}{ 4 \pi \varepsilon_0 d } \times {\frac{d}{ \hbar c }} = \frac{e^2}{ 4 \pi \varepsilon_0 \hbar c } .</math>}}

| The ratio of the velocity of the electron in the first circular orbit of the [[Bohr model of the atom]], which is {{math|{{sfrac|1|4π{{var|ε}}{{sub|0}}}}{{sfrac|''e''{{sup|2}}|''ħ''}}}}, to the [[speed of light]] in vacuum, {{mvar|c}}.<ref>
{{cite book
 |author=Sommerfeld, A. |author-link=Arnold Sommerfeld
 |title=Atombau und Spektrallinien |language=de
 |place=Braunschweig, DE
 |publisher=Friedr. Vieweg & Sohn
 |edition=2
 |year=1921
 |pages=241–242, Equation&nbsp;8
 |url=https://archive.org/stream/atombauundspekt00sommgoog?ref=ol#page/n261/mode/2up
 |quote=Das Verhältnis <math>v_{1}/c</math> nennen wir {{mvar|α}}. |trans-quote=The ratio <math>v_{1}/c</math> we call {{mvar|α}}.
}}
{{cite book |url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015078632786&view=1up&seq=233 |title=English translation|year=1923 |publisher=Methuen & co. }}
</ref> This is [[Arnold Sommerfeld|Sommerfeld]]'s original physical interpretation. Then the square of {{mvar|α}} is the ratio between the [[Hartree energy]] ({{nowrap|1=27.2 eV = twice the [[Rydberg constant|Rydberg energy]]}} {{=}} approximately twice its ionization energy) and the [[electron]] [[rest energy]] (511&nbsp;keV).
| <math>\alpha^2</math> is the ratio of the potential energy of the electron in the first circular orbit of the [[Bohr model of the atom]] and the energy {{math|''m''{{sub|e}}''c''{{sup|2}}}} equivalent to the mass of an electron. Using the [[virial theorem]] in the [[Bohr model of the atom]] <math>U_\text{el} = 2 U_\text{kin},</math> which means that <math> U_\text{el} = m_\text{e} v_\text{e}^2 = m_\text{e} (\alpha c)^2 = \alpha^2 (m_\text{e} c^2).</math> Essentially this ratio follows from the electron's velocity being <math>v_\text{e} = \alpha c</math>.

| The two ratios of three characteristic lengths: the [[classical electron radius]] {{math|''r''{{sub|e}}}}, the [[reduced Compton wavelength]] of the electron {{math|''ƛ''{{sub|e}}}}, and the [[Bohr radius]] {{math|''a''{{sub|0}}}}: {{math|1=''r''{{sub|e}} = ''αƛ''{{sub|e}} = ''α''{{sup|2}}''a''{{sub|0}}}}.

| In [[quantum electrodynamics]], {{mvar|α}} is directly related to the [[coupling constant]] determining the strength of the interaction between [[electron]]s and [[photon]]s.<ref>{{cite book| last1=Riazuddin|first1=Fayyazuddin| title=A Modern Introduction to Particle Physics|publisher=World Scientific| pages=4 |edition=third |url=https://books.google.com/books?id=dbysnBTHF4QC| access-date=20 April 2017| isbn=9789814338837|year=2012}}</ref> The theory does not predict its value. Therefore, {{mvar|α}} must be determined experimentally. In fact, {{mvar|α}} is one of the empirical [[Standard Model#Theoretical aspects|parameters in the Standard Model]] of [[particle physics]], whose value is not determined within the Standard Model.

| In the [[electroweak theory]] unifying the [[weak interaction]] with [[electromagnetism]], {{mvar|α}} is absorbed into two other [[coupling constant]]s associated with the electroweak [[gauge theory|gauge fields]]. In this theory, the [[electromagnetic interaction]] is treated as a mixture of interactions associated with the electroweak fields. The strength of the [[electromagnetic interaction]] varies with the strength of the [[energy]] field.

| In the fields of [[electrical engineering]] and [[solid-state physics]], the fine-structure constant is one fourth the product of the characteristic [[impedance of free space]], <math> Z_0 = \mu_0 c ,</math> and the [[conductance quantum]], <math>G_0 = 2 e^2 / h</math>: <math>\alpha = \tfrac{1}{4} Z_0 G_0.</math>  The [[optical conductivity]] of [[graphene]] for visible frequencies is theoretically given by {{math|{{sfrac| {{var|π}} |4}}{{var|G}}{{sub|0}}}}, and as a result its light absorption and transmission properties can be expressed in terms of the fine-structure constant alone.<ref name="NairBlake2008">{{cite journal |last1=Nair |first1=R. R. |last2=Blake |first2=P. |last3=Grigorenko |first3=A. N. |last4=Novoselov |first4=K. S. |last5=Booth |first5=T. J. |last6=Stauber |first6=T. |last7=Peres |first7=N. M. R. |last8=Geim |first8=A. K. |year=2008 |title=Fine Structure Constant Defines Visual Transparency of Graphene |journal=[[Science (journal)|Science]] |volume=320 |issue=5881 |pages=1308 |bibcode=2008Sci...320.1308N |doi=10.1126/science.1156965 |pmid=18388259|arxiv=0803.3718 |s2cid=3024573 }}</ref> The absorption value for normal-incident light on graphene in vacuum would then be given by {{math|{{sfrac|π{{var|α}}| (1 + π{{var|α}}/2){{sup|2}}}} }} or 2.24%, and the transmission by {{math|{{sfrac|1|(1 + π{{var|α}}/2){{sup|2}}}}}} or 97.75% (experimentally observed to be between 97.6% and 97.8%). The reflection would then be given by {{math|{{sfrac| π{{sup|2}} {{var|α}}{{sup|2}}| 4 (1 + π{{var|α}}/2){{sup|2}}}}}}.

| The fine-structure constant gives the maximum positive charge of an atomic nucleus that will allow a stable electron-orbit around it within the Bohr model (element [[feynmanium]]).<ref>{{cite journal |last1=Chandrasekhar |first1=S. |title=On stars, their evolution and their stability |journal=Reviews of Modern Physics |date=1 April 1984 |volume=56 |issue=2 |pages=137–147 |doi=10.1103/RevModPhys.56.137 |bibcode=1984RvMP...56..137C |s2cid=2317589 }}</ref> For an electron orbiting an atomic nucleus with atomic number {{mvar|Z}} the relation is {{math| {{sfrac|{{var|m}}{{var|v}}{{sup|2}}|{{var|r}}}} {{=}} {{sfrac|1| 4π{{var|ε}}{{sub|0}}}} {{sfrac|{{var|Z}}{{var|e}}{{sup|2}}|{{var|r}}{{sup|2}}}} }}. The Heisenberg [[uncertainty principle]] momentum/position uncertainty relationship of such an electron is just {{math|{{var|m}}{{var|v}}{{var|r}} {{=}} {{var|ħ}}}}. The relativistic limiting value for {{mvar|v}} is {{mvar|c}}, and so the limiting value for {{mvar|Z}} is the reciprocal of the fine-structure constant, 137.<ref>
{{cite journal
 |last1=Bedford |first1=D.
 |last2=Krumm   |first2=P.
 |year=2004
 |title=Heisenberg indeterminacy and the fine structure constant
 |journal=[[American Journal of Physics]]
 |volume=72 |issue=7 |page=969
 |doi=10.1119/1.1646135 |bibcode=2004AmJPh..72..969B
}}</ref>

}}

When [[perturbation theory (quantum mechanics)|perturbation theory]] is applied to [[quantum electrodynamics]], the resulting [[perturbative]] expansions for physical results are expressed as sets of [[power series]] in {{mvar|α}}. Because {{mvar|α}} is much less than one, higher powers of {{mvar|α}} are soon unimportant, making the perturbation theory practical in this case. On the other hand, the large value of the corresponding factors in [[quantum chromodynamics]] makes calculations involving the [[strong nuclear force]] extremely difficult.